Exponential bounds for the logarithmic derivative of Whittaker functions

Abstract:

Some well-known results of Grönwall on logarithmic derivative of modified Bessel functions of the first kind concerning exponential bounds are extended to Whittaker functions of the first and second kind $M_{\kappa ,\mu }$ and $W_{\kappa ,\mu }$. Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function $W_{\kappa ,\mu },$ and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of $M_{\kappa ,\mu }.$ The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind.