The function $(\cosh\sqrt{at^2+b})$ is exponentially convex

TL;DR

This paper proves that the function osh is exponentially convex for positive parameters, providing three different proofs to establish this property.

Contribution

It introduces three distinct proofs demonstrating the exponential convexity of osh for positive a and b.

Findings

The function osh is exponentially convex.

Exponential convexity holds on the entire real axis.

Multiple proofs confirm the property.

Abstract

Given positive numbers $a$ and $b$, the function $\sqrt{at^2+b}$ is exponentially convex function of $t$ on the whole real axis. Three proofs of this result are presented.