Sharp estimates for the modulus and the argument of the totally monotonie functions

Abstract

We find sharp explicit estimates for the modulus and the argument of the totally monotonie functions. Let T denote the class of totally monotonie functions [formula] where [formula] and μ(t) is a probability measure on [0, 1]. The class T was introduced by Hausdorff [1] and studied by Wirths [2] and others. Sharp estimates for ƒ (z) | in implicit forms were proved by Wirths [2, p. 512, Corollary 2.1]. Now we will obtain sharp explicit estimates for I ƒ (z) | in another way. In addition, we find sharp estimates for arg (ƒ(z)/z), where "arg" everywhere is in the internal (-π, π]. (1) where (2)