For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when $p\to 0$, i.e., the geometric mean.

Furthermore, when $p,q$ are two real numbers, define $L_{p,q}(x,y)$ as the obvious fixed point satisfying the equation $$L_{p,q}(x,y) = L_{p,q}(M_p(x,y),M_q(x,y))$$ —meaning that $L_{p,q}(x,y)$ is defined as the common limit of the sequences $(x_n)$ and $(y_n)$ such that $(x_0,y_0) = (x,y)$ and $(x_{n+1},y_{n+1}) = (M_p(x,y),M_q(x,y))$. (Showing convergence is easy, and evidently, $L_{p,q} = L_{q,p}$.)

Thus, $L_{0,1}(x,y)$ is the famous arithmetic-geometric mean.

Among other things, all these functions satisfy $\min(x,y) \leq F(x,y) \leq \max(x,y)$, as well as $F(y,x) = F(x,y)$ and $F(\lambda x, \lambda y) = \lambda F(x,y)$ — so we could just define them from $F(1,x)$, which is continuous and monotonically increasing with value $1$ at $1$. It is well-known that $M_p(x,y) < M_q(x,y)$ whenever $x\neq y$ and $p<q$, so of course $M_p(x,y) < L_{p,q}(x,y) < M_q(x,y)$ under those circumstances.

Surprisingly, I can find no reference to the $L_{p,q}$ in the literature except for $(p,q)=(0,1)$ (mentioned above), and $(p,q)=(-1,1)$ which this page points out is just $M_0$. More generally, we have $$L_{p,-p}(x,y) = M_0(x,y)$$ and $$L_{0,p}(x,y) = \big(L_{0,1}(x^p,y^p)\big)^{1/p}$$ —both are easy, and $L_{p,p}(x,x) = M_p(x,y)$ is trivial.

I couldn't find anything intelligent to say about $L_{1,2}$ (the "arithmetic-quadratic mean"), however, let alone the other $L_{p,q}$. Experimentally, it satisfies $L_{1,2}(x,y) \leq M_{3/2}(x,y)$, something which is probably easy to prove and not interesting enough to try, so that's not my question. Instead, my question will be:

Can $L_{1,2}$, or more generally $L_{p,q}$ (with $0<p<q$), be expressed in closed form using elementary functions and the arithmetic-geometric mean $L_{0,1}$? Or at least using "standard" special functions (e.g. elliptic integrals)?

(I realize that the answer "no" might be exceedingly difficult to prove, so I'll be liberal in what I accept as an answer: perhaps the question should really be "please say something interesting about $L_{1,2}$ or $L_{p,q}$ in general".)

  • $\begingroup$ Nothing useful to add, just wanted to say that this is a very interesting proposition $\endgroup$ – Yuriy S Jun 11 '16 at 21:53

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