On the characterization properties of certain hypergeometric functions in the open unit disk

Abstract

Abstract Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z 1 F 2 ( a ; b , c ; z ) $z{}_{1}F_{2}(a;b,c;z)$ in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the F 1 0 ( − ; c ; z ) ${}_{0}F_{1}(-;c;z)$ function are also mentioned. We further consider a boundedness property of the function F 2 1 ( a ; b , c ; z ) $_{1}F_{2}(a;b,c;z)$ in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out.

Keywords

• Lommel function
• Sturve function
• Bessel function
• Analytic univalent function
• Starlike function
• Close-to-convex function