Open Access

A general solution of the Fekete-Szegö problem

Boundary Value Problems20132013:98

DOI: 10.1186/1687-2770-2013-98

Received: 24 January 2013

Accepted: 9 April 2013

Published: 22 April 2013

Abstract

In the paper we introduce general classes of analytic functions defined by the Hadamard product. The Fekete-Szegö problem is completely solved in these classes of functions. Some consequences of the main results for new or well-known classes of functions are also pointed out.

MSC:30C45, 30C50, 30C55.

Keywords

analytic functions Fekete-Szegö problem subordination Hadamard product

1 Introduction

Let A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq1_HTML.gif denote the class of functions which are analytic in U = { z C : | z | < 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq2_HTML.gif and let A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq3_HTML.gif denote the class of functions f A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq4_HTML.gif normalized by f ( 0 ) = f ( 0 ) 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq5_HTML.gif. Each function f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq6_HTML.gif can be expressed as
f ( z ) = z + n = 2 a n z n ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ1_HTML.gif
(1)

By S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq7_HTML.gif, we denote the class of functions f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq6_HTML.gif, which are univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq8_HTML.gif.

A typical problem in geometric function theory is to study a functional made up of combinations of the coefficients of the original function. Usually, there is a parameter over which the extremal value of the functional is needed. The paper deals with one important functional of this type: the Fekete-Szegö functional. The classical Fekete-Szegö functional is defined by
Λ μ ( f ) = a 3 μ a 2 2 ( 0 < μ < 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equa_HTML.gif

and it is derived from the Fekete-Szegö inequality. The problem of maximizing the absolute value of the functional Λ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq9_HTML.gif in subclasses of normalized functions is called the Fekete-Szegö problem. The mathematicians who introduced the functional, M. Fekete and G. Szegö [1], were able to bound the classical functional in the class S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq7_HTML.gif by 1 + 2 exp { 2 μ 1 μ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq10_HTML.gif. Later Pfluger [2] used Jenkin’s method to show that this result holds for complex μ such that Re μ 1 μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq11_HTML.gif. Keogh and Merkes [3] obtained the solution of the Fekete-Szegö problem for the class of close-to-convex functions. Ma and Minda [4, 5] gave a complete answer to the Fekete-Szegö problem for the classes of strongly close-to-convex functions and strongly starlike functions. In the literature, there exists a large number of results about inequalities for Λ μ ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq12_HTML.gif corresponding to various subclasses of A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq3_HTML.gif (see, for instance, [123]).

In the paper, we consider the classes of functions which generalize these subclasses of functions.

We say that a function g A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq13_HTML.gif is subordinate to a function G A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq14_HTML.gif , and write g ( z ) G ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq15_HTML.gif (or simply g G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq16_HTML.gif), if and only if there exists a function
ω Ω : = { ω A ˜ : | ω ( z ) | | z | ( z U ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equb_HTML.gif
such that g = G ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq17_HTML.gif. In particular, if G is univalent in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq18_HTML.gif we have the following equivalence:
g ( z ) G ( z ) [ g ( 0 ) = G ( 0 ) g ( U ) G ( U ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equc_HTML.gif
For functions f , g A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq19_HTML.gif of the forms
f ( z ) = n = 1 a n z n and g ( z ) = n = 1 b n z n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equd_HTML.gif
by f g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq20_HTML.gif we denote the Hadamard product (or convolution) of f and g, defined by
( f g ) ( z ) = n = 1 a n b n z n ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Eque_HTML.gif
Let α be complex parameter and let Φ = ( ϕ , φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq21_HTML.gif, Ψ = ( ψ , χ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq22_HTML.gif, P = ( p , q ) A ˜ × A ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq23_HTML.gif be of the form
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equf_HTML.gif
By W α ( Φ , Ψ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq24_HTML.gif we denote the class of functions f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq6_HTML.gif such that
( φ f ) ( z ) ( χ f ) ( z ) 0 ( z U { 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equg_HTML.gif
and
( 1 α ) ϕ f φ f + α ψ f χ f p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equh_HTML.gif
Moreover, let us put
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equi_HTML.gif
It is clear that the class M α ( φ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq25_HTML.gif contains functions f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq6_HTML.gif such that
( 1 α ) z ( φ f ) ( z ) ( φ f ) ( z ) + α ( 1 + z ( φ f ) ( z ) ( φ f ) ( z ) ) p ( z ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equj_HTML.gif
We denote by CW ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq26_HTML.gif the class of functions f A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq6_HTML.gif for which there exist a function g S ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq27_HTML.gif such that ( φ f ) ( z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq28_HTML.gif ( z U { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq29_HTML.gif) and
ϕ f φ g p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equk_HTML.gif

Moreover, let us denote CW ( Φ ; p ) : = CW ( Φ ; p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq30_HTML.gif.

