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!



Inequalities involving Gamma function
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. 


