Convexity of polylogarithms

Question

I want to prove the following proposition:

The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$.

And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for $p\ge 1$ (wiki on polylogarithm). I didn't find any direct results on this subject.

For bounded $w$ it's possible to build a plot in, say, Mathematica, and see that the proposition is true there. For $w\to\infty$ I can find the asymptotical behaviour: $c_1w+c_2/w +\mathcal O(w^{-2})$, $c_i>0$.

Hence, two questions. Is this development sufficient to guarantee convexity? And what to do on the region where the asymptotics isn't true yet? Of course, I can build a plot on a sufficiently large compact, but to correctly choose the latter seems to be quite a complex task, too.

Computing the second derivative doesn't yield anything useful, or, at least, I can't prove that the sign is constant.

$\frac{\left(-Li_p(-e^w)\right)^{1/p-2}\left(p Li_{p-2}(-e^w)Li_p(-e^w)-(p-1)(-Li_{p-1}(-e^w))^2\right)}{p^2}$

I'd be glad to hear all suggestions on the possible reasoning for my proposition.

Edit (16.05.2013) I managed to prove the following statement:

Suppose the function $\theta(r)=\eta(r)\eta(r-2)-\frac{r-1}{r}\eta^2(r-1)$ is positive for $p$ and $p-1$, where $\eta$ - Dirichlet's eta function, $p\ge 2$. If the function $w\to(- Li_{p-1}(-e^w))^{\frac{1}{p-1}}$ is convex on $\mathbb R_+$, then so is $w\to(- Li_p(-e^w))^{1/p}$ .

Edit (17.05.2013) By a method, analogous to my previous edit, one can show that the hypothesis of convexity is true for all integer $p>0$.

Answer 1

One can show that this fact holds for $p\ge 2$. The convexity is equivalent to the positivity of the second derivative, which equals (up to a positive factor) to

$$\frac{p\mathcal F_p}{\mathcal F_{p-1}}-\frac{(p-1)\mathcal F_{p-1}}{\mathcal F_{p-2}},$$ where $\mathcal F_p(w) = -Li_p(-e^w)$. Now we need to use the form

$$\mathcal F_p(w)=\frac{1}{\Gamma(p)}\int_{0}^{\infty}\frac{t^{p-1}dt}{e^{t-w}+1}.$$ We are now in the frame of my other question (see here) with $d\mu(t) = \frac{1}{e^{t-w}+1}$. There we showed that the application $$ p\to \frac{\int_0^\infty t^p \mathrm d\mu(t)}{\int_0^\infty t^{p-1}\mathrm d\mu(t) },\quad p>1$$is monotone, which implies that $$\frac{p\mathcal F_p}{\mathcal F_{p-1}}-\frac{(p-1)\mathcal F_{p-1}}{\mathcal F_{p-2}}>0$$ and therefore $w\to (\mathcal F_p(w))^{1/p}$ is convex for $p\ge 2$.

Answer 2

I find by literature search the following theorem: For the complex number $z$ with $|z| \le 1$ the following infinite series (the series defining the polylogarithm): $$ \sum_{k=1}^{\infty} z^k/k^{\alpha} $$ is convex in $z$ for all $\alpha \ge 0$.

This is proved in the paper J.L. Lewis, Convexity of a certain series, J. Lond. Math. Soc. 27(3) (1983), pp. 435–446.

But I do not have access to that paper. If I can get hold on the paper I will see if I can post a proof here!