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We prove analogs of certain operator inequalities, including Hölder-McCarthy inequality, Kantorovich inequality, and Heinz-Kato inequality, for positive operators on the Hilbert space in terms of the Berezin symbols and the Berezin number of operators on the reproducing kernel Hilbert space.

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We introduce the notion -cluster points, investigate the relation between -cluster points and limit points of sequences in the topology induced by random 2-normed spaces, and prove some important results.