An inequality like Bernoulli inequality

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 $$(1+s)^p\ge 1+s^p+p(s+s^{p-1})-Cs^{\frac p2}\ \ \ \mbox{for}\ s\ge0.$$

Now, I guess if the following inequality holds:

if $\alpha>1, \beta>1$, then there exists $C>0$ independent of $s,t$ such that

$$(1+s)^\alpha(1+t)^\beta\ge 1+s^\alpha t^\beta+\alpha s+\beta t+\alpha s^{\alpha-1}t^\beta+\beta s^\alpha t^{\beta-1} -C s^{\frac\alpha2}t^{\frac\beta2}$$ 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?

• What does it mean that "(0.2) becomes (0.1) with $p=α+β$"? – Taroccoesbrocco Apr 21 '18 at 16:55
• This problem may be difficult, since the argument for single variable is not suitable for two variables and there is a difference between the exponent of $p$ and $\alpha$ and $\beta$. – Andrew Chang Apr 26 '18 at 0:33