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Certain Subclasses of Starlike Functions of Complex Order Involving Generalized Hypergeometric Functions
Making use of the generalized hypergeometric functions, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients and obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness and convexity, and neighborhood results for the class . In particular, we obtain integral means inequalities for the function that belongs to the class in the unit disc.
Let denote the class of functions of the form
which are analytic in the open unit disc and normalized by Also denote by the subclass of consisting of functions of the form
For positive real values of and the generalized hypergeometric function is defined by
where denotes the set of all positive integers and is the Pochhammer symbol defined by
The notation is quite useful for representing many well-known functions such as the exponential, the Binomial, the Bessel, the Laguerre polynomial, and others; for example, see .
Let be a linear operator defined by
unless otherwise stated. For notational simplicity, we can use a shorter notation for in the sequel. The linear operator is called Dziok-Srivastava operator (see ), includ (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer , Owa , Ruscheweyh , and Srivastava and Owa . Motivated by Goodman [7, 8], Rønning [9, 10] introduced and studied the following subclasses of A function is said to be in the class , uniformly -starlike functions if it satisfies the condition
and is said to be in the class uniformly -convex functions if it satisfies the condition
The interesting geometric properties of these function classes were extensively studied by Kanas et al., in [11–14]. Motivated by Altintas et al. , Murugusundaramoorthy and Srivastava , and Murugusundaramoorthy and Magesh , now, we define a new subclass uniformly starlike functions of complex order.
For , and , we let be the class of functions satisfying (1.2) with the analytic criterion
where is given by (1.6).
By suitably specializing the values of , , and the class , leads to various new subclasses of starlike functions of complex order. As for illustrations, we present some examples for the cases.
Example 1.1. If with , and of the form of (1.2), then
Example 1.2. If with , and of the form of (1.2), then where is called Ruscheweyh derivative of order defined by
The main object of this paper is to study some usual properties of the geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness, and convexity for the class Further, we obtain neighborhood results and integral means inequalities for aforementioned class.
2. Basic Properties
First we obtain the necessary and sufficient condition for functions in the class
Theorem 2.1. The necessary and sufficient condition for of the form of (1.2) to be in the class is where , and
Proof. Assume that , then Letting along the real axis, we have Hence, by maximum modulus theorem, the simple computational leads the desired inequality Conversely, suppose that (2.1) is true for Then if That is if which completes the proof.
Corollary 2.2. Let the function defined by (1.2) belong to Then and, , with equality for
Next we state the following theorem on extreme points for the class without proof.
Theorem 2.3 (Extreme Points). Let Then if and only if can be expressed in the form , where and
3. Close-to-Convexity, Starlikeness, and Convexity
We determine the radii of close-to-convexity, starlikeness, and convexity results for functions in the class in the following theorems.
Theorem 3.1. Let Then is close-to-convex of order in the disc , where
Proof. Let belong to It is known  that is close-to-convex of order , if it satisfies the condition For the left-hand side of (3.2) we have The last expression is less than if Using the fact that if and only if we can say that (3.2) is true if Or, equivalently, which completes the proof.
Theorem 3.2. Let Then the following are given. is starlike of order in the disc where is convex of order in the unit disc , where Each of these results is sharp for the extremal function given by (2.10).
Proof. Let It is known  that is starlike of order , if it satisfies the condition
For the left-hand side of (3.10) we have
The last expression is less than if
Using the fact that if and only if
we can say that (3.10) is true if
which yields the starlikeness of the family.
Using the fact that is convex if and only if is starlike, we can prove (2), on lines similar to those the proof of (1).
4. Integral Means
Lemma 4.1 (see ). If the functions and are analytic in with , then
Theorem 4.2. Let If and is nondecreasing sequence, then, for and one has where , and
5. Inclusion Relations Involving
Particularly for the identity function , we have
Theorem 5.1. Let where Then
Now we determine the neighborhood for each of the class which we define as follows. A function is said to be in the class if there exists a function such that
Theorem 5.2. If and where then
Proof. Suppose that , then we find from (5.6) that which implies the coefficient inequality Next, since , we have so that provided that is given precisely by (5.8). Thus by definition, for given by (5.8), which completes the proof.
By suitably specializing the various parameters involved in Theorems 2.1 to 5.2, we can state the corresponding results for the new subclasses defined in Examples 1.1 to 1.4 and also for many relatively more familiar function classes.
The authors would like to record their sincerest thanks to the referee(s) for their valuable suggestions and comments.
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