The following inequality is from the proof that the $L^p$ norm is Gâteaux differentiable for $ 1 < p<\infty$ (from "Analysis" by Lieb and Loss).
Let $a$, $b\in\mathbb{C}$ and $-1\leq t\leq 1$, $t\not=0.$ Then $$|a|^p-|a-b|^p\leq\frac{1}{t}(|a+tb|^p-|a|^p) \leq |a+b|^p-|a|^p.$$
I managed to prove the second inequality for positive $t$ by writing $a+tb=(1-t)a+t(a+b)$and using the convexity of $\cdot^p$. From this the first inequality follows for negative $t$ by substituting $-b$ for $b$. The same trick would finish the proof, if I could prove either the second inequality for negative $t$ or the first inequality for positive $t$.