  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 $\lambda\ne\mu$.
