Open Access

Strong differential subordination properties for analytic functions involving the Komatu integral operator

Boundary Value Problems20132013:44

DOI: 10.1186/1687-2770-2013-44

Received: 27 November 2012

Accepted: 21 January 2013

Published: 4 March 2013

Abstract

The purpose of the present paper is to investigate some strong differential subordination and superordination implications involving the Komatu integral operator which are obtained by considering suitable classes of admissible functions. The sandwich-type theorems for these operators are also considered.

MSC:30C80, 30C45, 30A20.

Keywords

strong differential subordination strong differential superordination univalent function convex function Komatu integral operator

1 Introduction

Let denote the class of analytic functions in the open unit disk U = { z C : | z | < 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq1_HTML.gif. For a positive integer n and a C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq2_HTML.gif, let
H [ a , n ] = { f H : f ( z ) = a + a n z n + a n + 1 z n + 1 + } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equa_HTML.gif

and let H 0 H [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq3_HTML.gif. We also denote by A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq4_HTML.gif the subclass of H [ a , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq5_HTML.gif with the usual normalization f ( 0 ) = f ( 0 ) 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq6_HTML.gif.

Let f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif and F ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq8_HTML.gif be members of . The function f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif is said to be subordinate to F ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq8_HTML.gif, or F ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq8_HTML.gif is said to be superordinate to f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif, if there exists a function w ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq9_HTML.gif analytic in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, with w ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq11_HTML.gif and | w ( z ) | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq12_HTML.gif, and such that f ( z ) = F ( w ( z ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq13_HTML.gif. In such a case, we write f ( z ) F ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq14_HTML.gif or f F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq15_HTML.gif. If the function F is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, then f ( z ) F ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq14_HTML.gif if and only if f ( 0 ) = F ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq16_HTML.gif and f ( U ) F ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq17_HTML.gif (cf. [1]).

Following Komatu [2], we introduce the integral operator L c λ : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq18_HTML.gif defined by
L c λ f ( z ) : = c λ Γ ( λ ) 0 1 t c 2 ( log 1 t ) λ 1 f ( t z ) d t ( Re { c } > 0 ; λ 0 ; f A ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ1_HTML.gif
(1.1)
where the symbol Γ stands for the gamma function. We also note that the operator L c λ f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq19_HTML.gif defined by (1.1) can be expressed by the series expansion as follows:
L c λ f ( z ) = z + k = 2 ( c c + k 1 ) λ a k z k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ2_HTML.gif
(1.2)
Obviously, we have, for λ , μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq20_HTML.gif,
L c λ ( L c μ f ( z ) ) = L c λ + μ f ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equb_HTML.gif
Moreover, from (1.2), it follows that
z ( L c λ + 1 f ( z ) ) = c L c λ f ( z ) ( c 1 ) L c λ + 1 f ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ3_HTML.gif
(1.3)

In particular, the operator L 2 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq21_HTML.gif is closely related to the multiplier transformation studied earlier by Flett [3]. Various interesting properties of the operator L 2 λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq22_HTML.gif have been studied by Jung et al. [4] and Liu [5].

To prove our results, we need the following definitions and theorems considered by Antonimo [6, 7] and Oros [8, 9].

Definition 1.1 ([6], cf. [7, 8])

Let H ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq23_HTML.gif be analytic in U × U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq24_HTML.gif and let f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif be analytic and univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif. Then the function H ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq23_HTML.gif is said to be strongly subordinate to f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif, or f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif is said to be strongly superordinate to H ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq23_HTML.gif, written as H ( z , ζ ) f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq25_HTML.gif, if for ζ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq26_HTML.gif, H ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq23_HTML.gif as the function of z is subordinate to f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq7_HTML.gif. We note that H ( z , ζ ) f ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq25_HTML.gif if and only if H ( 0 , ζ ) = f ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq27_HTML.gif and H ( U × U ¯ ) f ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq28_HTML.gif.

Definition 1.2 ([8], cf. [1])

Let ϕ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq29_HTML.gif and let h ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq30_HTML.gif be univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif. If p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif is analytic in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif and satisfies the (second-order) differential subordination
ϕ ( p ( z ) , z p ( z ) , z p ( z ) ; z , ζ ) h ( z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ4_HTML.gif
(1.4)

then p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif is called a solution of the strong differential subordination. The univalent function q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq32_HTML.gif is called a dominant of the solutions of the strong differential subordination, or more simply a dominant, if p ( z ) q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq33_HTML.gif for all p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif satisfying (1.4). A dominant q ˜ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq34_HTML.gif that satisfies q ˜ ( z ) q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq35_HTML.gif for all dominants q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq32_HTML.gif of (1.4) is said to be the best dominant.

