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Strong differential subordination properties for analytic functions involving the Komatu integral operator
© Cho; licensee Springer. 2013
Received: 27 November 2012
Accepted: 21 January 2013
Published: 4 March 2013
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.
and let . We also denote by the subclass of with the usual normalization .
Let and be members of ℋ. The function is said to be subordinate to , or is said to be superordinate to , if there exists a function analytic in , with and , and such that . In such a case, we write or . If the function F is univalent in , then if and only if and (cf. ).
In particular, the operator is closely related to the multiplier transformation studied earlier by Flett . Various interesting properties of the operator have been studied by Jung et al.  and Liu .
Let be analytic in and let be analytic and univalent in . Then the function is said to be strongly subordinate to , or is said to be strongly superordinate to , written as , if for , as the function of z is subordinate to . We note that if and only if and .
then is called a solution of the strong differential subordination. The univalent function is called a dominant of the solutions of the strong differential subordination, or more simply a dominant, if for all satisfying (1.4). A dominant that satisfies for all dominants of (1.4) is said to be the best dominant.
Recently, Oros  introduced the following strong differential superordinations as the dual concept of strong differential subordinations.
then is called a solution of the strong differential superordination. An analytic function is called a subordinant of the solutions of the strong differential superordination, or more simply a subordinant, if for all satisfying (1.5). A univalent subordinant that satisfies for all subordinants of (1.5) is said to be the best subordinant.
and are such that for . Further, let the subclass of for which be denoted by and .
Definition 1.4 ()
for , , and . We write as .
Definition 1.5 ()
for , , and . We write as .
For the above two classes of admissible functions, Oros and Oros proved the following theorems.
Theorem 1.1 ()
Theorem 1.2 ()
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 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. [11–14] and Cho et al. .
2 Subordination results
Firstly, we begin by proving the subordination theorem involving the integral operator defined by (1.1). For this purpose, we need the following class of admissible functions.
for , , and .
which evidently completes the proof of Theorem 2.1. □
If is a simply connected domain, then for some conformal mapping h of onto Ω. In this case, the class is written as . The following result is an immediate consequence of Theorem 2.1.
Our next result is an extension of Theorem 2.1 to the case where the behavior of q on is not known.
Proof Theorem 2.1 yields . The result is now deduced from . □
for some , or
there exists such that for all .
Proof The proof is similar to that of [, Theorem 2.3d] and so is omitted. □
The next theorem yields the best dominant of the differential subordination (2.7).
q is univalent in and for some , or
q is univalent in and there exists such that for all .
and q is the best dominant.
Proof Following the same arguments as in [, 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 , , and in view of Definition 2.1, the class of admissible functions , denoted by , is described below.
whenever , , , and .
In the special case , the class is simply denoted by .
for , , , and . 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 defined by (1.1), is investigated in this section. For this purpose, the class of admissible functions is given in the following definition.
for , , and .
which evidently completes the proof of Theorem 3.1. □
If is a simply connected domain, then for some conformal mapping h of onto Ω. In this case, the class is written as . Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 3.1.
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.
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.
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).
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