Univalence criteria for linear fractional differential operators associated with a generalized Bessel function

Abstract

In this paper our aim is to establish some generalizations upon the sufficient conditions for linear fractional differential operators involving the normalized forms of the generalized Bessel functions of the first kind to be univalent in the open unit disk as investigated recently by [{\sc E. Deniz, H. Orhan, H.M. Srivastava}, {\it Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions}, Taiwanese J. Math. {\bf 15} (2011), No. 2, 883-917] and [{\sc \'A. Baricz, B. Frasin}, {\it Univalence of integral operators involving Bessel functions}, Appl. Math. Letters {\bf 23} (2010), No. 4, 371--376]. Our method uses certain Luke's bounding inequalities for hypergeometric functions ${}_{p+1}F_p$ and ${}_pF_p$.