# How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let $$\begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*}$$ and $$\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.$$ In the literature, these two functions are respectively called the Euler gamma function and the digamma function. The derivatives $$\psi^{(k)}(z)$$ for $$k=1,2,3,4$$ are respectively called the trigamma, tetragamma, pentagamma, and hexagamma functions. For non-specific positive integer $$n\in\mathbb{N}$$, we call $$\psi^{(n)}(z)$$ the polygamma function.

I have proved that, for $$n\in\mathbb{N}$$ and $$\alpha>0$$, the ratio $$\frac{\psi^{(n)}(x+\alpha)}{\psi^{(n)}(x)}$$ is increasing from $$(0,\infty)$$ onto $$(0,1)$$. This is equivalent to saying that, for $$n\in\mathbb{N}$$ and $$\alpha>0$$, the function $$\Phi_{n,\alpha}(x)=\frac{1}{1-\frac{\psi^{(n)}(x+\alpha)}{\psi^{(n)}(x)}}$$ is increasing from $$(0,\infty)$$ onto $$(1,\infty)$$.

After considering the increasing property of $$\Phi_{n,\alpha}(x)$$, I would like to ask the following question.

For $$n\in\mathbb{N}$$ and $$\alpha>0$$, is the function $$\Phi_{n,\alpha}(x)$$ convex on $$(0,\infty)$$?

The answer to this question should be YES.

This problem is perhaps much easy for you, but it is very difficult for me right now. Please have a try to solve it. Thank you very mcuh.

I have asked this question at the site on the ResearchGate, but till now I haven't gotten any correct and useful answer.

I have proved that, for $$n\in\mathbb{N}$$, the function $$\Phi_{n,1}(x)$$ is convex on $$(0,\infty)$$.
In the paper "H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), no. 2, 181--221; available online at http://dx.doi.org/10.1515/form.2004.009 ", the function $$x^c\lvert\psi^{(n)}(x)\lvert$$ for $$n\in\mathbb{N}$$ and $$c\in\mathbb{R}$$ was proved to be convex on $$(0,\infty)$$ if and only if either $$c\le n$$, or $$c=n+1$$, or $$c\ge n+2$$. This is the main base of my proof.
The next special case I want to prove is the possibly correct conclusion that the function $$\Phi_{1,1/2}(x)$$ is convex on $$(0,\infty)$$.