Mapping Properties of Certain Linear Operator Associated with Hypergeometric Functions

abstract: The main object of the present paper is to find some sufficient conditions in terms of hypergeometric inequalities so that the linear operator denoted by H a,b,c μ.δ maps a certain subclass of close-to-convex function Rτ (A,B) into subclasses of k-uniformly starlike and k-uniformly convex functions k− ST(β) and k−UCV(β) respectively. Further, we consider an integral operator and discuss its properties. Our results generalize some relevant results.

Note that p F q (z) is an entire function if p < q + 1. However, if p = q + 1, then p F q (z) is analytic in U. Also, if p = q + 1 and ℜ( q j=1 β j − p j=1 α j ) > 0, then p F q (z) converges on ∂U. In particular, the function is the familiar Gaussian hypergeometric function. The hypergeometric function 2 F 1 (a, b; c; z) has been extensively studied by various authors and play an important role in the Geometric Function Theory. It is useful in unifying various functions by givien appropriate values to the parameters a, b and c. Further, the series (1.9) may be regarded as a generalization of the elementary geometric series. It reduces to the geometric series in two cases. When a = c and b = 1 and when b = c and a = 1. It is worthy to mention here that the function 2 F 1 (a, b; c; z) is symmetric in a and b and the series (1.9) terminates if at least one of the numerator parameters a and b is zero or negative integer. For recent expository work on hypergeometric function see [4,7,12,22]. It is well-known that 2 F 1 (a, b; c; z) is the solution of the second order homogeneous differential equation Note that the behavior of the hypergeometric function 2 F 1 (a, b; c; z) near z = 1 is classified into three cases according as ℜ(c − a − b) is positive, zero or negative. By Gauss summation formula we get has a series expansion of the form Using normalized hypergeometric function z 2 F 1 (a, b; c; z) consider the function (see [25],with p=1) For a function f ∈ A given by (1.1) and g ∈ A given by the Hadamard product (or convolution) of f and g is defined by (1.14) We consider the linear operator H a,b,c µ,δ : A −→ A defined by mean of Hadamard product as Thus, for a function f ∈ A of the form (1.1), we have Kim and Shon (see [13] c is known as Hohlov operator (see [9]). Motivated by Sharma et al. [23] (also, see [1,16]), in this paper sufficient conditions in term of hypergeometric inequalities are found so that the linear operator defined by (1.16) maps a certain subclass of close-to-convex function R τ (A, B) into subclasses of k-uniformly starlike and k-uniformly convex functions k − ST(β) and k − UCV(β) respectively. Further, we consider an integral operator and discuss its properties.

Preliminaries Lemmas
To investigate our main results, we need each of the following lemmas: Lemma 2.3. (see [5]) Let the function f , given by (1.1) be a member of R τ (A, B).
The estimate in (2.3) is sharp.

Main Results
Unless otherwise mentioned, we assume throughout the sequel that Theorem 3.1. Let a, b, c ∈ R be such that a, b > 1 and c > a + b + 2. If the hypergeometric inequality is satisfied, then H a,b,c µ,δ maps the class R τ (A, B) into k − ST(β).

Proof:
Let the function f given by (1.1) be a member of R τ (A, B). By (1.16) we have In view of Lemma 2.1, it is sufficient to show that Since f ∈ R τ (A, B) by virtue of Lemma 2.3, it is again sufficient to show that Repeated applications of the relation Thus, in view of (3.2) if the hypergeometric inequality (3.1) is satisfied, the Corollary 3.2. Let a, b, c ∈ R be such that a, b > 1 and c > a + b + 1. If the hypergeometric inequality Further, by taking µ = 0 in Corollary 3.2, we get the following result:  Letting k = β = 0 in Theorem 3.1 we have the following result in form of a corollary: Corollary 3.5. Let a, b, c ∈ R be such that a, b > 0 and c > a + b + 2. If the hypergeometric inequality Putting k = 1 and β = 0 in Theorem 3.1, we get the following result: Corollary 3.7. Let a, b, c ∈ R be such that a, b > 1 and c > a + b + 2. If the hypergeometric inequality is satisfied, then H a,b,c µ,δ maps the class R τ (A, B) into SP. Remark 3.8. Letting δ = 0 in Corollary 3.7 we get the result due to Sharma et al. ( [23], Corollary 2, page 330). Theorem 3.9. Let a, b, c ∈ R be such that a, b > 0 and c > a + b + 3. If the hypergeometric inequality is satisfied, then H a,b,c µ,δ maps the class R τ (A, B) into k − UCV(β).

Proof:
Let the function f given by (1.1) be a member of R τ (A, B). By virtue of Lemma 2.2 and the coefficient inequality (2.3) it is sufficient to show that where n(n + 1) (a) n (b) n (c) n (1) n + (1 + k)µδ

Acknowledgement
The Authors would like to thank to the editor and anonymous referees for their comments and suggestions which improve the contents of the manuscript. Further, the present investigation of the first-named author is supported by CSIR research project scheme no. 25(0278)/17/EMR-II, New Delhi, India.