TITLE: How to show $ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$
QUESTION [6 upvotes]: When $p\geq 2$ and $0\leq x\leq1$, how does one show the inequalities
$$ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$$ and $$ 2(1+x^p)\leq (1+x)^p + (1-x)^p \ ?$$ 
The first one looks like a correction of the parallelogram law for powers greater than $p=2$, I've tried using that and rearranging but it hasn't worked. I've also tried differentiating but then I don't know how to handle the new expressions.
This is from a qualification exam on analysis. Basically the problem becomes trivial if one shows the inequalities.
Thank you!

REPLY [2 votes]: For the second one, the left hand side decreases as the value of P increases and the right hand side increases as the value of P increases. So if you show the smallest case is true(P=2 and x should be chosen to maximize the left hadn side)  then I think that would be prove the second inequality. Of course you would also need to prove the LHS decreases and the RHS increases as P increases.