Certain Geometric Properties of Generalized Dini Functions

Abstract

We are mainly interested in some geometric properties for the combinations of generalized Bessel functions of the first kind and their derivatives known as Dini functions. In particular, we study the starlikeness of order α, convexity of order α, and close-to-convexity of order ((1+α)/2) for normalized Dini function. We also study close-to-convexity with respect to certain star-like functions. Further, we obtain conditions on generalized Dini function to belong to the Hardy space Hp.

New properties of the generalized Dini function

“…Remark 2.9. Similar results to Theorems 2.1 and 2.8 can be found for the generalized Dini functions d ν,a,b,c in Theorem 6 (iv) and Theorem 18 in [7].…”

Certain Geometric Properties of Lommel and Hyper-Bessel Functions

“…For details, we refer to [4][5][6][7][8][9]. Certain conditions for close-to-convexity of some special functions such as Bessel functions, q-Mittag-Leffler functions, Wright functions, and Dini functions have been determined by many mathematicians with different methods (for details, see [4,[10][11][12][13]). We need the following Lemmas to prove our results.…”