Geometric properties of generalized Bessel functions. (English) Zbl 1156.33302

Summary: Our aim is to establish some geometric properties (like univalence, starlikeness, convexity and close-to-convexity) for the generalized Bessel functions of the first kind. In order to prove our main results, we use the technique of differential subordinations developed by S. S. Miller and P. T. Mocanu [J. Differ. Equations 67, 199–211 (1987; Zbl 0633.34005)] and some classical results of S. Ozaki [Sci. Rep. Tokyo Bunrika Daigaku, Sect. A 2, 167–188 (1935; Zbl 0012.02402; JFM 61.0353.02)] and L. Fejér [Acta Litt. Sci. Szeged 8, 89–115 (1937; Zbl 0016.10803; JFM 63.0249.04)].


33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)