I have been study the minimal (maximal) of a $f-$divergence. Fumio Hiai introduced the $\widehat{S}_f (\rho \| \sigma)$ divergence in his article.
$$\widehat{S}_f (\rho \| \sigma) := \text{Tr} \sigma^{1/2} f(\sigma^{-1/2} \rho \sigma^{-1/2}) \sigma^{1/2}$$
This fomular can be rewrite as follow
$$\widehat{S}_f (\rho \| \sigma) = \text{Tr} \sigma f(\sigma^{-1/2} \rho \sigma^{-1/2})$$
For $\sigma, \rho \in \mathbb{D}_n$ ($\mathbb{D}_n$ is the set of density matrices of order $n$), $U \in \mathfrak{U_n}$ with $\mathfrak{U_n}$ be the set of unitary matrices. In the article, Hiai assume that $f: (0; +\infty) \rightarrow \mathbb{R}$ is a continuous function such that the limits
$$f(0^+) := \displaystyle\lim_{x \searrow 0} f(x) \text{ and } f'(+\infty) := \displaystyle\lim_{x \rightarrow +\infty} \dfrac{f(x)}{x}$$
exist in $\mathbb{R} \cup \{\pm\infty\}$, and they are not both infinity with opposite signs. Then he showed that
$$f(x) = f(0^+) + ax + bx^2 + \displaystyle\int_{(0,+\infty)} \left( \dfrac{x}{1+s} - \dfrac{x}{x+s} \right) d \mu_f (s)$$
I would like to find the minimal and maximal of $\widehat{S}_f (\rho \| U^* \sigma U)$ with the function $f$ above, that is find
$$\displaystyle\min_{U \in \mathfrak{U_n}} \widehat{S}_f (\rho \| U^* \sigma U) \text{ and } \displaystyle\max_{U \in \mathfrak{U_n}} \widehat{S}_f (\rho \| U^* \sigma U)$$.
I think maybe $f(\sigma^{-1/2} \rho \sigma^{-1/2}) = f(\sigma^{-1} \rho)$, but I have no proof. To find the extremely values, we can use some characterizations of operator monotone and operator convex functions in this book and the von Neumann trace inequality to calculate.
Thank for read, if you have some idea, please let me know. Thank all!
