Asymptotic analysis of the geometric properties of some special functions

Abstract

KEYWORDS: Bessel functions; q-Bessel functions; convex functions; radius of convexity; Rayleigh sums; asymptotic expansion; Euler-Rayleigh inequalities; Weierstrassian factorization; asymptotic inversion; ordinary potential polynomials; J-fraction; continued fraction; Coulomb wave functions; starlikeness; Coulomb differential equation; Mittag-Leffler expansion; radius of univalence; radius of starlikeness; starlike functions; spirallike functions; Coulomb zeta function; asymptotic expansion; zeros of Coulomb wave functions; Rayleigh sums. In this thesis, we introduce a new class of functions denoted as J , containing functions such as f for which the ratio zf′(z)/f(z) can be expressed in J-fraction form. Our focus extends to develop methods for determining the radius of a disk mapped into a starlike domain by functions belonging to the class J . Notably, we establish that normalized regular Coulomb wave functions and Bessel functions are members of the class J . We derive a lower bound for the radius of starlikeness of these functions under specific parameter conditions. Subsequently, we investigated the radius of starlikeness and univalence for two normalized Coulomb wave functions with negative Sommerfeld parameters. Our investigation also encloses the study of spiralikeness for Coulomb wave functions with complex parameters. To achieve these results, we employ the methodologies developed by Robertson [75] and Brown [22,23], wherein they explore the geometric properties of solutions to secondorder differential equations. Using the relationship between Coulomb wave functions and Bessel functions, we introduce a new generalized normalized Bessel functions. We derive their radius of starlikeness and univalence using two distinct methods. Our work extends to obtain the asymptotic expansion for the radius of starlikeness of Coulomb wave functions, specifically the smallest (in modulus) zero of the derivative of the function. We deduce bounds for the radius of starlikeness and present the Laurent series expansion of Rayleigh sums associated with zeros of regular Coulomb wave functions, employing a method applicable to Bessel functions. Further, we derive an asymptotic expansion for the radius of convexity of normalized Bessel functions, unveiling a Laurent series expansion for Rayleigh sums associated with Dini functions. We used slightly modified technique as compared to the method employed in [11] to obtain the asymptotic expansions. Finally, we employ a similar methodology to obtain asymptotic expansion for the radius of starlikeness for q-Bessel functions.