Some monotonicity properties and inequalities for the (p, k)-gamma function

Abstract

In this paper, the authors present some complete monotonicity properties and some inequalities involving the (p, k)-analogue of the Gamma function. The established results provide the (p, k)-generalizations for some results known in the literature. Γ p,k (k) = 1.

Extension of complete monotonicity results involving the digamma function

“…Results similar to the results of this paper, and concerning the the polygamma and Nielsen's beta functions, can be found in [16] and [17]. Other monotonicity properties concerning the (p, k)-digamma function can also be found in the recent works [11], [18] [25] and [26].…”

Some monotonicity properties and inequalities for the generalized digamma and polygamma functions

“…Very recently, Nantomah, Prempeh and Twum [ 35 ] introduced a -analogue of the gamma and digamma functions defined for , and as and It is obvious that . Some important identities and inequalities involving these functions may be found in [ 30 , 34 , 35 ].…”

Complete monotonicity related to the k-polygamma functions with applications

“…This means that kψ k (x) + ln ψ k (x) is strictly increasing on (0, ∞). By [31] for x > 0 and 0 < k ≤ 1, we have…”

“…It would be natural to generalize the properties of classical functions to the k-gamma, digamma and polygamma functions. It is well known that the k-analogues of the digamma and polygamma functions satisfy the following recursive formula and series identities (see [16,31,32]):…”