I'm trying to show that some function is convex. It all comes down to showing that some integral is bounded below by zero. Here is the problem:
Let $\kappa,\phi,\psi>0$ be real numbers. Let $f,F:[0,T]\rightarrow\mathbb{R}$ be some function where $\int_0^T f(s) \, ds < \infty$, and where $T>0$. Let's also define $F(t) = F_0 + \int_0^t f(s) \, ds$ where $F_0\in\mathbb{R}$.
Assuming all of the above, I want to find the necessary requirements on $\phi,\Psi,\kappa$ so that $$ \int_0^T \left\{ \kappa \,\left(f(t)\right)^2 + \phi \, \left( F(t) \right)^2 + 2\Psi \, f(t) F(t) \right\} \, dt \geq 0 \;. $$ I know that a sufficient condition is for $\sqrt{\kappa\phi}\geq\Psi$... Are there necessary conditions on these constants? Ideally I'd like to avoid requirements on $f,F_0$.