In particular, the classes
M α : = M α ( z 1 z ; 1 + z 1 z ) , S : = M 0 , S c : = M 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equl_HTML.gif

are the well-known classes of α-convex Mocanu functions [24], starlike functions and convex functions, respectively. The class C : = W ( ( z ( 1 z ) 2 , z 1 z ) ; 1 + z 1 z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq31_HTML.gif is the well-known class of close-to convex functions with argument β = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq32_HTML.gif.

The object of the paper is to solve the Fekete-Szegö problem in the defined classes of functions. Moreover, we find sharp bounds for the second and third coefficient in these classes. Some remarks depicting consequences of the main results are also mentioned.

2 The main results

The following lemmas will be required in our present investigation.

Lemma 1 [3]

If ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq33_HTML.gif, ω ( z ) = n = 1 c n z n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq34_HTML.gif ( z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq35_HTML.gif), then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equm_HTML.gif
The result is sharp. The functions
ω ( z ) = z , ω a ( z ) = z z + a 1 + a ¯ z ( z U , | a | < 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equn_HTML.gif

are the extremal functions.

Theorem 1 Let
( 1 α ) ( β k α k ) + α ( δ k γ k ) 0 ( k = 2 , 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equo_HTML.gif
If f W α ( Φ , Ψ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq36_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ2_HTML.gif
(2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ3_HTML.gif
(3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ4_HTML.gif
(4)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ5_HTML.gif
(5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ6_HTML.gif
(6)

The results are sharp.

Proof Let f W α ( Φ , Ψ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq36_HTML.gif. Then there exists a function ω Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq33_HTML.gif, ω ( z ) = n = 1 c n z n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq37_HTML.gif ( z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq35_HTML.gif), such that
( 1 α ) ϕ f φ f + α ψ f χ f = p ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ7_HTML.gif
(7)
It is easy to verify that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ8_HTML.gif
(8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ9_HTML.gif
(9)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equp_HTML.gif
Thus, by (7), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ10_HTML.gif
(10)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ11_HTML.gif
(11)
which by Lemma 1 gives sharp estimation (2). Let μ be a complex number. Then, by (10) and (11) we obtain
a 3 μ a 2 2 = p 1 ( 1 α ) ( β 3 α 3 ) + α ( δ 3 γ 3 ) { c 2 γ c 1 2 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equq_HTML.gif
where γ is defined by (6). Thus, by Lemma 1, we have (4). Let functions f 1 , f 2 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq38_HTML.gif satisfy the conditions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equr_HTML.gif

Then the functions belong to the class W α ( Φ , Ψ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq39_HTML.gif and they realize the equality in the estimation (4). Thus, the results are sharp. Putting μ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq40_HTML.gif in (4) we get the sharp estimation (3). □

Theorem 2 Let α 1 2 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq41_HTML.gif. If f M α ( φ , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq42_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equs_HTML.gif
where
β = p 2 p 1 + ( 1 + 3 α ) ( 1 + α ) 2 p 1 , γ = 2 ( 1 + 2 α ) α 3 p 1 ( 1 + α ) 2 α 2 2 μ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equt_HTML.gif

The results are sharp.

Proof Let f M α ( φ , p ) = W α ( Φ , Ψ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq43_HTML.gif, where
χ ( z ) = ϕ ( z ) = z φ ( z ) , ψ ( z ) = z ( z φ ( z ) ) ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equu_HTML.gif
Since
β n = γ n = n α n , δ n = n 2 α n ( n = 2 , 3 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equv_HTML.gif

the results follow from Theorem 1. □

If we put α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq44_HTML.gif in Theorem 1, then we obtain the following theorem.

Theorem 3 Let β k α k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq45_HTML.gif ( k = 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq46_HTML.gif). If f W ( Φ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq47_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equw_HTML.gif
where
β = p 2 p 1 + α 2 p 1 β 2 α 2 , γ = β 3 α 3 ( β 2 α 2 ) 2 p 1 μ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equx_HTML.gif

The results are sharp.

