An integral inequality involving exponentials and $H^1(\mathbb{R}^2)$

Question

Let $\beta, \alpha>1$, and $u \in H^1(\mathbb{R}^2$). I'm trying to show that $$\int_{\mathbb{R}^2}\left(e^{\beta u^2(x)} -1\right)^{2\alpha}dx \leq \int_{\mathbb{R}^2}(e^{2\alpha \beta u^2(x)}-1)dx$$ I tried to use a series expansion of the exponential and the fact that $u \in L^q(\mathbb{R}^2)$ for every $q\geq 2 $, but did not make much progress. Any help is much appreciated.

Answer 1

This has nothing to do with $L^p$ or $H^1$, this follows simply from the estimate $(x-y)^\alpha \leq x^\alpha - y^\alpha$ for $0\leq y \leq x$ and $\alpha>1$ (if we were lazy, we would only consider $y=1$ which is all we need here).

To see the above inequality, note that this is equivalent to showing that $$f(s)=1-s^\alpha -(1-s)^\alpha\geq 0$$ for $s\in [0,1]$ (divide the original inequality by $x^\alpha$ for $x\neq 0$). The new inequality follows immediately from $f(0)=f(1)$ and $$f''(s) = \alpha (\alpha-1)(-s^{\alpha-2}-(1-s)^{\alpha-2})<0$$ for $s\in (0,1)$.