The function $(b^x-a^x)/x$: Ratio's properties

TL;DR

This paper investigates the properties of a class of exponential functions, establishing conditions for their monotonicity and convexity, with applications in mathematical analysis.

Contribution

It provides new necessary and sufficient conditions for the monotonicity and convexity of a generalized exponential function involving multiple parameters.

Findings

Derived conditions for monotonicity of the functions.

Established criteria for logarithmic convexity and concavity.

Analyzed the functions' properties on the real line.

Abstract

In the paper, after reviewing the history, background, origin, and applications of the functions $\frac{b^{t}-a^{t}}{t}$ and $\frac{e^{-\alpha t}-e^{-\beta t}}{1-e^{-t}}$, we establish sufficient and necessary conditions such that the special function $\frac{e^{\alpha t}-e^{\beta t}}{e^{\lambda t}-e^{\mu t}}$ are monotonic, logarithmic convex, logarithmic concave, 3-log-convex and 3-log-concave on $\mathbb{R}$, where $\alpha,\beta,\lambda$ and $\mu$ are real numbers satisfying $(\alpha,\beta)\ne(\lambda,\mu)$, $(\alpha,\beta)\ne(\mu,\lambda)$, $\alpha\ne\beta$ and $\lambda\ne\mu$.