Upper bounds for the moduli of zeros of Hermite–Padé approximations for a set of exponential functions

Abstract

In this paper, we establish upper bounds for the moduli of zeros of Hermite–Padé approximations of type I for a system of exponential functions \(\left\{ {{e^{{\lambda _{{p^z}}}}}} \right\}_{p = 0}^k\), where \(\left\{ {{\lambda _p}} \right\}_{p = 0}^k\) are various arbitrary complex numbers. The proved statements supplement and generalize well-known results due to Saff and Varga, as well as those due to Stahl and Wielonsky, on the behavior of zeros of Hermite–Padé approximations for a set of exponential functions \(\left\{ {{e^{pz}}} \right\}_{p = 0}^k\).