SUBADDITIVE AND SUPERADDITIVE PROPERTIES OF POLYGAMMA FUNCTIONS

FENG QI AND BAI-NI GUO

Abstract. In this short paper, it is proved that the function |ψ^{(i)}(e^{x})| for i ∈ ℕ is subadditive in (ln θ_{0}, ∞) and superadditive in (−∞, ln θ_{0}), where θ_{0} ∈ (0, 1) is the unique root of equation 2|ψ^{(i)}(θ)| = |ψ^{(i)}(θ^{2})|.

1. Introduction

Recall [2, 3, 4] that a function f defined on an interval I is said to be subadditive on I if f(x + y) ≤ f(x) + f(y) holds for all x, y ∈ I such that x + y ∈ I. If f(x + y) ≥ f(x) + f(y), then f is called superadditive on I.

The subadditive and superadditive functions play an important role in the theory of differential equations, in the study of semi-groups, in number theory, and also in the theory of convex bodies. A lot of literature for the subadditive and superadditive functions can be found in [2, 3] and the references therein.

It is well known that the classical Euler’s gamma function Γ can be defined for x > 0 as Γ(x) = ∫_{0}^{∞} e^{−t}t^{x−1} dt. The digamma or psi function ψ is defined as the logarithmic derivative of Γ and ψ^{(i)} for i ∈ ℕ are called polygamma functions.

In [3], a subadditive property of the gamma function Γ was proved: The function [Γ(x)]^{α} is subadditive in (0, ∞) if and only if \frac{ln 2}{ln Δ} ≤ α ≤ 0, where Δ = min_{x≥0} \frac{Γ(2x)}{Γ(x)}.

In [4], a subadditive property of the psi function was obtained: The function ψ(a + e^{x}) is subadditive in (−∞, ∞) if and only if a ≥ c_{0}, where c_{0} is the unique positive zero of ψ(x).

In this short paper, we would like to discuss the subadditive and superadditive properties of the polygamma functions ψ^{(i)}(x) for i ∈ ℕ.

Our main result is the following Theorem 1.

Theorem 1. The function |ψ^{(i)}(e^{x})| for i ∈ ℕ is superadditive in (−∞, ln θ_{0}) or subadditive in (ln θ_{0}, ∞), where θ_{0} ∈ (0, 1) is the unique root of equation 2|ψ^{(i)}(θ)| = |ψ^{(i)}(θ^{2})|.

2. Proof of Theorem 1

In [5], the monotonicity of the function x^{α}|ψ^{(i)}(x+β)| was researched thoroughly, which is a generalization of the corresponding results in [1, 4], as follows:

(1) The function x^{α}|ψ^{(i)}(x)| in (0, ∞) is strictly increasing if and only if α ≥ i + 1 and strictly decreasing if and only if 0 ≤ α ≤ i.

Date: This manuscript was finalized on 16 June 2006.

2000 Mathematics Subject Classification. Primary 33B15; Secondary 26D07.

Key words and phrases. subadditive function, superadditive function, polygamma function.

This paper was typeset using AMS-L^{A}TEX.

(2) For β ≥ \frac{1}{2}, the function x^{α}|ψ^{(i)}(x + β)| is strictly increasing in [0, ∞) if and only if α ≥ i.

(3) Let δ : (0, ∞) → (0, \frac{1}{2}) be defined by

δ(t) = \frac{e^{t}(t − 1) + 1}{(e^{t} − 1)^{2}}  (1)

for t ∈ (0, ∞) and δ^{−1} : (0, \frac{1}{2}) → (0, ∞) stand for the inverse function of δ. If 0 < β < \frac{1}{2} and

α ≥ i + 1 − [\frac{e^{δ^{−1}(β)}}{e^{δ^{−1}(β)} − 1} + β − 1]δ^{−1}(β),  (2)

then the function x^{α}|ψ^{(i)}(x + β)| is strictly increasing in (0, ∞). It is noted that

0 < [\frac{e^{δ^{−1}(β)}}{e^{δ^{−1}(β)} − 1} + β − 1]δ^{−1}(β) < 1  (3)

for β ∈ (0, 1), since lim_{β→0^{+}}[βδ^{−1}(β)] = 0.

