Is this function convex or concave

Question

Consider the following function:

$$f:(0, \infty)^2 \rightarrow \mathbb{R}: (\phi,\psi) \rightarrow \frac{\phi}{\psi}$$

Is this function convex or concave? (Or neither?)

I tried by calculating the Hessian matrix: $$H_f (\phi,\psi) = \begin{pmatrix} 0 & - \frac{1}{\psi^2} \\ - \frac{1}{\psi^2} & \frac{2\phi}{\psi^3} & \\ \end{pmatrix} $$ but I could not determine whether it was positve or negativ definite. Any ideas?

Answer 1

So, you're using the fact that a $C^2$ function

$$f \; \text{is convex} \; \iff \; f'' \; \text{is positive semidefinite}$$

and you can use the fact that a symmetric matrix $A$ is

$$\text{ positive semi-definite} \; \iff \text{All principal minors}\; \Delta_k \geq 0$$

You can also see that the principal minors of your Hessain are $0, \dfrac{2\phi}{\psi^3}, -\dfrac{1}{\psi^4}$ which guarantees $f$ is not convex.