The Best Constants in the Strong Convexity of |ξ| p and in the Strong Monotonicity of |ξ| p-2 ξ
Abstract
<jats:p> We prove that the function <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\xi\in \mathbb{R}^N\rightarrow \vert \xi\vert^p</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mi mathvariant="normal">∣</mml:mi> <mml:mi>ξ</mml:mi> <mml:msup> <mml:mi mathvariant="normal">∣</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is strongly convex for every <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">p</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">p\in]1,+\infty[</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">[</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , 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 <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\xi\in \mathbb{R}^N\to \vert \xi\vert^{p-2}\xi</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mi mathvariant="normal">∣</mml:mi> <mml:mi>ξ</mml:mi> <mml:msup> <mml:mi mathvariant="normal">∣</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>ξ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is strongly monotone and to identify the explicit value of the best constant. </jats:p>