Gurland's ratio for the gamma function

Very accurate estimates of the polygamma functions

“…Our inequalities (1.3) for n = 1 are stronger than the double inequality stated by Merkle [32], for x > 0,…”

A class of logarithmically completely monotonic functions related to the q-gamma function and applications

“…2. Let n = 2 and p k = 1/2, k = 1, 2 in inequalities (28) we obtain the lower bounds for the q-analogue for Gurland's ratio [6] as follows x x+1/2 y y+1/2 . The left hand side inequalities of (38) has proved first by Mortici [8] and the right hand side of inequalities (38) is new.…”

Supplements to a class of logarithmically completely monotonic functions associated with the gamma function

“…holds, where the middle term in (6) is called Gurland's ratio [16]. The left-hand side inequality in (6) (1) If β ∈ (0, ∞) and α ≤ 0, then f α,β,+1 (x) is logarithmically completely monotonic on (0, ∞);…”

“…for β ≥ 1 and x > y > 0 holds true if and only if α ≤ 1 2 ; (2) the inequality (16) for β ∈ (0, ∞) holds true also if α ≤ 0.…”