Iterated means $a_{n+1}=\sqrt{a_n \frac{b_n+c_n}{2}}$, $b_{n+1}$ and $c_{n+1}$ similar, closed form for general initial conditions?

Analogous to the Schwab-Borchardt mean:

\begin{align*} B(a,b) &= \frac{\sqrt{b^{2}-a^{2}}}{\cos^{-1} \frac{a}{b}} \\ &= B\left( \frac{a+b}{2}, \sqrt{\frac{(a+b)b}{2}} \right) \end{align*}

which can also be obtained by the iteration: $$ \left \{ \begin{array}{rcl} a_{n+1} &=& \frac{a_{n}+b_{n}}{2} \\ b_{n+1} &=& \sqrt{a_{n+1} b_{n}} \end{array} \right.$$

Now your mean is

\begin{align*} M(x,x,y) &= D(x,y) \\ &= D\left( \sqrt{\frac{x(x+y)}{2}} , \sqrt{xy} \right) \\ &= \frac{\sqrt[4]{x^{2}(y^{2}-x^{2})}}{\sqrt{\cos^{-1} \frac{x}{y}}} \end{align*}