The radius of starlikeness of regular Coulomb wave functions

Abstract

Motivated by the pioneering work of M.S. Robertson [Ro54] and R.K. Brown [Br60], [Br62], who examined the geometric properties of some normalised solutions of second-order homogeneous differential equations, in this paper we investigate the radii of univalence and starlikeness for two kind of normalised regular Coulomb wave functions. Moreover, a generalized normalised Bessel function is introduced, and its radius of starlikeness is studied by using two different approaches. In addition, the asymptotic behaviour, with respect to the large order, of the radius of starlikeness of one type of normalised Coulomb wave functions is considered, which is in fact the first zero of the derivative of the regular Coulomb wave function. We derive a complete asymptotic expansion for this radius of starlikeness and provide a recurrence relation for the coefficients of this expansion. The proof is based on Rayleigh sums of the zeros of Coulomb wave functions, asymptotic inversion and some basic results on regular Coulomb wave functions developed by \v{S}tampach and \v{S}\'{t}ov\'{\i}\v{c}ek [SS14].