Theorem 4 Let β 2 β 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq48_HTML.gif. If f CW ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq49_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ12_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ13_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ14_HTML.gif
(14)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ15_HTML.gif
(15)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ16_HTML.gif
(16)

The results (12) and (13) are sharp for A μ B μ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq50_HTML.gif and A 0 B 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq51_HTML.gif, respectively.

Proof Let f CW ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq52_HTML.gif. Then there exists a function g S ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq53_HTML.gif and functions ω , η Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq54_HTML.gif,
ω ( z ) = n = 1 c n z n , η ( z ) = n = 1 d n z n ( z U ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equy_HTML.gif
such that
ϕ f φ g = p ω , z g ( z ) g ( z ) = ( q η ) ( z ) ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ17_HTML.gif
(17)
Thus, by (8), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ19_HTML.gif
(19)
and by Lemma 1, we obtain the sharp estimation (14). Let μ be a complex number. Then, by (18), (19) and Lemma 1 we have
2 | β 3 | | a 3 μ a 2 2 | ( A C ) | d 1 | 2 + ( B C ) | c 1 | 2 + 2 C | d 1 | | c 1 | + D , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ20_HTML.gif
(20)
or equivalently
2 | β 3 | | a 3 μ a 2 2 | A | d 1 | 2 + B | c 1 | 2 C ( | d 1 | | c 1 | ) 2 + D , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ21_HTML.gif
(21)
where A = A μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq55_HTML.gif, B = B μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq56_HTML.gif, C = C μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq57_HTML.gif, D are defined by (15) and (16). Thus, we obtain
2 | β 3 | | a 3 μ a 2 2 | A | d 1 | 2 + B | c 1 | 2 + D , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ22_HTML.gif
(22)
and, in consequence, by Lemma 1 we have (12). It is easy to verify that the equality in (22) is attained by choosing c 1 = d 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq58_HTML.gif, c 2 = d 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq59_HTML.gif if A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq60_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq61_HTML.gif or c 1 = d 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq62_HTML.gif, c 2 = d 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq63_HTML.gif if A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq64_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq65_HTML.gif. Therefore, we consider functions f 1 , f 2 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq38_HTML.gif such that
( ϕ f 1 ) ( z ) ( φ g ) ( z ) = p ( z ) , z g ( z ) g ( z ) = q ( z ) ( z U ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equz_HTML.gif
and
( ϕ f 2 ) ( z ) ( φ g ) ( z ) = p ( z 2 ) , z g ( z ) g ( z ) = p ( z 2 ) ( z U ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equaa_HTML.gif

respectively. Then the functions belong to the class CW ( Φ ; p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq66_HTML.gif and they realize the equality in the estimation (12) for A B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq67_HTML.gif. Putting μ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq40_HTML.gif in (12), we get the sharp estimation (13). □

The following theorem gives the complete sharp estimation of the Fekete-Szegö functional in the class CW ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq68_HTML.gif.

Theorem 5 Let β 2 β 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq48_HTML.gif. If f CW ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq69_HTML.gif, then
| a 3 μ a 2 2 | { 1 2 | β 3 | ( A + B + D ) if 0 A C 0 B C if ( A C B C ) , D 2 | β 3 | if A 0 B 0 , 1 2 | β 3 | ( D + B C + C 2 C A ) if A < 0 B C , 1 2 | β 3 | ( D + A C + C 2 C B ) if B < 0 A C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ23_HTML.gif
(23)

where A = A μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq55_HTML.gif, B = B μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq56_HTML.gif, C = C μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq57_HTML.gif, D are defined by (15) and (16). The result is sharp.

Proof From Theorem 4, we have sharp estimation (23) for A B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq67_HTML.gif. Let now A C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq70_HTML.gif and B < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq71_HTML.gif. Then, by (20) and Lemma 1 we have
2 | β 3 | | a 3 μ a 2 2 | v ( | c 1 | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equab_HTML.gif
where
v ( x ) : = ( C B ) x 2 + 2 C x + ( A C ) + D . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equac_HTML.gif

Simply calculations give that the function v attains a maximum in the interval [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq72_HTML.gif at the point x = C C B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq73_HTML.gif. Thus, we have (23) for A C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq70_HTML.gif and B < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq71_HTML.gif. Moreover, the equality in (20) is attained by choosing the functions η ( z ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq74_HTML.gif, ω ( z ) = z z + a 1 + a ¯ z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq75_HTML.gif, for a = C C B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq76_HTML.gif, i.e. c 1 = C C B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq77_HTML.gif, d 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq78_HTML.gif and c 2 = 1 | a | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq79_HTML.gif, d 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq80_HTML.gif. Therefore, the result is sharp for A C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq70_HTML.gif and B < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq71_HTML.gif.

