Suppose $f(x,y)\geq 0$ is integrable and symmetric in $x$ and $y$, i.e.$f(x,y)=f(y,x)$. Consider the following nonlinear optimization problem $$\max F(a,b)=\int_0^a\int_0^bf(x,y)dxdy,$$ Subject to $ab=1$.
Do we have the inequality $F(1,1)\geq F(a,b)$? See here.
No, if $f(x,y)=e^{x+y}$ then, e.g., $F(2,\frac{1}{2})=e^{2.5}-e^2-e^{.5}+1\approx4.14>2.95\approx e^2-2e+1=F(1,1)$.