Abstract

We derive subordination and superordination results for a family of normalized analytic functions in the open unit disk defined by integral operators. We apply this to obtain sandwich results and generalizations of some known results.

1. Introduction

Let denote the class of analytic functions in the unit disk , and let be the subclass of of the form

Let be the subclass of of the form

If and are analytic and there exists a Schwarz function , analytic in withsuch that , then the function is called to and is denoted byIn particular, if the function is univalent in , the above subordination is equivalent to

Suppose and are analytic functions in and . If and are univalent and if satisfies the second-order superordination then is a solution of the differential superordination (1.6). Note that if is subordinate to , then is superordinate to . An analytic function is called if for all satisfying (1.6). A univalent subordinant that satisfies for all subordinants of (1.6) is said to be the . Miller and Mocanu [1] have obtained conditions on , , and for which the following implication holds:

Ali et al. [2] have obtained sufficient conditions for certain normalized analytic functions to satisfywhere and are given univalent functions in with and .

Recently, Shanmugam et al. [3, 4] have also obtained sandwich results for certain classes of analytic functions. Further subordination results can be found in [5–8].

2. Definitions and Preliminaries

Definition 2.1. For , Shams et al. [9] defined the following integral operator:For the operator, one easily getsAlso for and , Shams et al. [9] defined a class of functions , so thatThe family is a general family containing various new and known classes of analytic functions (see, e.g., [10, 11]).

Definition 2.2 (see [1]). Denote by the set of all functions that are analytic and injective on , whereand are such that for

We will require certain results due to Miller and Mocanu [1, 12], Bulboacă [13], and Shanmugam et al. [4] contained in the following lemmas.

Lemma 2.3 (see [12]). Let be univalent in the unit disk , and let and be analytic in the domain containing with when . Set Suppose that
(i) is starlike univalent in ;(ii) for If is analytic in , with , , andthen and is the best dominant.

Lemma 2.4 (see [4]). Let be a convex univalent function in and with If is in andthen and is the best dominant.

Lemma 2.5 (see [12]). Let be univalent in , and let be analytic in a domain containing . If is starlike andthen and is the best dominant.

Lemma 2.6 (see [13]). Let be convex univalent in the unit disk , and let and be analytic in a domain containing . Suppose that
(i) for ;(ii) is starlike univalent in . If , with , and if is univalent in andthen and is the best subordinant.

Lemma 2.7 (see [1]). Let be convex univalent in and . Further assume that . If and is univalent in , thenwhich implies that and is the best subordinant.

The main object of this paper is to apply a method based on the differential subordination in order to derive several subordination results.

3. Subordination for Analytic Functions

Theorem 3.1. Let be univalent in the unit disk , , andIf satisfies the subordination
where is defined by (2.1), thenand is the best dominant.

Proof. ConsiderDifferentiating (3.4) with respect to logarithmically, we getNow, in view of (2.3), we obtain from (3.5) the following subordination:An application of Lemma 2.4, with and , leads to (3.3).

Taking in Theorem 3.1, we arrive at the following.

Corollary 3.2. Let and
. If andthenand is the best dominant.

Putting and in Theorem 3.1, we get the following corollary.

Corollary 3.3. Let and . If andthenand is the best dominant.

Theorem 3.4. Let be univalent in and , and such that . Let and suppose that satisfiesIfthenand is the best dominant.

Proof. Let us consider a function defined byNow, differentiating (3.14) logarithmically, we getBy settingit can be easily observed that is analytic in and that is analytic in . Also, we letFrom (3.11) we see that is starlike univalent in the unit disk , and from (3.18) we getAn application of Lemma 2.3 to (3.12) yields the result.

Putting , and in Theorem 3.4, we obtain the following corollary.

Corollary 3.5. If and for ,thenand is the best dominant.

By setting , and in Theorem 3.4, we get the following corollary.

Corollary 3.6. Suppose and let and . Ifthenand is the best dominant.

Remark 3.7. is univalent if and only if or (see [5]).

Again by setting , and , and by in Theorm 3.4, we get the following corollary.

Corollary 3.8. Suppose and is a nonzero complex number for whichThen, and is the best dominant.

The result contained in Corollary 3.8 was earlier given by Srivastava and Lashin [7].

Theorem 3.9. Let be univalent in the unit disk , and let , and . Suppose that satisfiesLetIfthenand is the best dominant.

Proof. Define a function byThen, a computation shows thatand henceSetand let From (3.26), we see that is starlike in and thatby the hypothesis (3.26) of Theorem 3.9. Thus, applying Lemma 2.3, the proof of Theorem 3.9 is completed.

By setting , we obtain the following corollary.

Corollary 3.10. Let and . Suppose thatIfthenand is the best dominant.