Let

f_{β}(x, y) = |ψ^{(i)}(β + x)| + |ψ^{(i)}(β + y)| − |ψ^{(i)}(β + xy)|  (4)

for x > 0 and y > 0, where β ≥ 0 and i ∈ ℕ. In order to show Theorem 1, it is sufficient to prove the positivity or negativity of the function f_{β}(x, y). Directly calculating yields

\begin{aligned}\frac{∂f_{β}(x, y)}{∂x} & = y|ψ^{(i+1)}(β + xy)| − |ψ^{(i+1)}(β + x)| \\ & = \frac{1}{x}[xy|ψ^{(i+1)}(β + xy)| − x|ψ^{(i+1)}(β + x)|]. \end{aligned}  (5)

From the monotonicity of the function x^{α}|ψ^{(i)}(x + β)| in [5] mentioned above, it follows easily that \frac{∂f_{0}(x,y)}{∂x} ⋛ 0 if and only if y ⋚ 1. This means that the function f_{0}(x, y) is strictly increasing for y < 1 and strictly decreasing for y > 1 in x ∈ (0, ∞). Since lim_{x→∞} f_{0}(x, y) = |ψ^{(i)}(y)| > 0, then for y > 1 the function f_{0}(x, y) is positive in x ∈ (0, ∞).

For y < 1, by the increasingly monotonicity of f_{0}(x, y), it is deduced that

(1) if x > 1, then f_{0}(1, y) = |ψ^{(i)}(1)| < f_{0}(x, y) < |ψ^{(i)}(y)|,

(2) if x < 1, then f_{0}(x, y) < f_{0}(1, y) = |ψ^{(i)}(1)|,

(3) if y < x < 1, then f_{0}(y, y) < f_{0}(x, y),

(4) if x < y < 1, then f_{0}(x, x) < f_{0}(x, y). This implies that

f_{0}(θ, θ) = 2|ψ^{(i)}(θ)| − |ψ^{(i)}(θ^{2})| < f_{0}(x, y)  (6)

for y < 1, where θ < 1 with θ < x and θ < y.

Since f_{0}(θ, θ) is strictly increasing in θ ∈ (0, 1) such that f_{0}(1, 1) = |ψ^{(i)}(1)| > 0 and lim_{θ→0^{+}} f_{0}(θ, θ) = −∞, then the function f_{0}(θ, θ) has a unique zero θ_{0} ∈ (0, 1) and f_{0}(θ, θ) > 0 for 1 > θ > θ_{0}.

In conclusion, the function f_{0}(x, y) is positive for all x, y > θ_{0} or negative for 0 < x, y < θ_{0}. The proof of Theorem 1 is complete.

References

[1] H. Alzer, Mean-value inequalities for the polygamma functions, Aequationes Math. 61 (2001), no. 1, 151–161.

[2] H. Alzer and S. Koumandos, Sub- and superadditive properties of Fejér’s sine polynomial, Bull. London Math. Soc. 38 (2006), no. 2, 261–268.

[3] H. Alzer and S. Ruscheweyh, A subadditive property of the gamma function, J. Math. Anal. Appl. 285 (2003), 564–577.

[4] P. Gao, A subadditive property of the digamma function, RGMIA Res. Rep. Coll. 8 (2005),

   no. 3, Art. 9; Available online at http://rgmia.vu.edu.au/v8n3.html.

[5] F. Qi, S. Guo and B.-N. Guo, Note on a class of completely monotonic functions involving the polygamma functions, RGMIA Res. Rep. Coll. 10 (2007), no. 1, Art. 5; Available online

   at http://rgmia.vu.edu.au/v10n1.html.

(F. Qi) Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

   E-mail address: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@msn.com, qifeng618@qq.com,
qifeng@hpu.edu.cn, fengqi618@member.ams.org
   URL: http://rgmia.vu.edu.au/qi.html

(B.-N. Guo) School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

   E-mail address: guobaini@hpu.edu.cn