Two logarithmically completely monotonic functions connected with gamma function

Abstract

Abstract. In this paper, the logarithmically complete monotonicity results of the functions [Γ(1 + x)] y /Γ(1 + xy) and Γ(1 + y)[Γ(1 + x)] y /Γ(1 + xy) are established.

Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic

“…In particular, many complete monotonicity and logarithmically complete monotonicity properties related to the gamma function, psi function, and polygamma function can be found in the literature [17,18,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. In 1997, Merkle [38] proved that F(x) = (2x) 2 (x) is strictly log-concave on (0, ∞).…”

A function involving gamma function and having logarithmically absolute convexity

“…The main aim of this paper is to generalize and extend some results obtained in [1,5,11,13,25,28,29].…”

“…In [25,28], the following logarithmically complete monotonicities, as generalizations of the decreasingly monotonic property in [29], are presented:…”

A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

“…In recent years, the logarithmically completely monotonic functions and their properties have been investigated extensively and explicitly in [3,5,[9][10][11][16][17][18]23,30,[34][35][36][38][39][40][41][42][43][44][45][46]50] and the references therein.…”