Applications of Schweitzer's Inequality

Preliminaries

P. Schweitzer proved the following inequality in a 1914 paper. (For a proof, see a separate page.)

For $0\lt m\lt M,\,$ and $x_k\in [m,M],\,$ for $k\in\overline{1,n},$

$\displaystyle \left(\frac{1}{n}\sum_{k=1}^nx_k\right)\left(\frac{1}{n}\sum_{k=1}^n\frac{1}{x_k}\right)\le\frac{(m+M)^2}{4mM}.$

Over the time, several other inequalities have been shown to follow from the above. Tastes may differ, but I found the derivations in Mitrinovic's book both charming and instructive. Three such derivations are reproduced below. I am indebted to N. N. Taleb for bringing Mitrinovic's book to my attention.

Reference

D. S. Mitrinovic, Analytic Inequalities, Springer 1970

Pólya-Szegö Inequality (1925)

Proof

Set $a=0,\,$ $b=\displaystyle \sum_{k=1}^na_kb_k;\,$ set $c_0=0,\,$ $c_t=\displaystyle \sum_{k=1}^ta_kb_k\,$ $t\in\overline{1,n}\,$ and define $f(x)\,$ piecewise: $\displaystyle f(x)=\frac{a_k}{b_k},\,$ $x\in\ (c_{k-1},c_k),\,$ $k\in\overline{1,n}.$

We have $\displaystyle m=\frac{m_1}{M_2}\lt\frac{a_k}{b_k}\lt\frac{M_1}{m_2}=M,\,$ which reduces the required inequality to that of Schweitzer's.

Kantorovich's Inequality (1948)

For $0\lt m\lt\gamma_k\lt M,\,$ and real $u_k,\,$ $k\in\overline{1,n},$

$\displaystyle \left(\sum_{k=1}^{n}\gamma_ku_k^2\right)\left(\sum_{k=1}^{n}\frac{1}{\gamma_k}u_k^2\right)\le\frac{1}{4}\left(\sqrt{\frac{M}{m}}+\sqrt{\frac{m}{M}}\right)^2\left(\sum_{k=1}^{n}u^2_{k}\right)^2.$

Proof

Set $a=0,\,$ $b=\displaystyle \sum_{k=1}^nu_k^2;\,$ set $c_0=0,\,$ $c_t=\displaystyle \sum_{k=1}^tu_k^2\,$ $t\in\overline{1,n}\,$ and define $f(x)\,$ piecewise: $\displaystyle f(x)=\gamma_k,\,$ $x\in\ (c_{k-1},c_k),\,$ $k\in\overline{1,n},\,$ where $0\lt m\lt\gamma_k\lt M,\,$ $k\in\overline{1,n}.$

Now apply Schweitzer's inequality.

Greub-Rheinboldt Inequality (1959)

For $0\lt m_1\lt a_k\lt M_1,\,$ $0\lt m_2\lt b_k\lt M_2,\,$ $k\in\overline{1,n},$

$\displaystyle \left(\sum_{k=1}^{n}a_k^2u_k^2\right)\left(\sum_{k=1}^{n}b_k^2u_k^2\right)\le\frac{(M_1M_2+m_1m_2)^2}{4m_1m_2M_1M_2}\left(\sum_{k=1}^{n}a_kb_ku^2_{k}\right)^2.$

Proof

The Greub-Rheinboldt Inequality is a consequence of that of Kantorovich. This becomes immediately obvious with the substitutions: $\displaystyle \gamma_k=\frac{a_k}{b_k}\,$ and $u_k=\sqrt{a_kb_k}u_k,\,$ where

$0\lt m_1\lt a_k\lt M_1,\,$ $0\lt m_2\lt b_k\lt M_2,\,$ $k\in\overline{1,k}.\,$

We then have

$\displaystyle m=\frac{m_1}{M_2}\le\gamma_k\le\frac{M_1}{m_2}=M$

and Kantorovich's equality yields the Greub-Rheinboldt Inequality.