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

In the paper, after reviewing the history, background, origin, and applications of the functions $\frac{b^{t}-a^{t}}{t}$ and $\frac{e^{-αt}-e^{-βt}}{1-e^{-t}}$, we establish sufficient and necessary conditions such that the special function $\frac{e^{αt}-e^{βt}}{e^{λt}-e^{μt}}$ are monotonic, logarithmic convex, logarithmic concave, 3-log-convex and 3-log-concave on $\mathbb{R}$, where $α,β,λ$ and $μ$ are real numbers satisfying $(α,β)\ne(λ,μ)$, $(α,β)\ne(μ,λ)$, $α\neβ$ and $λ\neμ$.

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