Maximization of functionals in $H^\omega [a,b]$

The structure and the properties of the extremal functions in the problem
$$ \int _a^b h(t)\psi (t)\,dt\to \sup, \qquad h\in H^\omega [a,b], $$
are described in the case when $\psi$ is an integrable function with zero mean and finitely many points of sign change on $[a,b]$ and $H^\omega [a,b]$ is the class of absolutely integrable functions on $[a,b]$ with modulus of continuity majorized by a fixed convex modulus of continuity $\omega$.