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 . For a positive integer n and , let
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.[1]).
Following Komatu [2], we introduce the integral operator defined by
where the symbol Γ stands for the gamma function. We also note that the operator defined by (1.1) can be expressed by the series expansion as follows:
Moreover, from (1.2), it follows that
In particular, the operator is closely related to the multiplier transformation studied earlier by Flett [3]. Various interesting properties of the operator 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 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 .
Let and let be univalent in . If is analytic in and satisfies the (secondorder) differential subordination
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 [9] introduced the following strong differential superordinations as the dual concept of strong differential subordinations.
Let and let be analytic in . If and are univalent in for and satisfy the (secondorder) strong differential superordination
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.
Denote by the class of functions q that are analytic and injective on , where
and are such that for . Further, let the subclass of for which be denoted by and .
Definition 1.4 ([8])
Let Ω be a set in ℂ, and n be a positive integer. The class of admissible functions consists of those functions that satisfy the admissibility condition
Definition 1.5 ([9])
Let Ω be a set in ℂ and with . The class of admissible functions consists of those functions that satisfy the admissibility condition
For the above two classes of admissible functions, Oros and Oros proved the following theorems.
Theorem 1.1 ([8])
Theorem 1.2 ([9])
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 sandwichtype 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 defined by (1.1). For this purpose, we need the following class of admissible functions.
Definition 2.1 Let Ω be a set in ℂ, , and . The class of admissible functions consists of those functions that satisfy the admissibility condition
whenever
and
then
Proof Define the function in by
From (2.2) with the relation (1.3), we get
Further computations show that
Define the transformation from to ℂ by
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 is equivalent to the admissibility condition for . Therefore, by Theorem 1.1, or
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.
then
Our next result is an extension of Theorem 2.1 to the case where the behavior of q on is not known.
Corollary 2.3Letandqbe univalent inwith. Letfor somewhere. Ifsatisfies
then
Proof Theorem 2.1 yields . The result is now deduced from . □
Theorem 2.4Lethandqbe univalent inwithand setand. Letsatisfy one of the following conditions:
(2) there existssuch thatfor all.
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 inand let. Suppose that the differential equation
has a solutionqwithand satisfies one of the following conditions:
(2) qis univalent inandfor some, or
(3) qis univalent inand there existssuch thatfor all.
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 , , and in view of Definition 2.1, the class of admissible functions , denoted by , is described below.
Definition 2.2 Let Ω be a set in ℂ, , and . The class of admissible functions consists of those functions such that
then
In the special case , the class is simply denoted by .
then
Corollary 2.8Let, and letbe an analytic function inwithfor. Ifsatisfies
then
Proof This follows from Corollary 2.6 by taking and , where . To use Corollary 2.6, we need to show that , that is, the admissible condition (2.10) is satisfied. This follows since
for , , , and . Hence, by Corollary 2.6, we deduce the required results. □
3 Superordination and sandwichtype 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.
Definition 3.1 Let Ω be a set in ℂ, with , and . The class of admissible functions consists of those functions that satisfy the admissibility condition
whenever
and
implies
Proof From (2.7) and (3.1), we have
From (2.5), we see that the admissibility condition for is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence, , and by Theorem 1.2, or
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 3.2Let, hbe analytic inand. If, and
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 inand. Suppose that the differential equation
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 3.4Letandbe analytic functions in, be a univalent function in, withand. If, and
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).
References

Miller, SS, Mocanu, PT: Differential Subordination, Theory and Application, Marcel Dekker, New York (2000)

Komatu, Y: Distortion Theorems in Relation to Linear Integral Operators, Kluwer Academic, Dordrecht (1996)

Flett, TM: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl.. 38, 746–765 (1972). Publisher Full Text

Jung, IB, Kim, YC, Srivastava, HM: The Hardy space of analytic functions associated with certain oneparameter families of integral operators. J. Math. Anal. Appl.. 176, 138–147 (1993). Publisher Full Text

Liu, JL: A linear operator and strongly starlike functions. J. Math. Soc. Jpn.. 54, 975–981 (2002). Publisher Full Text

Antonino, JA: Strong differential subordination to BriotBouquet differential equations. J. Differ. Equ.. 114, 101–105 (1994). Publisher Full Text

Antonino, JA: Strong differential subordination and applications to univalency conditions. J. Korean Math. Soc.. 43, 311–322 (2006)

Oros, GI, Oros, G: Strong differential subordination. Turk. J. Math.. 33, 249–257 (2009)

Oros, GI: Strong differential superordination. Acta Univ. Apulensis, Mat.Inform.. 19, 101–106 (2009)

Miller, SS, Mocanu, PT: Subordinants of differential superordinations. Complex Var. Theory Appl.. 48, 815–826 (2003). Publisher Full Text

Ali, RM, Ravichandran, V, Seenivasagan, N: Subordination and superordination of the LiuSrivastava operator on meromorphic functions. Bull. Malays. Math. Soc.. 31, 193–207 (2008)

Ali, RM, Ravichandran, V, Seenivasagan, N: Differential subordination and superordination of analytic functions defined by the multiplier transformation. Math. Inequal. Appl.. 12, 123–139 (2009)

Ali, RM, Ravichandran, V, Seenivasagan, N: Differential subordination and superordination of analytic functions defined by the DziokSrivastava linear operator. J. Franklin Inst.. 347, 1762–1781 (2010). Publisher Full Text

Ali, RM, Ravichandran, V, Seenivasagan, N: On subordination and superordination of the multiplier transformation of meromorphic functions. Bull. Malays. Math. Soc.. 33, 311–324 (2010)

Cho, NE, Kwon, OS, Srivastava, HM: Strong differential subordination and superordination for multivalently meromorphic functions involving the LiuSrivastava operator. Integral Transforms Spec. Funct.. 21, 589–601 (2010). Publisher Full Text