Recently, Oros [9] introduced the following strong differential superordinations as the dual concept of strong differential subordinations.

Definition 1.3 ([9], cf. [10])

Let φ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq36_HTML.gif and let h ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq30_HTML.gif be analytic in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif. If p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif and φ ( p ( z ) , z p ( z ) , z p ( z ) ; z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq37_HTML.gif are univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif for ζ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq26_HTML.gif and satisfy the (second-order) strong differential superordination
h ( z ) φ ( p ( z ) , z p ( z ) , z p ( z ) ; z , ζ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ5_HTML.gif
(1.5)

then p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif is called a solution of the strong differential superordination. An analytic function q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq32_HTML.gif is called a subordinant of the solutions of the strong differential superordination, or more simply a subordinant, if q ( z ) p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq38_HTML.gif for all p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif satisfying (1.5). A univalent subordinant q ˜ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq34_HTML.gif that satisfies q ( z ) q ˜ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq39_HTML.gif for all subordinants q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq32_HTML.gif of (1.5) is said to be the best subordinant.

Denote by Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq40_HTML.gif the class of functions q that are analytic and injective on U ¯ E ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq41_HTML.gif, where
E ( q ) = { ξ U : lim z ξ q ( z ) = } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equc_HTML.gif

and are such that q ( ξ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq42_HTML.gif for ξ U E ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq43_HTML.gif. Further, let the subclass of Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq44_HTML.gif for which q ( 0 ) = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq45_HTML.gif be denoted by Q ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq46_HTML.gif and Q ( 0 ) Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq47_HTML.gif.

Definition 1.4 ([8])

Let Ω be a set in , q ( z ) Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq48_HTML.gif and n be a positive integer. The class of admissible functions Ψ n [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq49_HTML.gif consists of those functions ψ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq50_HTML.gif that satisfy the admissibility condition
ψ ( r , s , t ; z , ζ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equd_HTML.gif
whenever r = q ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq51_HTML.gif, s = k ξ q ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq52_HTML.gif and
Re { t s + 1 } k Re { ξ q ( ξ ) q ( ξ ) + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Eque_HTML.gif

for z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif, ξ U E ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq54_HTML.gif, ζ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq26_HTML.gif and k n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq55_HTML.gif. We write Ψ 1 [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq56_HTML.gif as Ψ [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq57_HTML.gif.

Definition 1.5 ([9])

Let Ω be a set in and q H [ a , n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq58_HTML.gif with q ( z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq59_HTML.gif. The class of admissible functions Ψ n [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq60_HTML.gif consists of those functions ψ : C 3 × U ¯ × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq61_HTML.gif that satisfy the admissibility condition
ψ ( r , s , t ; ξ , ζ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equf_HTML.gif
whenever r = q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq62_HTML.gif, s = z q ( z ) / m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq63_HTML.gif for z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif and
Re { t s + 1 } 1 m Re { z q ( z ) q ( z ) + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equg_HTML.gif

for z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif, ξ U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq64_HTML.gif, ζ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq65_HTML.gif and m n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq66_HTML.gif. We write Ψ 1 [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq67_HTML.gif as Ψ [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq68_HTML.gif.

For the above two classes of admissible functions, Oros and Oros proved the following theorems.

Theorem 1.1 ([8])

Let ψ Ψ n [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq69_HTML.gif with q ( 0 ) = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq45_HTML.gif. If p H [ a , n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq70_HTML.gif satisfies
ψ ( p ( z ) , z p ( z ) , z 2 p ( z ) , z 2 p ( z ) ; z , ζ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equh_HTML.gif

then p ( z ) q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq33_HTML.gif.

Theorem 1.2 ([9])

Let ψ Ψ n [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq71_HTML.gif with q ( 0 ) = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq45_HTML.gif. If p Q ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq72_HTML.gif and
ψ ( p ( z ) , z p ( z ) , z 2 p ( z ) ; z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equi_HTML.gif
is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif for ζ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq26_HTML.gif, then
Ω { ψ ( p ( z ) , z p ( z ) , z 2 p ( z ) ; z , ζ ) : z U , ζ U ¯ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equj_HTML.gif

implies q ( z ) p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq38_HTML.gif.

