Abstract

We obtain sufficient conditions for the univalence, starlikeness, and convexity of a new integral operator defined on the space of normalized analytic functions in the open unit disk. Some subordination results for the new integral operator are also given. Several corollaries follow as special cases.

1. Introduction

Let be the open unit disk and the class of all functions of the formwhich are analytic in and satisfy the condition Consider the class of functions which are univalent in .

A domain is convex if the line segment joining any two points in lies entirely in , while a domain is starlike with respect to a point if the line segment joining any point of to lies inside .

A function is starlike if is a starlike domain with respect to origin and convex if is convex.

Analytically, is starlike if and only ifand is convex if and only ifThe classes consisting of starlike and convex functions are denoted by and , respectively. We denote by and the classes consisting of starlike and convex functions of order , , characterized, respectively, by

If and are analytic functions in , we say that is subordinate to , written as , if there is a function analytic in , with , , for all , such that for all . If is univalent, then if and only if and (see Miller and Mocanu [1]).

Using subordinations, Owa et al. [2] have defined the following subclass of .

A function is said to be in the class if it satisfiesfor some real and

For the class , Owa et al. proved the following.

Theorem 1 (see [2]). If satisfiesfor some real and , then

Breaz et al. [3] have introduced the following subclasses and of . A function is said to be in the class if it satisfies inequality (6) for some real and

A function is said to be a member of the class if it satisfiesfor some real and

Recently, Frasin and Jahangiri [4] defined the family , , consisting of functions satisfying the conditionWe note that and (see [5]).

In 1975, Pfaltzgraff [6] introduced the operatorand proved that if is a complex number and

During the last several years many authors have employed different methods to study generalizations of Pfaltzgraff [6] integral operator which maps subsets of into .

Pascu and Pescar [7] studied the operatorwhere is a positive integer.

The next operator was defined by Breaz et al. in [8] and has the formwhere and for all

Further extensions of these operators were obtained in [9–11].

Recently, starlikeness and convexity properties for certain general families of integral operators in the open unit disk were given in [12–14].

In the present paper, we introduce the new integral operator defined byfor the functions and the complex number

We observe that if we take , we obtain Pfaltzgraff [6] integral operator.

In the first section of this paper, our purpose is to derive univalence conditions, starlikeness properties, and the order of convexity for the integral operator introduced in (13). The object of the second section of this paper is to discuss some properties for the integral operator with the above classes , and .

The following results will be required in our investigation.

Lemma 2 (Mocanu and Şerb [15]). Let , the positive solution of equationIf and thenThe edge is sharp.

Theorem 3 (Becker [16]). If the function f is regular in the unit disk U, , andfor all , then the function f is univalent in .

Lemma 4 (the general Schwarz Lemma [17]). Let the function be regular function in the disk , with for fixed . If has one zero with multiplicity order bigger than for , then The equality case holds only if , where is constant.

Lemma 5 (see [18]). Let the functions and be analytic in with and let be a real number. If the function maps the unit disk onto a region which is starlike with respect to the origin, the inequalityimplies that

2. Univalence, Starlikeness, and Convexity Properties of the Integral Operator

The univalence condition for the operator defined in (13) is proved in the next theorem, by using the Becker univalence criterion.

Theorem 6. Let be a complex number, the positive solution of (14), , and . Ifthen the integral operator is in the class .

Proof. Let the function be as follows:Then, we haveFrom (26) we get By using (22) and applying Lemma 2, we have which implies thatLet us consider the functionThen we haveFrom (29), (31), and (23) we obtainFrom Theorem 3 and (32), we yield that the function is in the class .

If we put in Theorem 6, we obtain the following.

Example 7. Let be a complex number, the positive solution of (14), , and . Ifthen the integral operator is in the class .

In the following theorem we give sufficient conditions such that the integral operator

Theorem 8. Let with , , and in the class Let be a positive real number such thatIfthen the integral operator is in the class .

Proof. For the function defined in (25) we haveLettingwe find thatThus,Since , , by applying the Schwarz Lemma, we haveBy using the hypothesis and (41), we obtain that is,Therefore, applying Lemma 5, we find thatThis completes the proof of the theorem.

Letting in Theorem 8, we have the following.

Corollary 9. Let with , , and in the class Let be a positive real number such thatIfthen the integral operator is in the class .

Letting in Corollary 9, we obtain the following.

Corollary 10. Let with , , and . Let be a positive real number such thatIfthen the integral operator is in the class .

Since is a starlike function, from Corollary 10, we obtain the following.

Example 11. Let with , . Ifthen the integral operator is in the class .

Theorem 12. Let be a complex number, , and If for , , and , then the integral operator is in the class , where

Proof. Let the function be defined in (25). Then, we obtain Therefore,Since , , applying the Schwarz Lemma, we have From (54) we obtain This evidently completes the proof.

Letting in Theorem 12, we have the following.

Corollary 13. Let be a complex number, , with in the class , If , for , , and then the integral operator is as follows: where and .

Letting in Corollary 13, we obtain the following.

Corollary 14. Let be a complex number, , with a starlike function in U. If , for , , and then the integral operator is convex in , where

Taking in Corollary 14, we obtain the following.

Example 15. Let be a complex number and function . If then the integral operator is convex in , where .

3. Subordination Results

In view of the results due to Breaz et al. [3], we obtain some subordination results of the above integral operator .

Theorem 16. If and , , and , then for This implies that .