On exponential type entire functions without zeros in the open lower half-plane

TL;DR

This paper establishes conditions under which exponential type entire functions lack zeros in the open lower half-plane, linking these conditions to inequalities involving the functions' real and imaginary parts and their derivatives, and connecting to positive definite functions.

Contribution

It provides new sufficient conditions and inequalities characterizing zero-free regions of exponential type entire functions in the lower half-plane.

Findings

Derived exact inequalities involving real and imaginary parts of functions and derivatives

Established a connection to positive definite functions

Provided sufficient conditions for zero-free regions in the lower half-plane

Abstract

We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the real axis is deduced. A connection is established to the positive definite functions.