In the present paper, making use of the differential subordination and superordination results of Oros and Oros [8, 9], we determine certain classes of admissible functions and obtain some subordination and superordination implications of multivalent functions associated with the Komatu integral operator L c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq73_HTML.gif defined by (1.1). Additionally, new differential sandwich-type theorems are obtained. We remark in passing that some interesting developments on differential subordination and superordination for various operators in connection with the Komatu integral operator were obtained by Ali et al. [1114] and Cho et al. [15].

2 Subordination results

Firstly, we begin by proving the subordination theorem involving the integral operator L c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq74_HTML.gif defined by (1.1). For this purpose, we need the following class of admissible functions.

Definition 2.1 Let Ω be a set in , q Q 0 H [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq75_HTML.gif, Re { c } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq76_HTML.gif and λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq77_HTML.gif. The class of admissible functions Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq78_HTML.gif consists of those functions ϕ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq79_HTML.gif that satisfy the admissibility condition
ϕ ( u , v , w ; z , ξ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equk_HTML.gif
whenever
u = q ( ζ ) , v = k ζ q ( ζ ) + ( c 1 ) q ( ζ ) c , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equl_HTML.gif
and
Re { c 2 w ( c 1 ) 2 u c v ( c 1 ) u 2 ( c 1 ) } k Re { ζ q ( ζ ) q ( ζ ) + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equm_HTML.gif

for z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif, ζ U E ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq80_HTML.gif, ξ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq81_HTML.gif and k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq82_HTML.gif.

Theorem 2.1 Let ϕ Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq83_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies
{ ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z ) : z U , ξ U ¯ } Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ6_HTML.gif
(2.1)
then
L c λ + 1 f ( z ) q ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equn_HTML.gif
Proof Define the function p ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq31_HTML.gif in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif by
p ( z ) : = L c λ + 1 f ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ7_HTML.gif
(2.2)
From (2.2) with the relation (1.3), we get
L c λ f ( z ) = z p ( z ) + ( c 1 ) p ( z ) c . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ8_HTML.gif
(2.3)
Further computations show that
L c λ 1 f ( z ) = z 2 p ( z ) + ( 2 c 1 ) z p ( z ) + ( c 1 ) 2 p ( z ) c 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ9_HTML.gif
(2.4)
Define the transformation from C 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq85_HTML.gif to by
u = r , v = s + ( c 1 ) r c , and w = t + ( 2 c 1 ) s + ( c 1 ) 2 r c 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ10_HTML.gif
(2.5)
Let
ψ ( r , s , t ; z , ξ ) = ϕ ( u , v , w ; z , ξ ) = ϕ ( r , s + ( c 1 ) r c , t + ( 2 c 1 ) s + ( c 1 ) 2 r c 2 ; z , ξ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ11_HTML.gif
(2.6)
Using equations (2.2), (2.3) and (2.4), from (2.6), we obtain
ψ ( p ( z ) , z p ( z ) , z 2 p ( z ) ; z , ξ ) = ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ12_HTML.gif
(2.7)
Hence, (2.1) becomes
ψ ( p ( z ) , z p ( z ) , z 2 p ( z ) ; z , ξ ) Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equo_HTML.gif
Note that
t s + 1 = c 2 w ( c 1 ) 2 u c v ( c 1 ) u 2 ( c 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equp_HTML.gif
and so the admissibility condition for ϕ Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq86_HTML.gif is equivalent to the admissibility condition for ψ Ψ [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq87_HTML.gif. Therefore, by Theorem 1.1, p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq88_HTML.gif or
L c λ + 1 f ( z ) f ( z ) q ( z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equq_HTML.gif

which evidently completes the proof of Theorem 2.1. □

If Ω C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq89_HTML.gif is a simply connected domain, then Ω = h ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq90_HTML.gif for some conformal mapping h of U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif onto Ω. In this case, the class Φ L [ h ( U ) , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq91_HTML.gif is written as Φ L [ h , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq92_HTML.gif. The following result is an immediate consequence of Theorem 2.1.

Theorem 2.2 Let ϕ Φ L [ h , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq93_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) h ( z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ13_HTML.gif
(2.8)
then
L c λ + 1 f ( z ) q ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equr_HTML.gif

Our next result is an extension of Theorem 2.1 to the case where the behavior of q on U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq94_HTML.gif is not known.

