For a real number $p$, let $M_p(a, b)$ denote the usual power mean of order $p$ of positive real numbers $a$ and $b$. Further, let $H=M_{; ; ; -1}; ; ; $ and $He_{; ; ; \alpha}; ; ; = \alpha M_0 + (1 - \alpha) M_1$ for $\alpha \in [0, 1]$. We prove that the double mixed-means inequality \[ M_{; ; ; -\frac{; ; ; \alpha}; ; ; {; ; ; 2}; ; ; }; ; ; (a, b) \leq \frac{; ; ; 1}; ; ; {; ; ; 2}; ; ; [H(a, b) + He_{; ; ; \alpha}; ; ; (a, b)] \leq M_{; ; ; \frac{; ; ; \ln 2}; ; ; {; ; ; \ln 4 - \ln (1 - \alpha)}; ; ; }; ; ; (a, b) \] holds for all $\alpha \in [0, 1]$ and positive real numbers $a$ and $b$, with equality only for $a = b$, and that the orders of power means involved in its left-hand and right-hand side are optimal.