Starlikeness of Bessel functions and their derivatives

Abstract

Abstract. In this paper necessary and sufficient conditions are obtained for the starlikeness of Bessel functions of the first kind and their derivatives of the second and third order by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and Mittag-Leffler expansions for the derivatives of Bessel functions of the first kind, as well as some results on the zeros of these functions.

Certain Geometric Properties of Generalized Dini Functions

“…Geometric properties of special functions such as hypergeometric functions, Bessel functions, Struve functions, Mittag-Leffler functions, Wright functions, and some other related functions are an ongoing part of research in geometric function theory. We refer to some geometric properties of these functions [1][2][3][4][5][6][7][8][9][10] and references therein.…”

Radii of starlikeness and convexity of regular Coulomb wave functions

“…Motivated by the beautiful properties of special functions, in the recent years there was a vivid interest on geometric properties of special functions, like Bessel, Struve and Lommel functions of the first kind; see the papers [1,2,3,4,5,6,7,8,9,10,11,18,19] and the references therein. The determination of the radii of starlikeness and convexity of some normalized forms of these special functions was studied in details by the first author and his collaborators.…”

The radius of uniform convexity of Bessel functions

“…Baricz and Szász [3] proved that f ν (z) and g ν (z) are convex in U ⇔ ν ≥ 1, and h ν (z) is convex in U ⇔ ν ≥ ν * * ≃ −0.1438..., where ν * * is the unique root of the equation (2ν − 4)J ν+1 (1) + 3J ν (1) = 0. Furthermore, Baricz and Szász [6], Baricz et al [7] and Baricz et al [8] obtained necessary and sufficient conditions for the starlikeness and close-to-convexity of the function h ν (z) and its derivatives, some special combinations of Bessel functions and their derivatives, and the functions f ν (z), g ν (z) and derivatives of h ν (z) in U, respectively, by using a result of Shah and Trimble (see [38,Theorem 2]) about transcendental entire functions with univalent derivatives. In this section, we deal with the uniform convexity of the normalized Bessel functions f ν (z), g ν (z) and h ν (z) in U.…”