Christian Berg: Nevanlinna-Pick functions and some of their applications

The Nevanlinna-Pick functions, i.e., the holomorphic functions mapping the upper half-plane into itself, have many applications.

Some of these will be discussed based on joint work with Henrik L.
Pedersen. It will be shown that (log Gamma(z+1))/(z log(az)) is Nevanlinna-Pick if a\ge 1.

This has lead to improvements of results by Alzer, Anderson, Qiu, Vamanamurthy and Vourinen. Other Nevanlinna-Pick functions related to certain conformal mappings have lead to a proof of the Chen-Rubin conjecture about the behaviour of the median in the gamma distribution as function of the mean value.