A Diaz--Metcalf type inequality for positive linear maps and its applications
Abstract
We present a Diaz--Metcalf type operator inequality as a reverse Cauchy-Schwarz inequality and then apply it to get the operator versions of P\'olya-Szeg\"o's, Greub-Rheinboldt's, Kantorovich's, Shisha-Mond's, Schweitzer's, Cassels' and Klamkin-McLenaghan's inequalities via a unified approach. We also give some operator Gr\"uss type inequalities and an operator Ozeki-Izumino-Mori-Seo type inequality. Several applications are concluded as well.
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