Exercises

Apr 20'25

Given a convex function [math]f:\mathbb R\to\mathbb R[/math], prove that we have the following Jensen inequality, for any [math]x_1,\ldots,x_N\in\mathbb R[/math], and any [math]\lambda_1,\ldots,\lambda_N \gt 0[/math] summing up to [math]1[/math],

[[math]] f(\lambda_1x_1+\ldots+\lambda_Nx_N)\leq\lambda_1f(x_1)+\ldots+\lambda_Nx_N [[/math]]

with equality when [math]x_1=\ldots=x_N[/math]. In particular, by taking the weights [math]\lambda_i[/math] to be all equal, we obtain the following Jensen inequality, valid for any [math]x_1,\ldots,x_N\in\mathbb R[/math],

[[math]] f\left(\frac{x_1+\ldots+x_N}{N}\right)\leq\frac{f(x_1)+\ldots+f(x_N)}{N} [[/math]]

and once again with equality when [math]x_1=\ldots=x_N[/math]. Prove also that a similar statement holds for the concave functions, with all the inequalities being reversed.

BBot

Apr 20'25

Prove that for [math]p\in(1,\infty)[/math] we have the following Hölder inequality

[[math]] \left|\frac{x_1+\ldots+x_N}{N}\right|^p\leq\frac{|x_1|^p+\ldots+|x_N|^p}{N} [[/math]]

and that for [math]p\in(0,1)[/math] we have the following reverse Hölder inequality

[[math]] \left|\frac{x_1+\ldots+x_N}{N}\right|^p\geq\frac{|x_1|^p+\ldots+|x_N|^p}{N} [[/math]]

with in both cases equality precisely when [math]|x_1|=\ldots=|x_N|[/math].

BBot

Apr 20'25

Develop the theory of the gamma function, defined as

[[math]] \Gamma(z)=\int_0^\infty x^{z-1}e^{-x}dx [[/math]]

notably by establishing the following formula, for any [math]N\in\mathbb N[/math],

[[math]] \Gamma(N)=(N-1)! [[/math]]

and then comment on the formulae for the volumes and areas of spheres.