New properties of the divided difference of psi and polygamma functions (Q2050217)

The authors define the functions \[ \xi_{n}(x)=\frac{n \phi_{n}(x)}{\phi_{n+1}(x)}-x\quad\text{and}\quad\eta_{n}(x)=\left[\frac{\phi_{n}(x)}{(n-1) !}\right]^{-1 / n}-x \] and study some of their properties. Here \(-\phi_n(x)\) is the divided difference of the functions \(\psi_{n-1}(x+p)\) and \(\psi_{n-1}(x+q)\), where \(\psi_n=(-1)^{n-1}\psi^{(n)}\), and \(\psi^{(n)}\) stands for the \(n\)th polygamma function. For example, it turns out that, under certain conditions, \[ \lim _{n \rightarrow \infty} \xi_{n}(x)=\min \{p, q\}. \] It is also true, that \(\xi_n(x)\) is a convex function on \((-\min \{p, q\},\infty)\) iff \(p=q\). Similar statement holds true, mutatis mutandis, for \(\eta_n(x)\). After proving these results, the paper contains applications of them in Section 4. These applications involve, for example, sharp inequalities for above defined \(\phi_n(x)\) maps. For instance, if \(n \in \mathbb{N}\), and \(|p-q| \gtrless 1\), then the inequalities \[ x+\frac{p+q-1}{2} \gtrless \frac{n \phi_{n}(x)}{\phi_{n+1}(x)} \gtrless \frac{(n+1) \phi_{n+1}(x)}{\phi_{n+2}(x)} \gtrless x+\min \{p, q\} \] hold for every \(x\in(-\min \{p, q\},\infty)\). This yields, in the particular case when \(p=q=0\), an inequality for the \(\psi_n\) function: \[ x-\frac{1}{2}<\frac{n \psi_{n}(x)}{\psi_{n+1}(x)}<\frac{(n+1) \psi_{n+1}(x)}{\psi_{n+2}(x)}<x \] for positive argument \(x\). Another application is the presentation of the answer to Qi's and Agarwal's question about the monotonicity and convexity properties of the function \[ f_{p, q ; \alpha, \beta}(x)=\left[\frac{\Gamma(x+p)}{\Gamma(x+q)}\right]^{\alpha /(p-q)}-\beta x. \]