Next, let A < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq81_HTML.gif and C B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq82_HTML.gif. Then, by (20) and Lemma 1 we have
2 | β 3 | | a 3 μ a 2 2 | v ˜ ( | d 1 | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equad_HTML.gif
where
v ˜ ( x ) : = ( C A ) x 2 + 2 C x + ( B C ) + D . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equae_HTML.gif

Since the function v ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq83_HTML.gif attains a maximum in the interval [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq84_HTML.gif at the point x = C C A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq85_HTML.gif, we have the estimation (23) for A < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq81_HTML.gif and C B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq82_HTML.gif. The equality in (20) is attained by choosing the functions ω ( z ) = z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq86_HTML.gif, η ( z ) = z z + a 1 + a ¯ z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq87_HTML.gif, for a = C C A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq88_HTML.gif, i.e. c 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq89_HTML.gif, d 1 = C C A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq90_HTML.gif and c 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq91_HTML.gif, d 2 = 1 | a | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq92_HTML.gif.

Finally, let us assume ( 0 A C B 0 ) ( 0 B C A 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq93_HTML.gif. Then, by (20) we have
2 | β 3 | | a 3 μ a 2 2 | F ( | c 1 | , | d 1 | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ24_HTML.gif
(24)
where
F ( x , y ) = ( C A ) x 2 ( C B ) y 2 + 2 C x y + D . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equaf_HTML.gif
Since F is the continuous function on T : = [ 0 , 1 ] × [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq94_HTML.gif, by (24) we have
2 | β 3 | | a 3 μ a 2 2 | max F ( T ) = max F ( T K ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equ25_HTML.gif
(25)
where K is the set of critical points of the function F in T. It is easy to verify that
K T = { if  C 2 ( C A ) ( C B ) A = C , { ( x , y ) int T : x = C C A y } if  C 2 = ( C A ) ( C B ) A C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equag_HTML.gif
If C 2 = ( A C ) ( B C ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq95_HTML.gif, then
F ( C C A y , y ) = C 2 ( C A ) ( C B ) A C y 2 + D = D ( y [ 0 , 1 ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equah_HTML.gif
Moreover, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equai_HTML.gif
Thus, we obtain
max F ( T K ) = A + B + D , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equaj_HTML.gif

which by (25) gives (23) for ( 0 A C B 0 ) ( 0 B C A 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq93_HTML.gif. The equality in (24) is attained by choosing c 1 = d 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq58_HTML.gif and c 2 = d 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq59_HTML.gif. Therefore, the result is sharp and the proof is completed. □

Putting μ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq96_HTML.gif in Theorem 5 we obtain the following theorem.

Theorem 6 Let β 2 β 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq48_HTML.gif. If f CW ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq49_HTML.gif, then
| a 3 | { 1 2 | β 3 | ( A + B + D ) if 0 A C 0 B C if ( A C B C ) , D 2 | β 3 | if A 0 B 0 , 1 2 | β 3 | ( D + B C + C 2 C A ) if A < 0 B C , 1 2 | β 3 | ( D + A C + C 2 C B ) if B < 0 A C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equak_HTML.gif

where A = A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq97_HTML.gif, B = B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq98_HTML.gif, C = C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq99_HTML.gif, D are defined by (15) and (16). The result is sharp.

3 Applications

If we put α = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq100_HTML.gif and α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq44_HTML.gif in Theorem 2, then we obtain the following two corollaries.

Corollary 1 Let α 2 α 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq101_HTML.gif. If f S c ( φ , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq102_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equal_HTML.gif
where
γ = 3 α 3 p 1 2 α 2 2 μ p 1 p 2 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equam_HTML.gif

The results are sharp.

Corollary 2 Let α 2 α 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq101_HTML.gif. If f S ( φ , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq103_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equan_HTML.gif
where
γ = 2 α 3 p 1 α 2 2 μ p 1 p 2 p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equao_HTML.gif

The results are sharp.

