Let $x, y \in \mathbb{R}^N$ be distinct and $p > 1$. According to the paper
$\bullet$ Lindqvist, Peter; On the equation $\mbox{div}(|\nabla u|^{pā2}\nabla u) + \lambda|u|^{pā2}u=0$. Proc. Amer. Math. Soc. 109 (1990), no. 1, 157ā164.
the inequality below $$|x|^p - |y|^p > p|y|^{p-2} \cdot \langle y, x - y \rangle,$$ is classical and is proved by a map convexity argument $w \mapsto |w|^p$.
Question: In the literature, does this inequality have a specific name? Is your demonstration easier than it looks? I look forward to any reference or proposed solution. Thanks!
