In 2009, J. C. Kuang defined the bivariate mean $K_{\omega_1,\omega_2; p}(a, b)$ of positive numbers $a, b$ with three real parameters $\omega_1$, $\omega_2$ and $p$ as $$ K_{\omega_1,\omega_2; p}(a, b)\coloneq \cases \left[\dfrac{\omega_1A(a^p, b^p) +\omega_2G(a^p, b^p)}{\omega_1+\omega_2}\right]^{1/p},& p\ne0,\\ G(a,b), & p=0, \endcases $$ where $(a, b) \in (0,\infty)^2$, $\omega_1,\omega_2 \in [0, \infty)$ satisfy $\omega_1 + \omega_2 \neq 0$, $A(a, b) = \frac{1}{2}(a+b)$, and $G(a, b) = \sqrt{ab}$. \par In the paper under review, the authors find and apply necessary and sufficient conditions for the bivariate mean $K_{\omega_1,\omega_2; p}(a, b)$ to be Schur convex or Schur harmonically convex respectively. \par Theorem 1. Let $p \in \Bbb{R}$ and let $\omega_1,\omega_2 \in [0, \infty)$ be such that $\omega_1 + \omega_2 \neq 0$. Then \roster \item the bivariate mean $K_{\omega_1,\omega_2; p}(a, b)$ is Schur convex with respect to $(a, b) \in (0,\infty)^2$ if and only if $(\omega_1,\omega_2; p) \in S_1$, \item the bivariate mean $K_{\omega_1,\omega_2; p}(a, b)$ is Schur concave with respect to $(a, b) \in (0,\infty)^2$ if and only if $(\omega_1,\omega_2; p) \in S_2$, \endroster where $S_1$ and $S_2$ are given by $$ \multline S_1 = \{(\omega_1,\omega_2; p): 2 \leq p, 0 < \omega_1, 0 \leq \omega_2 \leq \omega_1(p - 1)\}\\ \cup \{ (\omega_1,\omega_2; p): 1 \leq p < 2,\omega_2 = 0 < \omega_1\} \endmultline $$ and $$ \multline S_2 = \{(\omega_1,\omega_2; p): p \leq 2, 0 < \omega_1, \max \{0,\omega_1(p - 1)\} \leq \omega_2\}\\ \cup \{(\omega_1,\omega_2; p): p \in \Bbb{R}, 0 = \omega_1 < \omega_2\}. \endmultline $$ Theorem 2. Let $p \in \Bbb{R}$ and $\omega_1,\omega_2 \in [0, \infty)$ with $\omega_1 + \omega_2 \neq 0$. Then \roster \item the bivariate mean $K_{\omega_1,\omega_2; p}(a, b)$ is Schur harmonically convex with respect to $(a, b) \in (0,\infty)^2$ if and only if $(\omega_1,\omega_2; p) \in H_1$, \item the bivariate mean $K_{\omega_1,\omega_2; p}(a, b)$ is Schur harmonically concave with respect to $(a, b) \in (0,\infty)^2$ if and only if $(\omega_1,\omega_2; p) \in H_2$, \endroster where $H_1$ and $H_2$ are given by $$ H_1 = \{(\omega_1,\omega_2; p): -2 \leq p, \max \{0, -\omega_1(p + 1)\} \leq \omega_2\} $$ and $$ H_2 = \{(\omega_1,\omega_2; p): p \leq -2, 0 \leq \omega_2 \leq -\omega_1(p + 1)\}\cup \{(\omega_1,\omega_2; p): p \leq -1,\omega_2 = 0 < \omega_1\}. $$