Abstract

In this present paper, we introduce and explore certain new classes of uniformly convex and starlike functions related to the Liu–Owa integral operator. We explore various properties and characteristics, such as coefficient estimates, rate of growth, distortion result, radii of close-to-convexity, starlikeness, convexity, and Hadamard product. It is important to mention that our results are a generalization of the number of existing results in the literature.

1. Introduction

Let denote the complex plane and assume that Ap denotes the class of -valent function of the formwhich are analytic in the open unit disc and . Specially, for , we denote A=A1.

By , , and , the subclasses of Ap consist of all univalent, convex, and starlike functions . We also denote , the class of starlike function of order , . In 1991, Goodman [1, 2] introduced the classes and of uniformly starlike and uniformly convex functions, respectively. A function is uniformly starlike (uniformly convex) in if is in and has the property that, for every circular arc contained in , with center also in , the arc is starlike (convex) with respect to . A more useful representation of and was given in [3]; see [4, 5], for details:and

In 1999, for , Kanas and Wisniowska [6] introduced the classes and of -uniformly convex and uniformly starlike functions, respectively, see also [710].

Let denote the subclass of Ap consisting of functions of form (1) and satisfy the following inequality:

Also, let denote the subclass of Ap consisting of functions of form (1) and satisfy the following inequality:

It follows from (4) and (5) that

Notice that, and , for . The convolution (Hadamard product) for two functions , Ap, is defined bywhere is given in (1) and .

Taking from the above cited work and using Liu–Owa integral operator, we introduce the following class of -valent analytic function. In 2004, Liu and Owa [11] (see also [1214]) introduced the integral operator : as follows:and

For , given by (1), and using properties of gamma function, we have

Definition 1. For , , , and , a function is in class if and only ifWe also denote , where the class of functions of form (1) for which . For more details, see [1520].

1.1. Special Cases

Specializing parameters, , and , we obtain the following subclasses studied by various authors:

(1) [21]

(2) [21]

(3) [22]

(4) [23]

(5) [24]

(6) [25]

2. Main Results for the Class

2.1. Coefficient Estimates

In this section, we obtain a necessary and sufficient condition for functions in the classes .

Theorem 1. A function given by (1) is in the class ifwhereand

Proof. It suffices to show that inequality (11) holds true. As we know,Then, inequality (11) may be written aswhich can be written as , whereandThen, we haveNow,Also,Using (23) and (24), then we can obtain the following inequality:The last expression is bounded below by 0 ifwhich complete the proof.

Theorem 2. Let be given by (1) and in ; then, if and only ifwhere , , , and are given by (13)–(16), respectively.

Proof. In view of Theorem 2, we need only to show that satisfies coefficient inequality (27). If , then, by definition, we haveSince is a function of form (1) with the argument properties given in the class and setting in the above inequality, we haveLetting (29) leads to the desired inequality:The function,is an external function for (27).

Corollary 1. Let the function defined by (1) be in the class ; then,with equality in (32), is attained for the function given by (31).

Theorem 3. Let the function with argument property as in class . Define andwhere and . Then, is in the class if and only if it can be expressed aswhere and .

Proof. Assume thatThen, by Theorem 2, . It follows thatConversely, assume that the function defined by (1) belongs to the class ; then,Setting , and , then and this completes the proof.

2.2. Growth and Distortion Result

In this section, we find a growth and distortion bound for functions in the classes .

Theorem 4. Let the function be defined by (1) in the class ; then, for ,andwhere equalities (38) and (39) hold for the function given by (27), for .

Proof. From Theorem 2, we haveThe last inequality follows from Theorem 2 Thus,Similarly,Now, by differentiating (1), we obtainandUsing Theorem 2 in (44), we haveor, equivalentlyUsing (46) into (43) and (44) yields inequality (39).

2.3. Radii of Close-to-Convexity, Starlikeness, and Convexity

In this section, we obtain the radii of close-to-convexity, starlikeness, and convexity for functions in the classes .

Theorem 5. Let ; then,
(i) is starlike of order in the disc , where(ii) is convex of order in the disc , where

These results are sharp for the extremal function given by (31).

Proof. (i) Given and is starlike of order , we haveFor the left-hand side of (49), we haveThe last expression is less than ifUse the fact that if and only ifWe can say (49) is true ifor, equivalently,which is required.
(ii) Using the fact that is convex if and only if is starlike, we can prove (ii) on similar lines to the proof of (i).

2.4. Modified Hadamard Product

Let the function be defined by

Then, we define the modified Hadamard product of and by

Now, we prove the following.

Theorem 6. Let given by (55) be in the class ; then, for

Proof. We need to prove the largest , such thatFrom Theorem 2, we haveBy Cauchy–Schwarz inequality, we haveThus, it is sufficient to show thatThat is,