Prove or give a counter-example for the following:
$\frac{2}{\gamma}[\sqrt{(1+\gamma (n-1))(1+\gamma (s -1))}-(1+\gamma (s -1))] \leq n-s$
where $n,s$ natural numbers with $n \geq 2$, $0 and $\gamma$ a real in (0,1).
Prove or give a counter-example for the following:
$\frac{2}{\gamma}[\sqrt{(1+\gamma (n-1))(1+\gamma (s -1))}-(1+\gamma (s -1))] \leq n-s$
where $n,s$ natural numbers with $n \geq 2$, $0 and $\gamma$ a real in (0,1).
If $f(\gamma)$ is your left side, $ \frac{df}{d\gamma} = {\frac {-2-\gamma\,s+2\,\gamma-\gamma\,n+2\,\sqrt { \left( 1+\gamma\,n -\gamma \right) \left( 1+\gamma\,s-\gamma \right) }}{{\gamma}^{2} \sqrt { \left( 1+\gamma\,n-\gamma \right) \left( 1+\gamma\,s-\gamma \right) }}}$ The limit of this as $\gamma \to 0+$ is $-(n-s)^2/4$, which is negative. If we set the numerator equal $0$, subtract the square root term from both sides, square and simplify we find $\gamma^2 (n-s)^2 = 0$. So $f'(\gamma) < 0$ on $(0,\infty)$, and in particular $f(\gamma) < \lim_{\gamma \to 0+} f(\gamma) = n-s$.