  It is known that the $L^{2}$-norms of a harmonic function over spheres
satisfies some convexity inequality strongly linked to the Almgren's frequency
function. We examine the $L^{2}$-norms of harmonic functions over a wide class
of evolving hypersurfaces. More precisely, we consider compact level sets of
smooth regular functions and obtain a differential inequality for the
$L^{2}$-norms of harmonic functions over these hypersurfaces. To illustrate our
result, we consider ellipses with constant eccentricity and growing tori in
$\mathbf{R}^3.$ Moreover, we give a new proof of the convexity result for
harmonic functions on a Riemannian manifold when integrating over spheres. The
inequality we obtain for the case of positively curved Riemannian manifolds
with non-constant curvature is slightly better than the one previously known.
