On Clarkson's inequality in the real case

Abstract

The best constant in a generalized complex Clarkson inequality is Cp,q (ℂ) = max {21–1/p , 21/q , 21/q –1/p +1/2} which differs moderately from the best constant in the real case Cp,q (ℝ) = max {21–1/p , 21/q ,Bp,q }, where . For 1 < q < 2 < p < ∞ the constant Cp,q (ℝ) is equal to Bp,q and these numbers are difficult to calculate in general. As applications of the generalized Clarkson inequalities the (p, q)-Clarkson inequalities in Lebesgue spaces, in mixed norm spaces and in normed spaces are presented. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)