Year: 2025
Type: article
Abstract: Let ?q,n = (?1)n?1 ?(n)q for n ? N? {0} and q ? (0, 1), where ?(n) q is the q-polygamma functions. In this paper, by means of the monotonicity of means and two classes of completely monotonicity functions, we establish lower and upper bounds for the means I?q,n,w (a, b) defined, for b > a > 0 and w being positive and integrable on [a, b], by I?q,n,w (a, b) = ??1 q,n (?ba w(x)?q,n (x) dx/?ba w(x) dx); and prove that the sequence {I?q,n,w (a,b)} n?0 is decreasing with lim n?? I?q,n,w (a, b) = a. Moreover, we show that, for a, b ? R with a ? b, the function x 7? ??1 q,n (?ba w(t)?q,n (x + t) dt/?ba w(t) dt)?x is increasing from (?min {a, b} ,?) onto ( min {a, b}, ?2), where ?2 = logq (?ba w(t) qtdt/?ba w(t) dt). These generalize some known results.
Source: Filomat
DOI: 10.2298/fil2503051y
Authors: Zhen-Hang Yang, Jing-Feng Tian
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4412898358