Let σ² be the variance and μ₄ the fourth moment of a symmetric probability distribution.

We will prove that for distributions with non-negative characteristic function the inequality μ₄ ≥ 2σ⁴ holds and that μ₄ - 2σ⁴ if and only if the characteristic function f is given by f(x) = cos²(ax).   for some   . For symmetric unimodal distributions we have μ₄ ≥ (9/5)σ⁴ and μ₄ = (9/5)σ⁴ if and only if the characteristic function f is given by f(x) = (sin(ax))/ax,   for some   .

The products of variances of adjoint positive definite densities have a greatest lower bound A. There is a self-adjoint distribution such that σ⁴ = Λ. We will prove that for such distributions the equality μ₄ ≤ 2 + σ⁴ holds.