Abstract
In this paper we present and investigate an invertible family of lower and upper bounds for the logarithmic derivative M′(a,b,z)/M(a,b,z) of Kummer's function M(a,b,z), when 0<a<b. The derived bounds are theoretically well-defined, asymptotically precise, numerically accurate, and easy to compute. Moreover, we extend the list of known bounds for the logarithmic derivative of Kummer's function and improve the state of the art lower bound on a significant part of the domain. Furthermore, we show a connection to the family of Amos bounds for modified Bessel function ratios. Finally, via closed-form integration we derive bounds for the logarithm of Kummer's function log(M(a,b,z)), which can be used in a wide range of applications where computing Kummer's function in standard precision could easily underflow or overflow.
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