Choosing the function p in Theorems 1-6, we can obtain several new results.

Let a, b be complex number, | b | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq104_HTML.gif, a b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq105_HTML.gif, and let
p ( z ) = 1 + a z 1 + b z ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equap_HTML.gif
It is clear, that
p ( z ) = 1 + ( a b ) z b ( a b ) z 2 + ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equaq_HTML.gif

Thus, by Theorems 1-3 and 5, we obtain the following four corollaries.

Corollary 3 Let ( 1 α ) ( β k α k ) + α ( δ k γ k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq106_HTML.gif ( k = 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq46_HTML.gif). If f W α ( Φ , Ψ ; 1 + a z 1 + b z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq107_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equar_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equas_HTML.gif

The results are sharp.

Corollary 4 Let α 1 2 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq41_HTML.gif. If f M α ( φ , 1 + a z 1 + b z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq108_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equat_HTML.gif
where
β = b + ( 1 + 3 α ) ( 1 + α ) 2 ( a b ) , γ = 2 ( 1 + 2 α ) ( a b ) α 3 ( 1 + α ) 2 α 2 2 μ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equau_HTML.gif

The results are sharp.

Corollary 5 Let β k α k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq109_HTML.gif ( k = 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq110_HTML.gif). If f W ( Φ ; 1 + a z 1 + b z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq111_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equav_HTML.gif
where
β = b + ( a b ) α 2 β 2 α 2 , γ = ( β 3 α 3 ) ( a b ) ( β 2 α 2 ) 2 μ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equaw_HTML.gif

The results are sharp.

Corollary 6 Let β 2 β 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq48_HTML.gif. If f CW ( Φ ; 1 + a z 1 + b z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq112_HTML.gif, then
| a 3 μ a 2 2 | { | b a | 2 | β 3 | ( D + A + B ) if 0 A C 0 B C if ( A C B C ) , | b a | 2 | β 3 | D if A 0 B 0 , | b a | 2 | β 3 | ( D + B C + C 2 C A ) if A < 0 B C , | b a | 2 | β 3 | ( D + A C + C 2 C B ) if B < 0 A C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equax_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equay_HTML.gif

The result is sharp.

Let 0 < θ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq113_HTML.gif and let
p ( z ) = ( 1 + z 1 z ) θ ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equaz_HTML.gif
It is easy to verify, that
p ( z ) = 1 + 2 θ z + θ ( θ + 1 ) z 2 + ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equba_HTML.gif

Thus, by Theorems 1-3 and 5, we obtain the following four corollaries.

Corollary 7 Let ( 1 α ) ( β k α k ) + α ( δ k γ k ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq106_HTML.gif ( k = 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq46_HTML.gif). If f W α ( Φ , Ψ ; ( 1 + z 1 z ) θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq114_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbb_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbc_HTML.gif

The results are sharp.

Corollary 8 Let α 1 2 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq41_HTML.gif. If f M α ( φ , ( 1 + z 1 z ) θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq115_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbd_HTML.gif
where
β = 1 + θ 2 + 2 ( 1 + 3 α ) ( 1 + α ) 2 θ , γ = 4 ( 1 + 2 α ) θ α 3 ( 1 + α ) 2 α 2 2 μ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Eqube_HTML.gif

The results are sharp.

Corollary 9 Let β k α k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq109_HTML.gif ( k = 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq46_HTML.gif). If f W ( Φ ; ( 1 + z 1 z ) θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq116_HTML.gif, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbf_HTML.gif
where
β = 1 + θ 2 + 2 α 2 θ β 2 α 2 , γ = 2 ( β 3 α 3 ) θ ( β 2 α 2 ) 2 μ β . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbg_HTML.gif

The results are sharp.

Corollary 10 Let β 2 β 3 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq48_HTML.gif. If f CW ( Φ ; ( 1 + z 1 z ) θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq117_HTML.gif, then
| a 3 μ a 2 2 | { θ | β 3 | ( D + A + B ) if 0 A C 0 B C if ( A C B C ) , θ D | β 3 | if A 0 B 0 , θ | β 3 | ( D + B C + C 2 C A ) if A < 0 B C , θ | β 3 | ( D + A C + C 2 C B ) if B < 0 A C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbh_HTML.gif
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbi_HTML.gif

The result is sharp.

