Inequalities between the second and fourth moments

Let sigma(2) be the variance and mu(4) the fourth moment of a symmetric pro bability distribution. We will prove that for distributions with non-negative characteristic funct ion the inequality mu(4) greater than or equal to 2 sigma(4) holds and that mu(4) = 2 sigma(4) if and only if the characteristic function f is given b y f(x) = cos(2)(ax), x is an element of R for some a is an element of R. Fo r symmetric unimodal distributions we have mu(4) greater than or equal to ( 9/5)sigma(4) and mu(4) = (9/5)sigma(4) if and only if the characteristic fu nction f is is given by f(x) = (sin(ax))/ax, x is an element of R for some a is an element of R. The products of variances of adjoint positive definite densities have a gre atest lower bound Lambda. There is a self-adjoint distribution such that si gma(4) = Lambda. We will prove that for such distributions the equality mu( 4) less than or equal to 2 + sigma(4) holds.