Convex solutions of a functional equation arising in information theory

Given a convex function f defined for positive real variables, the so-called Csiszár f -divergence is a function If defined for two n-dimensional probability vectors p = (p1, . . . , pn) and q = (q1, . . . , qn) as If (p, q) := n i=1 qif ( pi qi ). For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy the following functional equation: f (x) = xf ( 1 x ) for allx >0. In the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them. In doing so, existing usual f -divergences are recovered and new ones are proposed. © 2006 Elsevier Inc. All rights reserved.