In course of a particular research I ran into these two inequalities which I would like to have some help with.
$r,R>0$ for both the questions.
- Is there a function $M(r)$ for which this inequality is satisfied?
$\cos^{-1}\left (\frac{4M(r)}{r}-1 \right ) + \sqrt{1- \left ( \frac{4M(r)}{r}-1 \right )^2} > \pi$
- Let $t_0$ be the minimum positive solution of $a \cosh (\omega t) + b \sinh (\omega t) = 0$.
Then for $-B<0$ and $t>0$, is it possible to choose the values of variables $a,b$ and $g>0$ such that the expression multiplying $-B$ is negative for $t \leq t_0$ and $r>gR$?
$-B\frac {(a\sqrt{r^2-g^2R^2} \mp bgr\omega)\cosh (\omega t)+ (b\sqrt{r^2-g^2R^2} \mp agr\omega)\sinh(\omega t)}{r\sqrt{r^2-g^2R^2}(a \cosh(\omega t) + b \sinh (\omega t))^{2g+1}}$