Let $d_0$, $d_1$, $\rho$ positive matrices of $M_n$ of trace one. We consider the state $$ d_t=(1-t)d_0+td_1 $$ for any $0 \le t \le 1$. We suppose that the relative entropy $S(d_t|\rho)$ is finite for any $t \in [0,1]$. Consider the function $$f : [0,1] \to \mathbb{R}^+,\, t \mapsto S(d_t|\rho)=\mathrm{tr}( d_t \log d_t -d_t\log \rho).$$
I read in the book "Quantum information theory..." p.97 of Petz that $f$ is convex. I does not understand why.
Note that for any $0<\lambda<1$
$$\begin{eqnarray} d_{\lambda t_1 + (1-\lambda) t_2} & = & (1-\lambda t_1 - (1-\lambda) t_2)d_0+(\lambda t_1 + (1-\lambda) t_2)d_1 \\ & = & (1-\lambda + \lambda - \lambda t_1 - (1-\lambda) t_2)d_0 + (\lambda t_1 + (1-\lambda) t_2)d_1 \\ & = & (\lambda (1 - t_1) + (1-\lambda) (1- t_2))d_0 + (\lambda t_1 + (1-\lambda) t_2)d_1 \\ & = & \lambda ((1 - t_1) d_0 + t_1 d_1) + (1-\lambda) ((1- t_2)d_0 + t_2d_1) \\ & = & \lambda d_{t_1} + (1-\lambda) d_{t_2} \; .\end{eqnarray}$$
Combined with the convexity of the function $x\log x$, this suffices to show that $f$ is convex as well.