From Boyd and Vandenberghe Chapter 11; a generalized logarithm $\psi$ for a proper cone $K$ is any function that satisfies these two properties. 1) $\nabla^2 \psi(y) \prec 0$ for any $y \in K$, and 2) for a constant $s > 0$ and all $y \succ_K 0$: $\psi(sy) = \psi(y) + \theta \log y$, where $\theta$ is the degree of $\psi$.
The text then goes on to say that generalized logarithm satisfies the following property: $\nabla \psi(y) \succ_{K^*} 0$ (i.e., the gradient of the generalized log is contained in the dual cone).
How do I prove the above? Are there any references for the above property? I have looked in Rockafellar.