Abstract
In this article, we are mainly interested in finding the sufficient conditions under which Lommel functions and hyper-Bessel functions are close-to-convex with respect to the certain starlike functions. Strongly starlikeness and convexity of Lommel functions and hyper-Bessel functions are also discussed. Some applications are also the part of our investigation.
Highlights
Introduction and PreliminariesLet A denote the class of functions f of the form ∞ f (z) = z + ∑ an zn, (1)n =2 analytic in the open unit disc U = {z : |z| < 1} and S denote the class of all functions in A which ∼are univalent in U
N =2 analytic in the open unit disc U = {z : |z| < 1} and S denote the class of all functions in A which are univalent in U
Certain conditions for close-to-convexity of some special functions such as Bessel functions, q-Mittag-Leffler functions, Wright functions, and Dini functions have been determined by many mathematicians with different methods
Summary
To discuss the close-to-convexity of normalized Lommel functions with respect to the certain starlike functions, here we define modified form of the normalized Lommel functions where M =. We consider the hyper-Bessel function in terms of the hypergeometric functions defined below (for details see [3]). Since the function Jβ c is not in class A, the normalized hyper-Bessel function J β c is defined by. To discuss the close-to-convexity of normalized hyper-Bessel functions with respect to the certain starlike functions, here we define modified form of the normalized hyper-Bessel functions. Certain conditions for close-to-convexity of some special functions such as Bessel functions, q-Mittag-Leffler functions, Wright functions, and Dini functions have been determined by many mathematicians with different methods (for details, see [4,10,11,12,13]).
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