Abstract
In this paper, the normalized hyper-Bessel functions are studied. Certain sufficient conditions are determined such that the hyper-Bessel functions are close-to-convex, starlike and convex in the open unit disc. We also study the Hardy spaces of hyper-Bessel functions.
Highlights
Let H denote the class of functions that are analytic in U = {z : |z| < 1}, and A denote the class of functions f that are analytic in U having the Taylor series form ∞ f (z) = z + ∑ an zn, z ∈ U. (1) n =2The class S of univalent functions f is the class of those functions in A that are one-to-one in U .Let S ∗ denote the class of all functions f such that f (U ) is star-shaped domain with respect to origin while C denotes the class of functions f such that f maps U onto a domain which is convex
It is clear that S ∗ (0) = S ∗ and C (0) = C are the usual classes of starlike and convex functions respectively
Motivated by the above works, we study the geometric properties of hyper-Bessel function Hγc given by the power series (6)
Summary
A function f in A belongs to the class S ∗ (α) of starlike functions of order α if and only if Re (z f 0 (z) / f (z)) > α, α ∈. We consider the hyper-Bessel function in the form of the hypergeometric functions defined as z γ1 +γ2 +...+γc c +1. Consider the hyper-Bessel function Jγc which is defined by. Studied some geometric properties of hyper-Bessel function. They studied radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions. Motivated by the above works, we study the geometric properties of hyper-Bessel function Hγc given by the power series (6). Sufficient conditions on univalency of an integral operator defined by hyper-Bessel function is studied. < 12 [14], f ∈ U CV
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