Let Ω k = { u + i v : u > k ( u 1 ) 2 + v 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq118_HTML.gif, k > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq119_HTML.gif. Note that Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq120_HTML.gif is the convex domain contained in the right half plane, with 1 Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq121_HTML.gif. More precisely, it is the elliptic domain for k > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq122_HTML.gif, the hyperbolic domain for 0 < k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq123_HTML.gif and the parabolic domain for k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq124_HTML.gif.

Let us denote by h k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq125_HTML.gif the univalent function, which maps the unit disc U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq8_HTML.gif onto the conic domain Ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq120_HTML.gif with h k ( 0 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq126_HTML.gif. Obviously, the function h k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq125_HTML.gif is convex in U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq8_HTML.gif. It is easy to check that f W ( Φ ; h k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq127_HTML.gif if and only if
Re ( ( ϕ f ) ( z ) ( φ g ) ( z ) ) > k | ( ϕ f ) ( z ) ( φ g ) ( z ) 1 | ( z U ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbj_HTML.gif

The following lemma gives coefficients estimates for the function.

Lemma 2 [13]

Let h k = 1 + n = 1 p n z n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq128_HTML.gif ( z U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq129_HTML.gif). Then
p 1 = { 2 D 2 ( k ) 1 k 2 for 0 k < 1 , 8 π 2 for k = 1 , π 2 4 t ( 1 + t ) ( k 2 1 ) K 2 ( t ) for k > 1 , p 2 = { D 2 ( k ) + 2 3 p 1 for 0 k < 1 , 2 3 p 1 for k = 1 , 4 ( t 2 + 6 t + 1 ) K 2 ( t ) π 2 24 t ( 1 + t ) K 2 ( t ) p 1 for k > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_Equbk_HTML.gif

where D ( k ) = 2 π arcsin k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq130_HTML.gif and K 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq131_HTML.gif is the complete elliptic integral of first kind.

Using Lemma 1 in Theorems 1-5 we obtain the solutions of the Fekete-Szegö problem for the classes W α ( Φ , Ψ ; h k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq132_HTML.gif, M α ( φ , h k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq133_HTML.gif, W α ( Φ ; h k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq134_HTML.gif, CW α ( Φ ; h k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq135_HTML.gif.

Remark 1 The classes W α ( Φ , Ψ ; h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq136_HTML.gif, M α ( φ , h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq137_HTML.gif, CW α ( Φ ; P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-98/MediaObjects/13661_2013_Article_366_IEq138_HTML.gif reduced to well-known subclasses by judicious choices of the parameters; see, for example [128]. In particular, they generalize several well-known classes defined by linear operators, which were investigated in earlier works. Also, the obtained results generalize several results obtained in these classes of functions.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Institute of Mathematics, University of Rzeszów

