Inequalities involving \(q\)-analogue of multiple psi functions
One can define the \(q\)-analogue of the multiple Gamma functions formally the very same way as the multiple Gamma functions are defined: \[\Gamma_n(z;q)=G_n(z;q)^{-1^{n-1}}.\] Here \[\begin{aligned} \log G_{n}(z+1 ; q) &=-\left(\begin{array}{l} z \\ n \end{array}\right) \log (1-q)-\sum_{i=1}^{\infty}\left(\begin{array}{c} -k \\ n-1 \end{array}\right) \log \left(1-q^{z+k}\right) \\ &+\sum_{j=0}^{n-1} G_{n, j}(z)\left\{\sum_{k=1}^{\infty} k^{j} \log \left(1-q^{k}\right)\right\} \end{aligned},\] where the \(G_{n,j}(z)\) polynomials are defined as \[\left(\begin{array}{l} z-u \\ n-1 \end{array}\right)=\sum_{j=0}^{n-1} G_{n, j}(z) u^{j}.\] Taking logarithmic derivative in the definition of \(\Gamma_n(z;q)\), one arrives at the definition of the \(q\)-analogue of the multiple polygamma functions: \[\Psi_{n}^{(m)}(z ; q)=(\ln q)^{m+1} \sum_{k=1}^{\infty}\left(\begin{array}{c} n+k-2 \\ n-1 \end{array}\right) \frac{k^{m} q^{k z}}{1-q^{k}}, \quad 0<q<1, \quad m \geq n.\] The author derives convexity properties for these functions. For example, it is proven that \(\Psi_{n}^{(m)}(x ; q)\) is convex (concave) on \((0, \infty)\) if \(m\) is odd (even) for \(0<q<1\). Numerous other statements regarding superadditivity of the \(\Psi_{n}^{(m)}(x ; q)\) functions are given, too.
scientific article; zbMATH DE number 7226790