Is there a way to prove convexity of $$ f\left(\frac{x^{\mathsf T}\sigma}{\sqrt{x^{\mathsf T}\Sigma x }}\right)\sqrt{x^{\mathsf T}\Sigma x}$$ where $\sigma$ is a vector, $\Sigma$ is positive definite, $f$ is positive with $\frac{df}{dz}>0$?
PS: Finding the Hessian does not seem to work.