Prove
$$ |a + b|^p \leq (|a| + |b|)^p \leq (1 - \lambda)^{1 - p}|a|^p + \lambda^{1 - p} |b|^p $$
for any $0 < \lambda < 1$ and $p > 1$.
Of course the first inequality is obvious. I have tried applying the definition of convexity for $|t|^p$ but have had no luck. Also I get a feeling that Young's inequality appears somewhere.
Hint: Apply Jensen's inequality.
One ca express $$|a|+|b|=\lambda\frac{|a|}{\lambda} + (1-\lambda)\frac{|b|}{1-\lambda}$$
So, as $t\mapsto |t|^p$ is convex,
$$ \Big(|a|+|b|\Big)^p=\Big(\lambda\frac{|a|}{\lambda} + (1-\lambda)\frac{|b|}{1-\lambda}\Big)^p\leq \lambda \frac{|a|^p}{\lambda^p}+(1-\lambda)\frac{|b|^p}{(1-\lambda)^p} $$
