I am reading a paper in which the author stated an inequality as follows:
if $p>2$, then there is a positive constant $C$ independent of $s$ such that \begin{equation}(1+s)^p\ge 1+s^p+p(s+s^{p-1})-Cs^{\frac p2}\ \ \ \mbox{for}\ s\ge0.\end{equation}
Now, I guess if the following inequality holds:
if $\alpha>1, \beta>1$, then there exists $C>0$ independent of $s,t$ such that
\begin{equation}(1+s)^\alpha(1+t)^\beta\ge 1+s^\alpha t^\beta+\alpha s+\beta t+\alpha s^{\alpha-1}t^\beta+\beta s^\alpha t^{\beta-1} -C s^{\frac\alpha2}t^{\frac\beta2}\end{equation} for $s,t\ge0$.
Let $s=t$. Then the second inequality above becomes the first inequality with $p=\alpha+\beta$.
How to prove the second inequality?