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We investigate inequalities of M.G. Krein, Yu.V. Linnik and E.A. Gorin forpositive definite functions. Modifications and generalizations of theseinequalities are proved. We also prove that multipoint E.A. Gorin's inequalityfollows from two--point M.G. Krein's inequality.

Inequalities of M.G. Krein, Yu.V. Linnik and E.A. Gorin for positive definite functions