Refinements of Lower Bounds for Polygamma Functions

0 te dt (1) for x > 0, the psi function ψ(x) = Γ ′(x) Γ(x) and the polygamma functions ψ (x) for i ∈ N are a series of important special functions and have much extensive applications in many branches such as statistics, probability, number theory, theory of 0-1 matrices, graph theory, combinatorics, physics, engineering, and other mathematical sciences. In [9, Corollary 2], the inequality ψ(x)e < 1 (2) for x > 0 was deduced. In [5, Lemma 1.1] and [6, Lemma 1.1], the inequality (2) was recovered. In [1, Theorem 4.8], by the aid of the inequality [ψ(x)] + ψ′′(x) > 0 (3) for x > 0, the inequality (2) was generalized as (n− 1)! exp(−nψ(x+ 1)) < |ψ(x)| < (n− 1)! exp(−nψ(x)) (4) for x > 0 and n ∈ N, which can be rearranged as