On a functional equation connected with Rao’s quadratic entropy

Abstract:

We determine the general solution of the functional equation fxy, \[ f\left ( {\frac {{x + y}} {2}} \right ) + f\left ( {\frac {{x - y}} {2}} \right ) = 2f\left ( {\frac {x} {2}} \right ) + 2f\left ( {\frac {y} {2}} \right ) + \lambda f(x)f(y),\] /: [-$f:[ - 1,1] \to {\mathbf {R}}$. This equation was used by Lau in order to characterize Rao’s quadratic entropies. The general solution is obtained here as a special case of a more general result for $f$ mapping a neighborhood of $0$ in linear topological space into a field.