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I think one way is to verify if both $\frac{\partial^2 f}{\partial U^2}$ and $\frac{\partial^2 f}{\partial V^2}$ are postive semi-definite, is there other way to prove it?

Pei
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    Take $A=0$, and $U,V$ to be scalars. Show that the resulting function $(u,v) \mapsto u^4v^2$ is not convex. – copper.hat Aug 25 '20 at 07:25

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You can do a Positive definite matrix check, eigenvalue test. But for doing that you need to simplify the expression.

Gaurav
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