Sufficiency for nephroid starlikeness using hypergeometric functions

Starlikeness of the Gaussian Hypergeometric Functions

In this paper, we derive conditions on the parameters a, b, c so that the function zF (a, b; c; z) is starlike in D, where F (a, b; c; z) denotes the classical hypergeometric function. We give some consequences of our results including some mapping properties of convolution operator.

Certain Sufficiency Conditions on Gaussian Hypergeometric Functions

The author aims at finding certain conditions on a, b and c such that the normalized Gaussian hypergeometric function zF (a, b; c; z) given by F (a, b; c; z) = ∞ ∑ n=0 (a, n)(b, n) (c, n)(1, n) z, |z| < 1, is in certain subclasses of analytic functions. A particular operator acting on F (a, b; c; z) is also discussed.

Sufficiency for Gaussian Hypergeometric Functions to Be Uniformly Convex

Let F(a,b;c;z) be the classical hypergeometric function and f be a normalized analytic functions defined on the unit disk . Let an operator Ia,b;c(f ) be defined by [Ia,b;c(f )](z)= zF(a,b;c;z)∗f(z). In this paper the authors identify two subfamilies of analytic functions 1 and 2 and obtain conditions on the parameters a,b,c such that f ∈ 1 implies Ia,b;c(f )∈ 2.

Some Starlikeness Criterions for Analytic Functions

Copyright q 2010 Gejun Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We determine the condition on α, μ, β, and λ for which | 1−α z/f z μ α zf ′ z /f z z/f z − 1| < λ implies f z ∈ S∗ β , where S∗ β is the class of st...

Hypergeometric Functions