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New (probabilistic) derivation of Diaz-Metcalf and P\'{;o}; lya-Szeg\H{;o}; inequalities and consequences

Classical inequalities of Diaz-Metcalf and P\'olya-Szeg\H{;o}; are generalized to probabilistic setting which covers the initial deterministic (both discrete and integral) variants. From these two inequalities, by the probabilistic derivation method forther well known inequalities are obtained (that ones by Kantorovich, Rennie and Schweitzer).

Almost surely bounded random variable; Diaz-Metcalf inequality; discrete inequality; integral inequality; Kanotorovich inequality; mathematical expectation; P\'olya-Szeg\H o inequality; Rennie ineuquality; Schweitzer inequality

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