Corollary 2.3 Let Ω C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq95_HTML.gif and q be univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif with q ( 0 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq96_HTML.gif. Let ϕ Φ L [ Ω , q ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq97_HTML.gif for some ρ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq98_HTML.gif where q ρ ( z ) = q ( ρ z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq99_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equs_HTML.gif
then
L c λ + 1 f ( z ) q ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equt_HTML.gif

Proof Theorem 2.1 yields L c λ + 1 f ( z ) q ρ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq100_HTML.gif. The result is now deduced from q ρ ( z ) q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq101_HTML.gif. □

Theorem 2.4 Let h and q be univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif with q ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq102_HTML.gif and set q ρ ( z ) = q ( ρ z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq99_HTML.gif and h ρ ( z ) = h ( ρ z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq103_HTML.gif. Let ϕ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq104_HTML.gif satisfy one of the following conditions:
  1. (1)

    ϕ Φ L [ h , q ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq105_HTML.gif for some ρ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq98_HTML.gif, or

     
  2. (2)

    there exists ρ 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq106_HTML.gif such that ϕ Φ L [ h ρ , q ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq107_HTML.gif for all ρ ( ρ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq108_HTML.gif.

     
If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies (2.8), then
L c λ + 1 f ( z ) q ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equu_HTML.gif

Proof The proof is similar to that of [[1], Theorem 2.3d] and so is omitted. □

The next theorem yields the best dominant of the differential subordination (2.7).

Theorem 2.5 Let h be univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif and let ϕ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq104_HTML.gif. Suppose that the differential equation
ϕ ( q ( z ) , z q ( z ) + ( c 1 ) q ( z ) c , z 2 q ( z ) + ( 2 c 1 ) z q ( z ) + ( c 1 ) 2 q ( z ) c 2 ; z , ξ ) = h ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ14_HTML.gif
(2.9)
has a solution q with q ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq102_HTML.gif and satisfies one of the following conditions:
  1. (1)

    q Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq109_HTML.gif and ϕ Φ L [ h , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq93_HTML.gif,

     
  2. (2)

    q is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif and ϕ Φ L [ h , q ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq110_HTML.gif for some ρ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq111_HTML.gif, or

     
  3. (3)

    q is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif and there exists ρ 0 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq112_HTML.gif such that ϕ Φ L [ h ρ , q ρ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq113_HTML.gif for all ρ ( ρ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq108_HTML.gif.

     
If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies (2.8) and
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equv_HTML.gif
is analytic in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, then
L c λ + 1 f ( z ) q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equw_HTML.gif

and q is the best dominant.

Proof Following the same arguments as in [[1], Theorem 2.3e], we deduce that q is a dominant from Theorem 2.2 and Theorem 2.4. Since q satisfies (2.9), it is also a solution of (2.8) and therefore q will be dominated by all dominants. Hence, q is the best dominant. □

In the particular case q ( z ) = M z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq114_HTML.gif, M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq115_HTML.gif, and in view of Definition 2.1, the class of admissible functions Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq116_HTML.gif, denoted by Φ L [ Ω , M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq117_HTML.gif, is described below.

Definition 2.2 Let Ω be a set in , Re { c } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq118_HTML.gif, λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq77_HTML.gif and M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq115_HTML.gif. The class of admissible functions Φ L [ Ω , M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq119_HTML.gif consists of those functions ϕ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq79_HTML.gif such that
ϕ ( M e i θ , ( k + c 1 ) M e i θ c , L + [ ( 2 c 1 ) k + ( c 1 ) 2 ] M e i θ c 2 ; z , ξ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ15_HTML.gif
(2.10)

whenever z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif, ξ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq81_HTML.gif, Re { L e i θ } ( k 1 ) k M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq120_HTML.gif, θ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq121_HTML.gif and k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq82_HTML.gif.

Corollary 2.6 Let ϕ Φ L [ Ω , M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq122_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equx_HTML.gif
then
L c λ + 1 f ( z ) M z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equy_HTML.gif

In the special case Ω = q ( U ) = { w : | w | < M } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq123_HTML.gif, the class Φ L [ Ω , M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq124_HTML.gif is simply denoted by Φ L [ M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq125_HTML.gif.

