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

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 $\lambda \ne \mu$.

8 pages