Inégalité de Levinson

Pour les articles homonymes, voir Levinson.

En mathématiques, l'inégalité de Levinson est l'inégalité suivante, due à Norman Levinson, faisant intervenir des nombres strictement positifs. Soit $a>0$ et $f$ une fonction admettant une dérivée troisième sur l'intervalle $\left]0,2a\right[$ telle que $f'''(x)\geqslant 0$ pour tout $x\in \left]0,2a\right[$ .

Supposons que $0<x_{i}\leqslant a$ pour $i=1,\dots ,n$ et $0<p$ . Alors :

{\frac {\displaystyle \sum _{i=1}^{n}p_{i}f(x_{i})}{\displaystyle \sum _{i=1}^{n}p_{i}}}-f\left({\frac {\displaystyle \sum _{i=1}^{n}p_{i}x_{i}}{\displaystyle \sum _{i=1}^{n}p_{i}}}\right)\leqslant {\frac {\displaystyle \sum _{i=1}^{n}p_{i}f(2a-x_{i})}{\displaystyle \sum _{i=1}^{n}p_{i}}}-f\left({\frac {\displaystyle \sum _{i=1}^{n}p_{i}(2a-x_{i})}{\displaystyle \sum _{i=1}^{n}p_{i}}}\right).

L'inégalité de Ky Fan (en) est le cas particulier de l'inégalité de Levinson où

p_{i}=1,\ a={\frac {1}{2}},

f(x)=\log(x).

Références

(en) Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé « Levinson's inequality » (voir la liste des auteurs).
(en) Scott Lawrence et Daniel Segalman, « A generalization of two inequalities involving means », Proc. Amer. Math. Soc., vol. 35, n^o 1,‎ 1972, p. 96-100 (DOI 10.2307/2038448).
(en) Norman Levinson, « Generalization of an inequality of Ky Fan », J. Math. Anal. Appl., vol. 8,‎ 1964, p. 133-134 (DOI 10.1016/0022-247X(64)90089-7).

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