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I am having some difficulty in proving the following inequality: \begin{equation*} \frac{1-e^{-\gamma b}}{b^\eta}-\frac{1-e^{-\gamma s}}{s^\eta}\geq \gamma(1-\eta)\int^b_sy^{-\eta}e^{-\gamma y}dy \end{equation*} where $\gamma,\ \eta\in(0,1)$, $0\leq s\leq b$ and $b$ satisties:

\begin{equation*} \frac{(1-e^{-\gamma b})\eta}{e^{-\gamma b}}=(1-\eta)\gamma b \end{equation*}

Any suggestions on how to approach this would be appreciated. Any suggested literature on inequalities involving the gamma function would also be really appreciated. Thanks!

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Let $D(b)$ denote the difference between the left-hand side and the right-hand side of your displayed inequality. Then $D(s)=0$, and the inequality $D'(b)\le0$ can be easily shown to be equivalent to $e^{\gamma b}\ge1+\gamma b$, which latter is true. So, $D(b)\le0$ if $b\ge s\ge0$; that is, the opposite to your proposed inequality holds.

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