# convexity of the product of two entropy-like functions

Consider the functions $T_p(q)= \sum_i q_i^p$, where p>1 and q is a finite-dimensional vector satisfying $\sum_i q_i = 1, q_i >0$ (ie, a probability mass function). In information-theoretic terms, these are equivalent (up to constants) to the Tsallis entropies of q

$T_p$ is a convex function of $q$ for any $p>1$. This is also true of its generalization as a function of (Trace-normalized) Hermitian operators, ie $T_p(A) := Tr(A^p)$

The question is: are products of the form $(T_{p_1})^{r_1}\cdots(T_{p_n})^{r_n}$, where $r_i >1$, also convex?

Note that in general $T_{p_i}, T_{p_j}$ may define different partial orders, so it would appear that simple arguments that give sufficient conditions for the product of two convex functions to be convex, such as this one, cannot be used. On the other hand, I haven't found any counterexamples yet.

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