Is it possible to find $\beta>1,0<\epsilon<1$ such that the following inequality is true for large enough $n$:
$4\pi^2 \beta\big( 2(n-1)^{-\frac{1}{2}(\epsilon+1)}n^{\frac{3}{2}}+\frac{1}{2}\frac{n}{n-1}\sqrt{(\gamma-1)(n-1)^{1-\epsilon}\ln n}+\frac{n^2}{2(n-1)}\big)<\exp(\frac{C(n-1)^\epsilon}{n})$
I encountered this problem when I am doing a research problem on deriving certain nice probabilistic bound by choosing approapriate values of parameters. However, here is where I am getting stuck at. Could someone help me out?
Many thanks.
Since $\epsilon < 1, \frac{(n-1)^\epsilon}n \to 0$ as $n \to \infty$. Therefore the RHS of your inequality will converge to $1$.
The LHS has three terms all of which are positive, and for which the last obviously diverges to $\infty$ as $n \to \infty$. Therefore the entire left-hand side diverges to $\infty$ as $n \to \infty$.
So no, you cannot find an $\epsilon < 1$ or $\beta > 0$ for which your inequality holds for large $n$.