Concavity/ Convexity of power means function?

Question

Let $a_i > 0$ forall $1 \leq i \leq n$ and let $M(x) := \big(\frac{\sum_{i = 1}^n a_i^x}{n} \big)^{\frac{1}{x}}$ be the power means function. It is well known that the power means function is non-decreasing.

I am interested in the concavity properties of $M(x)$ but checking if $M''(x) \geq 0$ seems to be too difficult.

I was able to compute $M''(x)$. Let $M(x) = f(x)^{\frac{1}{x}}$ where $f(x) = \frac{\sum_{i=1}^n a_i^x}{n}$. Then I was able to find: $$ \frac{M''(x) f^2(x)}{\big(\frac{M(x)}{x^4} \big)} = (xff' - f \ln f)^2 + x^3(ff'' - f'^2) - 2x(xff' - f^2 \ln f) $$ where $'$ denotes derivative and number denotes powers. Checking the sign of $M''(x)$ is equivalent to the checking the sign of the RHS. This terms seem too hard to check for positivity. Can anyone help? I know for a fact that $M(x)$ is convex and concave on different intervals, but I can't find these intervals. Can anyone help?

Answer 1

All this is complicated.

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Too complicated $-$ consider alone $M''(x)=\ldots\;$
And paired with the resulting nescience, this excludes a canonical answer.

It is known that

• if $n=2$ and $a_1\neq a_2$, then $M(x)$ possesses one inflection point (not necessarily at $x=0$). On the left of it, $M(x)$ is convex. And concave to the right of it.
  See Nam & Minh: "Proof for a Conjecture on General means"
  (J of Inequalities in Pure & Applied Math., Vol. 9, 2008)
• if $n=3$, then more inflection points may pop up.
  A concrete instance having three inflection points (albeit in the weighted context) was proposed by Grant Keady , cf the first paragraph on page 2 of "Conjecture on General means"
  (J of Inequalities in Pure & Applied Math., Vol. 7, 2006).

The questions are as intriguing as old, you may also read H. Shniad, published 73 years ago
(Bulletin Amer. Math. Soc. 54, pp 770$-$776, 1948).