Starlikeness and convexity of generalized Bessel functions

Abstract

In this paper, we give sufficient conditions for the parameters of the normalized form of the generalized Bessel functions to be convex and starlike in the open unit disk. As an application of our main results, we solve a recent open problem concerning a subordination property of Bessel functions with different parameters. Moreover, we present a new inequality for the Euler gamma function, which we apply in order to have tight bounds for the generalized and normalized Bessel function of the first kind.

Keywords:

• starlike functions
• convex functions
• generalized Bessel functions
• gamma function

2000 Mathematics Subject Classification :

• 30C45
• 33C10

Acknowledgements

The research of Á. Baricz was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The authors are indebted to S. András for providing expert help with the Matlab 6.5 software.