A class of completely mixed monotonic functions involving the gamma function with applications

A Rational Approximation for the Complete Elliptic Integral of the First Kind

“…where (a) 0 = 1 for a = 0, (a) n = a(a + 1) • • • (a + n − 1) = Γ(a + n)/Γ(a) is the shifted factorial function and Γ(x) = ∞ 0 t x−1 e −t dt (x > 0) is the gamma function [1,2]. The famous Landen identities [3], p. 507 show that…”

Monotonicity properties for a ratio of finite many gamma functions

“…(1) If (-1) k f (k) (x) ≥ 0 for all k ≥ 0 and x ∈ (0, ∞), then we call f (x) a completely monotonic function on (0, ∞). See the review papers [22,31,36] and [35, Chapter IV]. f (x) is a completely monotonic function on (0, ∞), then we call f (x) a logarithmically completely monotonic function on (0, ∞).…”

On Cauchy–Schwarz inequality for N-tuple diamond-alpha integral

“…As we all know, inequality plays a very important and basic role in all mathematic areas, and it is also an indispensable and basic tool in engineering technology (see [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). The classic Cauchy-Schwarz inequality is an important cornerstone in some branches of mathematic areas.…”