Starlikeness and convexity results for hypergeometric functions

Abstract

a b s t r a c tThe objective of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions of order α type β. We also consider an integral operator related to the hypergeometric function.

Some Starlike and Convexity Properties Associated with P-Valent Hypergeometric Functions

“…Let a, b > −1, ab < 0, c > max{0, a + b}, z ∈ U, −1 ≤ B < A ≤ 1, 0 ≤ β < 1. Then For A = 1 − 2α and B = −1 in Corollary 1 and 2, we easily arrive at the recent results due to Mostafa [6], which evidently contains the results due to Silverman [9] for β = 1.…”

“…Since (a) n+k = (a) k (a + k) n , and using (1.6), we can write the equation (2 Remark 2.1 Putting p = 1, A = 1 − 2α, B = −1 in Theorems 1 and 2, we get the results given recently by Mostafa [6], which contains the results due to Silverman [9]. be an integral opeartor acting on 2 F 1 (a, b; c; z).…”

“…, (iii) S * (α, 1) = S * (α) and (iv) C * (α, 1) = C(α) The subclasses of class of S * (α, β) and C * (α, β) were introduced and studied by Gupta and Jain([4], see also [6]) while the subclasses S * (α) and C(α) were studied by Silverman [10]. Let 2 F 1 (a, b; c; z) be the(Gaussian)hypergeometric function defined by…”

“…Silverman [9] gave necessary and sufficient conditions for z 2 F 1 (a, b; c; z) to be in S * (α) and C(α) and also examined a linear operator acting on hypergeometric functions. For the other interesting developments for univalent and multivalent hypergeometric function the reader can refer the works of Carlson and Shaffer [1], Merkes and Scott [5], Ruscheweyh and Singh [8], Cho et al [2], Mostafa [6] and El-Ashwah et al [3]…”

Starlike and convex properties for Poisson distribution series

“…The classes S * (α, β) and C * (α, β), were introduced and studied by Gupta and Jain [2] (see [3]). Also, we note that for β = 1 the classes S * (α, β) and C * (α, β) reduce to the class of starlike and convex functions of order α(0 ≤ α < 1) (see [6]).…”