References

  1. Fekete M, Szegö G: Eine Bemerkung über ungerade schlichte Funktionen. J. Lond. Math. Soc. 1933, 8: 85-89.View ArticleMATH
  2. Pfluger A: The Fekete-Szegö inequality for complex parameters. Complex Var. Theory Appl. 1986, 7: 149-160. 10.1080/17476938608814195MathSciNetView ArticleMATH
  3. Keogh FR, Merkes EP: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969, 20: 8-12. 10.1090/S0002-9939-1969-0232926-9MathSciNetView ArticleMATH
  4. Ma W, Minda D: An internal geometric characterization of strongly starlike functions. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 1991, 45: 89-97.MathSciNetMATH
  5. Ma W, Minda D: Coefficient inequalities for strongly close-to-convex functions. J. Math. Anal. Appl. 1997, 205: 537-553. 10.1006/jmaa.1997.5234MathSciNetView ArticleMATH
  6. Abdel-Gawad HR, Thomas DK: The Fekete-Szegö problem for strongly close-to-convex functions. Proc. Am. Math. Soc. 1992, 114: 345-349.MathSciNetMATH
  7. Cho NE, Owa S: On the Fekete-Szegö and argument inequalities for strongly close-to-star functions. Math. Inequal. Appl. 2002, 5: 697-705.MathSciNetMATH
  8. Choi JH, Kim YC, Sugawa T: A general approach to the Fekete-Szegö problem. J. Math. Soc. Jpn. 2007, 59(3):707-727. 10.2969/jmsj/05930707MathSciNetView ArticleMATH
  9. Bhowmik B, Ponnusamy S, Wirths K-J: On the Fekete-Szegö problem for concave univalent functions. J. Math. Anal. Appl. 2011, 373: 432-438. 10.1016/j.jmaa.2010.07.054MathSciNetView ArticleMATH
  10. Darus M, Thomas DK: On the Fekete-Szegö theorem for close-to-convex functions. Math. Jpn. 1996, 44: 507-511.MathSciNetMATH
  11. Darus M, Shanmugam TN, Sivasubramanian S: Fekete-Szegö inequality for a certain class of analytic functions. Mathematica 2007, 49: 29-34.MathSciNetMATH
  12. Jakubowski ZJ, Zyskowska K: On an estimate of a functional in the class of holomorphic univalent functions. Math. Bohem. 1993, 118(3):281-296.MathSciNetMATH
  13. Kanas S: Coefficient estimates in subclasses of the Carathéodory class related to conic domains. Acta Math. Univ. Comen. 2005, 74: 149-161.MathSciNetMATH
  14. Kanas S, Darwish HE: Fekete-Szegö problem for starlike and convex functions of complex order. Appl. Math. Lett. 2010, 23: 777-782. 10.1016/j.aml.2010.03.008MathSciNetView ArticleMATH
  15. Koepf V: On the Fekete-Szegö problem for close-to-convex functions. Proc. Am. Math. Soc. 1987, 101: 89-95.MathSciNetMATH
  16. Koepf W: On the Fekete-Szegö problem for close-to-convex functions II. Arch. Math. 1987, 49: 420-433. 10.1007/BF01194100MathSciNetView ArticleMATH
  17. London RR: Fekete-Szegö inequalities for close-to-convex functions. Proc. Am. Math. Soc. 1993, 117: 947-950.MathSciNetMATH
  18. Mishra AK, Gochhayat P: The Fekete-Szegö problem for k -uniformly convex functions and for a class defined by the Owa-Srivastava operator. J. Math. Anal. Appl. 2008, 347: 563-572. 10.1016/j.jmaa.2008.06.009MathSciNetView ArticleMATH
  19. Orhan H, Deniz E, Răducanu D: The Fekete-Szegö problem for subclasses of analytic functions defined by a differential operator related to conic domains. Comput. Math. Appl. 2010, 59: 283-295. 10.1016/j.camwa.2009.07.049MathSciNetView ArticleMATH
  20. Orhan H, Răducanu D: Fekete-Szegö problem for strongly starlike functions associated with generalized hypergeometric functions. Math. Comput. Model. 2009, 50: 430-438. 10.1016/j.mcm.2009.04.014View ArticleMATH
  21. Sim YJ, Kwon OS, Cho NE, Srivastava HM: Some classes of analytic functions associated with conic regions. Taiwan. J. Math. 2012, 16: 387-408.MathSciNetMATH
  22. Srivastava HM, Mishra AK: Applications of fractional calculus to parabolic starlike and uniformly convex functions. Comput. Math. Appl. 2000, 39: 57-69.MathSciNetView ArticleMATH
  23. Srivastava HM, Mishra AK, Das MK: The Fekete-Szegö problem for a subclass of close-to-convex functions. Complex Var. Theory Appl. 2001, 44: 145-163. 10.1080/17476930108815351MathSciNetView ArticleMATH
  24. Mocanu PT: Une propriété de convexité g énéralisée dans la théorie de la représentation conforme. Mathematica (Cluj) 1969, 11(34):127-133.MathSciNetMATH
  25. Aouf MK, Dziok J: Distortion and convolutional theorems for operators of generalized fractional calculus involving Wright function. J. Appl. Anal. 2008, 14(2):183-192.MathSciNetView ArticleMATH
  26. Dziok J: Applications of the Jack lemma. Acta Math. Hung. 2004, 105: 93-102.MathSciNetView ArticleMATH
  27. Dziok J: Classes of functions defined by certain differential-integral operators. J. Comput. Appl. Math. 1999, 105: 245-255. 10.1016/S0377-0427(99)00014-XMathSciNetView ArticleMATH
  28. Dziok J: Inclusion relationships between classes of functions defined by subordination. Ann. Pol. Math. 2011, 100: 193-202. 10.4064/ap100-2-8MathSciNetView ArticleMATH

Copyright

© Dziok; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.