On Clarkson's inequality in the real case

The best constant in a generalized complex Clarkson inequality is C p,q (ℂ) = max {2 1–1/ p , 2 1/ q , 2 1/ q –1/ p +1/2 } which differs moderately from the best constant in the real case C p,q (ℝ) = max {2 1–1/ p , 2 1/ q , B p,q }, where . For 1 < q < 2 < p < ∞ the constant C p,q (ℝ) is equal to B p,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)