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 κ , μ M_{\kappa ,\mu } and W κ , μ W_{\kappa ,\mu } . Moreover, a complete monotonicity result is proved for the logarithmic derivative of the Whittaker function W κ , μ , W_{\kappa ,\mu }, and some monotonicity results with respect to the parameters and argument are shown for the logarithmic derivative of M κ , μ . M_{\kappa ,\mu }. The results extend and complement the known results in the literature about modified Bessel functions of the first and second kind.