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The 150th anniversary of the birth of the outstanding Russian mathematicianVladimir Andreevich Steklov falls on January 9, 2014. In this paper (it is apart of survey to be published in January 2014), we describe advances in one ofseveral areas influenced by Steklov's work.

V.A. Steklov and the Problem of Sharp (Exact) Constants in Inequalities of Mathematical Physics