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 sandwichtype theorems for these operators are also considered.
MSC: 30C80, 30C45, 30A20.
Keywords:
strong differential subordination; strong differential superordination; univalent function; convex function; Komatu integral operator1 Introduction
Let ℋ denote the class of analytic functions in the open unit disk
and let
Let
Following Komatu [2], we introduce the integral operator
where the symbol Γ stands for the gamma function. We also note that the operator
Obviously, we have, for
Moreover, from (1.2), it follows that
In particular, the operator
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
Let
then
Recently, Oros [9] introduced the following strong differential superordinations as the dual concept of strong differential subordinations.
Let
then
Denote by
and are such that
Definition 1.4 ([8])
Let Ω be a set in ℂ,
whenever
for
Definition 1.5 ([9])
Let Ω be a set in ℂ and
whenever
for
For the above two classes of admissible functions, Oros and Oros proved the following theorems.
Theorem 1.1 ([8])
Let
then
Theorem 1.2 ([9])
Let
is univalent in
implies
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
2 Subordination results
Firstly, we begin by proving the subordination theorem involving the integral operator
Definition 2.1 Let Ω be a set in ℂ,
whenever
and
for
Theorem 2.1Let
then
Proof Define the function
From (2.2) with the relation (1.3), we get
Further computations show that
Define the transformation from
Let
Using equations (2.2), (2.3) and (2.4), from (2.6), we obtain
Hence, (2.1) becomes
Note that
and so the admissibility condition for
which evidently completes the proof of Theorem 2.1. □
If
Theorem 2.2Let
then
Our next result is an extension of Theorem 2.1 to the case where the behavior of q on
Corollary 2.3Let
then
Proof Theorem 2.1 yields
Theorem 2.4Lethandqbe univalent in
(1)
(2) there exists
If
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 2.5Lethbe univalent in
has a solutionqwith
(1)
(2) qis univalent in
(3) qis univalent in
If
is analytic in
andqis 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
Definition 2.2 Let Ω be a set in ℂ,
whenever
Corollary 2.6Let
then
In the special case
Corollary 2.7Let
then
Corollary 2.8Let
then
Proof This follows from Corollary 2.6 by taking
for
3 Superordination and sandwichtype results
The dual problem of differential subordination, that is, differential superordination
of the Komatu integral operator
Definition 3.1 Let Ω be a set in ℂ,
whenever
and
for
Theorem 3.1Let
is univalent in
implies
Proof From (2.7) and (3.1), we have
From (2.5), we see that the admissibility condition for
which evidently completes the proof of Theorem 3.1. □
If
Theorem 3.2Let
is univalent in
implies
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 3.3Lethbe analytic in
has a solution
is univalent in
implies
andqis 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 sandwichtype theorem.
Theorem 3.4Let
is univalent in
implies
Competing interests
The author declares that they have no competing interests.
Authors’ contributions
The author worked on the results and he read and approved the final manuscript.
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. 20120002619).
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