New (probabilistic) proof of Diaz-Metcalf and Pólya-Szegö inequalities and some consequences. (Ukrainian, English) Zbl 1103.26019

Pogany, Tibor K.

Teor. Jmovirn. Mat. Stat. 70, 101-109 (2004); translation in Theory Probab. Math. Stat. 70, 113-122 (2005).

The paper is devoted to inequalities of the form \[ C (\mathbf{E}\xi\eta)^2>\text\textbf{E}\xi^2\text\textbf{E}\eta^2 \] for some constant \(C\). E.g., in the generalized Pólya-Szegö inequality (demonstrated in the paper) it is \[ C={1\over 4} \left( \sqrt{{m_1,m_2\over M_1 M_2}}+ \sqrt{{M_1,M_2\over m_1 m_2}} \right)^2, \]
where \(0<m_i\leq M_i\), and \(\Pr\{m_1\leq\xi\leq M_1\}=\Pr\{m_2\leq\eta\leq M_2\}\) and in the Diaz-Metcalf inequality it is
\[ C= \left( {m_1\over M_2}+{M_1\over m_2} \right) \mathbf{E}\xi^2 +{m_1 M_1\over m_2 M_2}\text\textbf{E}\eta^2. \]
These inequalities are used in proofs of the Rennie, Schweitzer and Kantorovich inequalities.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

26D15	Inequalities for sums, series and integrals
60E15	Inequalities; stochastic orderings

Keywords:

bounded random variables; Kantorovich inequality; Rennie inequality; Schweitzer inequality