Refinements of lower bounds for polygamma functions

In the paper, some lower bounds for polygamma functions are refined.


Introduction and main results
It is well-known that the classical Euler's gamma function for x > 0, the psi function ψ(x) = Γ ′ (x) Γ(x) and the polygamma functions ψ (i) (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.
2000 Mathematics Subject Classification. Primary 33B15; Secondary 26D07. Key words and phrases. refinement, lower bound, polygamma function, inequality. The first author was partially supported by the China Scholarship Council. This paper was typeset using A M S-L A T E X.
Furthermore, the function ψ (n) (x) was bounded in [4, Theorem 2.2] alternatively as which can be rewritten as for x > 0 and 1 ≤ k ≤ n − 1.
The main aim of this paper is to further refine the left-hand side inequalities in (6) and (8) or (7) and (9) as follows.

Lemmas
In order to prove Theorem 1, the following lemmas are needed.
Recall [13, Chapter XIII] and [26, Chapter IV] that a function f (x) is said to be completely monotonic on an interval I ⊆ R if f (x) has derivatives of all orders on I and holds for all k ≥ 0 on I. Recall also [3,18] that a function f is said to be logarithmically completely monotonic on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm ln f satisfies for k ∈ N on I. In [18,Theorem 4], it was proved that all logarithmically completely monotonic functions are also completely monotonic, but not conversely. This result was formally published when revising [15]. For more information, please refer to [7,20].
Proof. For the sake of convenience, denote the function (17) Direct calculation reveals that for t ∈ (0, ∞). Hence, by the limit (18) and the mathematical induction, we have which is equivalent to the inequality (16). It is obvious that the functions e 1/t and e 1/(t+1) are logarithmically completely monotonic on (0, ∞) and (−1, ∞) respectively, so they are also completely monotonic on (0, ∞) and (−1, ∞) respectively. This means that Equivalently, the signs of the functions are the same and they are opposite to for k ≥ 0 on (0, ∞). As a result, the sign of the function is opposite to the sign of the function for k ≥ 0 on (0, ∞). Therefore, from the inequality (20), it is obtained inductively that on (0, ∞) for k ≥ 0. Accordingly, by (19), it follows that Combining this with (21) shows that the function h(t) defined by (17) is completely monotonic on (0, ∞). The proof of Lemma 2 is complete.

Proof of Theorem 1
Now we are in a position to prove Theorem 1. Letting (10) and rearranging yields for t ∈ (0, ∞).

Remarks
Finally, we would like to supply several remarks on Theorem 1.
Remark 2. The inequality (10) would be invalid if n is big enough. In other words, the inequality (10) is valid not for all n ∈ N. Otherwise, the inequality should be valid on (0, ∞). However, the reversed inequality of (31) holds on (0, ∞). This situation motivates us to naturally pose an open problem: What is the largest positive integer n such that the inequality (10) holds on (0, ∞)?
Remark 3. Rewriting (2) and (10) for n = 1 leads to for x > 0, where stands for the logarithmic mean for positive numbers a and b. Since the logarithmic mean L(a, b) is strictly increasing with respect to both a > 0 and b > 0 and the psi function ψ(x) is also strictly increasing on (0, ∞), the inequalities (4), (6), (7), (10) and (32) stimulate us to naturally ask the following question: What are the best scalars p(n) ≥ 0 and q(n) > 0 such that the inequality is valid on (0, ∞)? Similarly, the inequalities (8), (9) and (11) motivate us to pose the following open problem: What are the best constants p(n, k) ≥ 0 and 0 < q(n, k) ≤ 1 such that the inequality holds on (0, ∞) for 1 ≤ k ≤ n − 1.  [2,17,21] respectively. The strict concavity and some other generalizations of the function in the inequality (36) was discussed in [21] recently.
Remark 5. The case n = 2 and k = 1 in (11) is on (0, ∞). This refines the inequality on (0, ∞), the special case n = 2 and k = 1 of the inequality (8). The inequality (38) was also refined in another direction and generalized in [19]. The inequality (3), a special case n = 1 of the inequality (26), has been generalized to the complete monotonicity and many other cases. For more information, please refer to [19,20] and closely-related references therein.
Remark 6. The generalized logarithmic mean L(p; a, b) of order p ∈ R for positive numbers a and b with a = b is defined in [8, p. 385] by It is known [23,24] that L(p; a, b) is strictly increasing with respect to p ∈ R. See also [11,14] and closely-related references therein. Furthermore, we can pose the following more general open problem: What are the best scalars λ(n), µ(n), p(n) and q(n) such that the inequality is valid on (0, ∞)? What are the best constants λ(n, k), µ(n, k), p(n, k) and q(n, k) such that the inequality k ψ (k) (L(λ(n, k); x, x + q(n, k))) holds on (0, ∞) for 1 ≤ k ≤ n − 1.
By the left-hand side inequality in (12), it follows that where u = 1 t . Hence k ≤ 1. The inequality (10) for n = 1 is proved.