How can this be proved? (Convex optimization)
Take $\Omega_1=\Omega_2=[0,0.5]$.
Let $p(x)=x^4$, so that $p''(x)=12x^2&gt;0$ on $(0,0.5)$.
Let $q(x)=0.5x$, which is linear in $x$, and $K=0.2&gt;0$.
For $\bar\gamma=1$, both objective functions attain their respective maximum at $x=0.5$. As the following figure shows, objective function $(1)$ (blue curve) has a higher maximum than that of objective function $(2)$ (red curve).

But for $\gamma=5&gt;\bar\gamma$, objective function $(1)$ attains a maximum value of approximately $0.276$, while objective function $(2)$ attains a higher maximum value of approximately $0.309$, contradicting the claim that $(1)$'s maximum would always be greater than $(2)$'s maximum for $\gamma&gt;\bar\gamma$.