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

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

2009-04-07

arxiv.org/abs/0904.1115v1

Mathematics

Classical Analysis and ODEs

8 pages

Scientific paper

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 $\lambda\ne\mu$.

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