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.