Abstract
The purpose of the present paper is to investigate some subordination and superordinationpreserving properties of certain nonlinear integral operators defined on the space of meromorphically pvalent functions in the punctured open unit disk. The sandwichtype theorems associated with these integral operators are established. Relevant connections of the various results presented here with those involving relatively simpler nonlinear integral operators are also indicated.
MSC: 30C45, 30C80.
Keywords:
differential subordination; differential superordination; meromorphic functions; integral operators; convex functions; closetoconvex functions; subordination (or Löwner) chain1 Introduction, definitions and preliminaries
Let denote the class of analytic functions in the open unit disk
Let f and F be members of ℋ. The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a function w analytic in , with
such that
In such a case, we write
Furthermore, if the function F is univalent in , then we have (cf.[1,2] and [3])
Definition 1 (Miller and Mocanu [1])
Let
and let h be univalent in . If is analytic in and satisfies the differential subordination
then is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination or, more simply, a dominant if
for all satisfying (1.1). A dominant that satisfies the following condition:
for all dominants q of (1.1) is said to be the best dominant.
Definition 2 (Miller and Mocanu [4])
Let
and let h be analytic in . If p and are univalent in and satisfy the differential superordination
then is called a solution of the differential superordination. An analytic function q is called a subordinant of the solutions of the differential superordination or, more simply, a subordinant if
for all satisfying (1.2). A univalent subordinant that satisfies the following condition:
for all subordinants q of (1.2) is said to be the best subordinant.
Definition 3 (Miller and Mocanu [4])
We denote by the class of functions f that are analytic and injective on , where
and are such that
Let denote the class of functions of the form
which are analytic in the punctured open unit disk . Let and be the subclasses of consisting of all functions which are, respectively, meromorphically starlike and meromorphically convex in (see, for details, [1,5]).
For a function , we introduce the following general integral operator defined by
Several members of the family of integral operators defined by (1) have been extensively studied by many authors (see, for example, [610]; see also [11] and [12]) with suitable restrictions on the parameters α, β, γ and δ, and for f belonging to some favored subclasses of meromorphic functions. In particular, Bajpai [6] showed that the integral operator belongs to the classes and , whenever f belongs to the classes and , respectively.
Making use of the principle of subordination between analytic functions, Miller et al.[13] and, more recently, Owa and Srivastava [14] obtained some interesting subordinationpreserving properties for certain integral operators. Moreover, Miller and Mocanu [4] considered differential superordinations as the dual concept of differential subordinations (see also [15]). It should be remarked that in recent years several authors obtained many interesting results involving various integral operators associated with differential subordination and superordination (for example, see [5,1618]). In the present paper, we obtain the subordination and superordinationpreserving properties of the general integral operator defined by (1) with the sandwichtype theorem.
The following lemmas will be required in our present investigation.
Lemma 1 (Miller and Mocanu [19])
Suppose that the function
satisfies the following condition:
for all realsand for alltwith
If the function
then
Lemma 2 (Miller and Mocanu [20])
Let
If
then the solution of the differential equation
is analytic inand satisfies the following inequality:
Lemma 3 (Miller and Mocanu [1])
Let
and let
If the functionqis not subordinate to, then there exist points
for which
Let
If R is the univalent function defined in by
then the open door function (see [1]) is defined by
Remark 1 The function defined by (1.4) is univalent in , where , and is the complex plane with slits along the halflines given by
Lemma 4 (Totoi [21])
andis defined by (1.4) with, then
and
whereis the integral operator defined by (1).
A function defined on is the subordination chain (or Löwner chain) if is analytic and univalent in for all , is continuously differentiable on for all and
Lemma 5 (Miller and Mocanu [4])
Let
Also set
If
is a subordination chain and
then the following subordination condition:
implies that
Furthermore, if
has a univalent solution, thenqis the best subordinant.
Lemma 6 (Pommerenke [22])
The function
with
Suppose thatis analytic infor alland thatis continuously differentiable onfor all. If the functionsatisfies the following inequalities:
and
for some positive constantsand, thenis a subordination chain.
2 Main results and their corollaries and consequences
We begin by proving a general subordination property involving the integral operator defined by (1), which is contained in Theorem 1 below.
Theorem 1Let, whereis defined by (1.5). Suppose also that
where
Then the subordination relation
implies that
whereis the integral operator defined by (1). Moreover, the function
is the best dominant.