Corollary 2.7 Let ϕ Φ L [ M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq126_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies
| ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) | < M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equz_HTML.gif
then
| L c λ + 1 f ( z ) | < M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equaa_HTML.gif
Corollary 2.8 Let c > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq127_HTML.gif, M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq115_HTML.gif and let C ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq128_HTML.gif be an analytic function in U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq129_HTML.gif with Re { ζ C ( ξ ) } 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq130_HTML.gif for ζ U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq131_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif satisfies
| c 2 L c λ 1 f ( z ) c L c λ f ( z ) ( c 1 ) 2 L c λ + 1 f ( z ) + C ( ξ ) | < ( c 1 ) M , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equab_HTML.gif
then
| L c λ + 1 f ( z ) | < M . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equac_HTML.gif
Proof This follows from Corollary 2.6 by taking ϕ ( u , v , w ; z , ξ ) = c 2 w c v ( c 1 ) 2 u + C ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq132_HTML.gif and Ω = h ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq133_HTML.gif, where h ( z ) = ( c 1 ) M z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq134_HTML.gif. To use Corollary 2.6, we need to show that ϕ Φ L [ Ω , M ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq135_HTML.gif, that is, the admissible condition (2.10) is satisfied. This follows since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equad_HTML.gif

for z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif, ξ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq81_HTML.gif, Re { L e i θ } ( k 1 ) k M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq136_HTML.gif, θ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq121_HTML.gif and k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq82_HTML.gif. Hence, by Corollary 2.6, we deduce the required results. □

3 Superordination and sandwich-type results

The dual problem of differential subordination, that is, differential superordination of the Komatu integral operator L c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq74_HTML.gif defined by (1.1), is investigated in this section. For this purpose, the class of admissible functions is given in the following definition.

Definition 3.1 Let Ω be a set in , q H [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq137_HTML.gif with q ( z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq138_HTML.gif, Re { c } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq76_HTML.gif and λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq77_HTML.gif. The class of admissible functions Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq139_HTML.gif consists of those functions ϕ : C 3 × U ¯ × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq140_HTML.gif that satisfy the admissibility condition
ϕ ( u , v , w ; ζ , ξ ) Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equae_HTML.gif
whenever
u = q ( z ) , v = z q ( z ) / m + ( c 1 ) q ( z ) c , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equaf_HTML.gif
and
Re { c 2 w ( c 1 ) 2 u c v ( c 1 ) u 2 ( c 1 ) } 1 m Re { z q ( z ) q ( z ) + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equag_HTML.gif

for z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq53_HTML.gif, ζ U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq131_HTML.gif, ξ U ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq141_HTML.gif and m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq142_HTML.gif.

Theorem 3.1 Let ϕ Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq143_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif, L c λ + 1 f ( z ) Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq144_HTML.gif and
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equah_HTML.gif
is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, then
Ω { ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ f ( z ) ; z , ξ ) : z U , ξ U ¯ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ16_HTML.gif
(3.1)
implies
q ( z ) L c λ + 1 f ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equai_HTML.gif
Proof From (2.7) and (3.1), we have
Ω { ψ ( p ( z ) , z p ( z ) , z 2 p ( z ) ; z , ξ ) : z U , ξ U ¯ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equaj_HTML.gif
From (2.5), we see that the admissibility condition for ϕ Φ L [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq145_HTML.gif is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence, ψ Ψ [ Ω , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq146_HTML.gif, and by Theorem 1.2, q p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq147_HTML.gif or
q ( z ) L c λ + 1 f ( z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equak_HTML.gif

which evidently completes the proof of Theorem 3.1. □

If Ω C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq148_HTML.gif is a simply connected domain, then Ω = h ( U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq90_HTML.gif for some conformal mapping h of U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif onto Ω. In this case, the class Φ L [ h ( U ) , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq149_HTML.gif is written as Φ L [ h , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq150_HTML.gif. Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.1.

Theorem 3.2 Let q H [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq151_HTML.gif, h be analytic in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif and ϕ Φ L [ h , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq152_HTML.gif. If f ( z ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq153_HTML.gif, L c λ + 1 f ( z ) Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq154_HTML.gif and
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equal_HTML.gif
is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, then
h ( z ) ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equ17_HTML.gif
(3.2)
implies
q ( z ) L c λ + 1 f ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equam_HTML.gif

Theorem 3.1 and Theorem 3.2 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2). The following theorem proves the existence of the best subordinant of (3.2) for certain ϕ.

