Take any real-valued function $u$ taking values in $[0,1]$. Fix any CDF on $\mathbb{R}$, say $F_0$. Consider a maximization problem $\max_{F \in \mathcal{F}} \int^1_0 u(x) dF(x)$ where $\mathscr{F}$ is the set of distributions $F$ such that $F_0$ is a mean-preserving spread of $F$.
Can we say the following?: if we multiply a convex function $h$ to $u$, then $f^*$ chosen under $u\cdot h$ is more dispersed than the distribution $f^{**}$ chosen under $u$.
I know adding a convex function $h$ to $u$ makes the solution more dispersive, but I'm curious whether the similar thing holds about multiplication.
I do not think the property you are asking for is true. In fact, take $u(x)=1-x$ and $h(x)=x$. Both are convex, but their product is not. Or take $u(x)=(1-x)^2$ and $h(x)=\frac{1}{u(x)}$. Then both functions are convex but their product is constant. Given that, $f^{*}$ chosen under $u\cdot h$ can be anything.