Quasi-convexity and Pseudo-convexity of a function in two variables with positive second partial derivatives

Question

Assume an infinitely differentiable smooth function with positive second order partial derivatives. That is, $$\dfrac{\partial^2\,f}{\partial\, x^2}>0\;,\text{ and }\;\dfrac{\partial^2\,f}{\partial\, y^2}>0.$$ Can we say anything about the pseudo-convexity or quasi-convexity of this function? In other words, can we prove or disprove that the function has a single$^1$ Minima?

$^1\;-$Strictly speaking, single minima or a contiguous set of Minimas with same functional value, lying on some $xy$ plane $z=c$.

Answer 1

No. The function $f(x,y) = x^2 + y^2 - 4xy$ is neither pseudo nor quasi-convex.