# Inequalities involving Gamma function

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!

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.