Best Constants in Kahane-Khintchine Inequalities for Complex Steinhaus Functions

for all z1; . . . ; zn 2 C and all n 1 . The constant p 2 is shown to be the best possible. The method of proof relies upon a combinatorial argument, Taylor expansion, and the central limit theorem. The result is additionally strengthened by showing that the underlying functions are Schur-concave. The proof of this fact uses a result on multinomial distribution of Rinott, and Schur’s proposition on sum of convex functions. The estimates obtained throughout are shown to be the best possible. The result extends and generalizes to provide similar inequalities and estimates for other Orlicz norms.