Proof Let us define the functions F and G, respectively, by
We first show that if the function q is defined by
then
From the definition of (1), we obtain
We also have
It follows from (2.7) and (2.8) that
Now, by a simple calculation with (2.9), we obtain the following relationship:
Thus, from (2.1), we have
and by using Lemma 2, we conclude that the differential equation (2.10) has a solution with
Put
where ρ is given by (2.2). From (2.1), (2.10) and (2.11), we obtain
We now proceed to show that
Indeed, from (2.11), we have
where
For ρ given by (2.2), we note that the coefficient of in the quadratic expression given by (2.14) is positive or equal to zero and also is a perfect square. Hence from (2.13), we see that (2.12) holds true. Thus, by using Lemma 1, we conclude that
That is, the function defined by (2.5) is convex in .
We next prove that the subordination condition (2.3) implies that
for the functions F and G defined by (2.5). Without loss of generality, we can assume that G is analytic and univalent on and that
We now consider the function defined by
We note that
and
Furthermore, since G is convex, by using the wellknown growth and distortion sharp inequalities for convex functions (see [23]), we can prove that the second condition of Lemma 6 is satisfied. Therefore, by virtue of Lemma 6, is a subordination chain. We observe from the definition of a subordination chain that
and
This implies that
We now suppose that F is not subordinate to G. Then, in view of Lemma 3, there exist points and such that
Hence we have
by virtue of the subordination condition (2.3). This contradicts the above observation that
Therefore, the subordination condition (2.3) must imply the subordination given by (2.15). Considering , we see that the function G is the best dominant. This evidently completes the proof of Theorem 1. □
We next prove a solution to a dual problem of Theorem 1 in the sense that the subordinations are replaced by superordinations.
Theorem 2Let, whereis defined by (1.5). Suppose also that
whereρis given by (2.2) andis univalent in, and
whereis the integral operator defined by (1). Then the superordination relation
implies that
Moreover, the function
is the best subordinant.
Proof Let the functions F and G be given by (2.5). We first note from (2.7) and (2.8) that
After a simple calculation, equation (2.17) yields the following relationship:
where the function q is given in (2.6). Then, by using the same method as in the proof of Theorem 1, we can prove that
that is, G defined by (2.5) is convex (univalent) in .
We next prove that the superordination condition (2.16) implies that
For this purpose, we consider the function defined by
Since G is convex and , we can prove easily that is a subordination chain as in the proof of Theorem 1. Therefore, according to Lemma 5, we conclude that the superordination condition (2.16) must imply the superordination given by (2.18). Furthermore, since the differential equation (2.17) has the univalent solution G, it is the best subordinant of the given differential superordination. Hence we complete the proof of Theorem 2. □
If we combine Theorem 1 and Theorem 2, then we obtain the following sandwichtype theorem.
Theorem 3Let (), whereis defined by (1.5). Suppose also that
whereρis given by (2.2) and the functionis univalent in, and
whereis the integral operator defined by (1). Then the subordination relation
implies that
Moreover, the functions
are the best subordinant and the best dominant, respectively.
The assumption of Theorem 3, that is, the functions
need to be univalent in , will be replaced by another set of conditions in the following result.
Corollary 1Let (), whereis defined by (1.5). Suppose also that the condition (2.19) is satisfied and
whereρis given by (2.2). Then the subordination relation
implies that
whereis the integral operator defined by (1). Moreover, the functions
are the best subordinant and the best dominant, respectively.
Proof In order to prove Corollary 1, we have to show that the condition (2.20) implies the univalence of and
By noting that from (2.2), we obtain from the condition (2.20) that ψ is a closetoconvex function in (see [24]), and hence is univalent in . Furthermore, by using the same techniques as in the proof of Theorem 3, we can prove the convexity (univalence) of F and so the details may be omitted. Therefore, by applying Theorem 3, we obtain Corollary 1. □
By setting in Theorem 3, we have the following result.
Corollary 2Let (), whereis defined by (1.5) with. Suppose also that
and the functionis univalent in, and
whereis the integral operator defined by (1) with. Then the subordination relation
implies that
Moreover, the functions
are the best subordinant and the best dominant, respectively.
If we take in Theorem 3, then we are easily led to the following result.
Corollary 3Let (), whereis defined by (1.5) with. Suppose also that
and the functionis univalent in, and
whereis the integral operator defined by (1) with. Then the subordination relation
implies that
Moreover, the functions
are the best subordinant and the best dominant, respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors jointly worked on the results and they 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 the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20120002619).
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