Theorem 3.3 Let h be analytic in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif and ϕ : C 3 × U × U ¯ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq155_HTML.gif. Suppose that the differential equation
ϕ ( q ( z ) , z q ( z ) + ( c 1 ) q ( z ) c , z 2 q ( z ) + ( 2 c 1 ) z q ( z ) + ( c 1 ) 2 q ( z ) c 2 ; z , ξ ) = h ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equan_HTML.gif
has a solution q Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq109_HTML.gif. If ϕ Φ L [ h , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq152_HTML.gif, f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif, L c λ + 1 f ( z ) Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq156_HTML.gif and
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equao_HTML.gif
is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, then
h ( z ) ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equap_HTML.gif
implies
q ( z ) L c λ + 1 f ( z ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equaq_HTML.gif

and q is the best subordinant.

Proof The proof is similar to that of Theorem 2.5 and so is omitted. □

Combining Theorem 2.2 and Theorem 3.2, we obtain the following sandwich-type theorem.

Theorem 3.4 Let h 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq157_HTML.gif and q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq158_HTML.gif be analytic functions in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, h 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq159_HTML.gif be a univalent function in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, q 2 Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq160_HTML.gif with q 1 ( 0 ) = q 2 ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq161_HTML.gif and ϕ Φ L [ h 2 , q 2 ] Φ L [ h 1 , q 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq162_HTML.gif. If f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq84_HTML.gif, L c λ + 1 f ( z ) H [ 0 , 1 ] Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq163_HTML.gif and
ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equar_HTML.gif
is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_IEq10_HTML.gif, then
h 1 ( z ) ϕ ( L c λ + 1 f ( z ) , L c λ f ( z ) , L c λ 1 f ( z ) ; z , ξ ) h 2 ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equas_HTML.gif
implies
q 1 ( z ) L c λ + 1 f ( z ) q 2 ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-44/MediaObjects/13661_2012_Article_309_Equat_HTML.gif

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Pukyong National University

References

  1. Miller SS, Mocanu PT: Differential Subordination, Theory and Application. Marcel Dekker, New York; 2000.
  2. Komatu Y: Distortion Theorems in Relation to Linear Integral Operators. Kluwer Academic, Dordrecht; 1996.MATHView Article
  3. Flett TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 1972, 38: 746-765. 10.1016/0022-247X(72)90081-9MATHMathSciNetView Article
  4. Jung IB, Kim YC, Srivastava HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J. Math. Anal. Appl. 1993, 176: 138-147. 10.1006/jmaa.1993.1204MATHMathSciNetView Article
  5. Liu JL: A linear operator and strongly starlike functions. J. Math. Soc. Jpn. 2002, 54: 975-981. 10.2969/jmsj/1191592000MATHView Article
  6. Antonino JA: Strong differential subordination to Briot-Bouquet differential equations. J. Differ. Equ. 1994, 114: 101-105. 10.1006/jdeq.1994.1142MATHMathSciNetView Article
  7. Antonino JA: Strong differential subordination and applications to univalency conditions. J. Korean Math. Soc. 2006, 43: 311-322.MATHMathSciNetView Article
  8. Oros GI, Oros G: Strong differential subordination. Turk. J. Math. 2009, 33: 249-257.MATHMathSciNet
  9. Oros GI: Strong differential superordination. Acta Univ. Apulensis, Mat.-Inform. 2009, 19: 101-106.MATHMathSciNet
  10. Miller SS, Mocanu PT: Subordinants of differential superordinations. Complex Var. Theory Appl. 2003, 48: 815-826. 10.1080/02781070310001599322MATHMathSciNetView Article
  11. Ali RM, Ravichandran V, Seenivasagan N: Subordination and superordination of the Liu-Srivastava operator on meromorphic functions. Bull. Malays. Math. Soc. 2008, 31: 193-207.MATHMathSciNet
  12. Ali RM, Ravichandran V, Seenivasagan N: Differential subordination and superordination of analytic functions defined by the multiplier transformation. Math. Inequal. Appl. 2009, 12: 123-139.MATHMathSciNet
  13. Ali RM, Ravichandran V, Seenivasagan N: Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava linear operator. J. Franklin Inst. 2010, 347: 1762-1781. 10.1016/j.jfranklin.2010.08.009MATHMathSciNetView Article
  14. Ali RM, Ravichandran V, Seenivasagan N: On subordination and superordination of the multiplier transformation of meromorphic functions. Bull. Malays. Math. Soc. 2010, 33: 311-324.MATHMathSciNet
  15. Cho NE, Kwon OS, Srivastava HM: Strong differential subordination and superordination for multivalently meromorphic functions involving the Liu-Srivastava operator. Integral Transforms Spec. Funct. 2010, 21: 589-601. 10.1080/10652460903494751MATHMathSciNetView Article

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