The Best Constants in the Strong Convexity of |ξ| ^p and in the Strong Monotonicity of |ξ| ^p-2 ξ

We prove that the function

is strongly convex for every

p p

,

p\in]1,+\infty[ p ∈ ] 1 , + ∞ [

, and we give a characterization, through the resolution of an algebraic equation, of the best constant which appears in the strong convexity inequality. The same proof allows us to prove that the function

\xi\in \mathbb{R}^N\to \vert \xi\vert^{p-2}\xi ξ ∈ R N → ∣ ξ ∣ p − 2 ξ

is strongly monotone and to identify the explicit value of the best constant.