eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2019-02-23 7 1 18 22 10.12691/tjant-7-1-4 TJANT2019714 article Jumei Zhang 1 Li Yin yinli7979@163.com 1 Wenyan Cui 1 School of Science, Binzhou University In the paper, we discuss a new k-generalization of the Nielsen's β-function. Later, we study the completely monotonicity, convexity and inequalities of the new function. http://pubs.sciepub.com/tjant/7/1/4/tjant-7-1-4.pdf Nielsen's ¦-function k-generalization inequality completely monotonic