Some transcendental equations on the Stieltjes cone

TL;DR

This paper studies a class of transcendental equations involving functions in the Stieltjes cone, establishing conditions for the existence and uniqueness of solutions, and providing insights into the zero distribution of special functions in complex sectors.

Contribution

It introduces a general approach to analyze transcendental equations in the Stieltjes cone, proving solution uniqueness and zero-free regions for certain special functions.

Findings

Each such equation has at most one solution in the complex plane cut along the negative real axis.

The unique solution is real and positive with an analytical bound.

The approach helps identify zero-free regions for special functions in complex sectors.

Abstract

A general class of transcendental equations in complex domain is considered for functions belonging to the Stieltjes cone. Under certain conditions each transcendental equation has no solution or one, at most, in the complex plane cut along the negative real axis. The unique solution is real valued and positive with an analytical bound. Particular cases consist in transcendental equations containing exponential, hyperbolic, power law, logarithmic and special functions. The present approach provides a simple way to prove that some special functions have no zero in certain sectors of the complex plane cut along the negative real axis.