Tamsui Oxford Journal of Mathematical Sciences 25(1) (2009) 9-14 Aletheia University
A Class of Completely Monotonic Functions Related to the Remainder of Binet’s Formula with Applications ∗ Senlin Guo† Department of Mathematics, Zhongyuan University of Technology, Zhengzhou City, Henan Province, 450007, China
and Feng Qi‡ Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China Received January 7, 2007, Accepted March 13, 2007.
Abstract In the note, the complete monotonicity of difference between remainders of Binet’s formula and the star-shaped and subadditive properties of the remainder of Binet’s formula are proved.
Keywords and Phrases: Completely monotonic function; Star-shaped function; Subadditive function; Remainder; Binet’s formula; Gamma function. ∗
2000 Mathematics Subject Classification. 26A51; 33B15. E-mail:
[email protected] ‡ E-mail:
[email protected] †
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1. Introduction It is well-known [1, 4, 6] that a function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and (−1)n f (n) (x) ≥ 0 for x ∈ I and n ≥ 0, a function f (x) is said to be star-shaped on (0, ∞) if f (αx) ≤ αf (x) for x ∈ (0, ∞) and all 0 < α < 1, a function f is said to be superadditive on (0, ∞) if f (x + y) ≥ f (x) + f (y) for all x, y > 0, and a function f is said to be subadditive if −f is superadditive. In [4, p. 453], it was presented that a star-shaped function must be superadditive. The noted Binet’s formula [3, p. 11] states that √ 1 ln x − x + ln 2π + θ(x) (1.1) ln Γ(x) = x − 2 R∞ for x > 0, where Γ(x) = 0 tx−1 e−t dt stands for Euler’s gamma function and Z ∞ 1 1 e−xt 1 − + dt (1.2) θ(x) = et − 1 t 2 t 0 is called the remainder of Binet’s formula (1.1). For real numbers p > 0, q ∈ R and r 6= 0, define fp,q,r (x) = r[θ(px) − qθ(x)],
x ∈ (0, ∞).
(1.3)
The aims of this note are to establish the complete monotonicity of fp,q,r (x) and the star-shaped and subadditive properties of θ(x). The main results are the following theorems. Theorem 1. If fp,q,r (x) is completely monotonic in (0, ∞), then either r > 0 and (p, q) ∈ D1 or r < 0 and (p, q) ∈ D2 , where D1 = (0, 1] × (−∞, 1] ∪ (1, ∞) × (−∞, 1)
(1.4)
D2 = (0, 1) × (1, ∞) ∪ [1, ∞) × [1, ∞).
(1.5)
and Theorem 2. If either r > 0 and q ≤ min{1, 1/p} or r < 0 and q ≥ max{1, 1/p}, then fp,q,r (x) is completely monotonic in (0, ∞). Theorem 3.
1. The function −θ(x) is star-shaped in (0, ∞).
2. The function θ(x) is subadditive in (0, ∞).
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2.A lemma In order to prove above theorems, the lemma below is necessary. Lemma 1.
1. The function δ(t) =
et
1 1 1 − + −1 t 2
(2.1)
for t > 0 is strictly increasing onto (0, 1/2); the derivative of δ(t) is strictly decreasing onto (0, 1/12). 2. If 0 < α < 1, then αδ(t) < δ(αt) < 1 · δ(t)
(2.2)
for t > 0. The constants α and 1 in (2.2) are the best possible. 3. If α > 1, inequality (2.2) is reversed and the constants α and 1 in (2.2) are also the best possible. Proof. The decreasing monotonicity of δ 0 (t) has been verified in [5]. From this, the increasing monotonicity of δ(t) can be deduced readily. For 0 < α < 1, it is clear that δ(αt) < δ(t) for t > 0. The right hand side inequality in (2.2) is proved. For 0 < α < 1, let g(t) = δ(αt) − αδ(t) for t > 0. Since limt→0+ δ(t) = 0, then limt→0+ g(t) = 0. Since δ 0 (t) is decreasing, then g 0 (t) = α[δ 0 (αt) − δ 0 (t)] > 0, and g(t) is strictly increasing, then g(t) = δ(αt) − αδ(t) > 0. The left hand side inequality in (2.2) follows. Since limt→∞ δ(t) = 1/2, then limt→∞ [δ(αt)/δ(t)] = 1. Since limt→0+ δ 0 (t) = 1/12, then limt→0+ [δ(αt)/δ(t)] = limt→0+ [αδ 0 (αt)/δ 0 (t)] = α by L’Hˆopital’s rule. Therefore, the constants α and 1 in (2.2) are the best possible. By a similar argument, the reversed inequality (2.2) for α > 1 can be proved. The proof of Lemma 1 is complete. 2
3. Proofs of theorems Now we are in a position to prove our theorems.
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Proof of Theorem 1. Utilization of (1.1) and straightforward computation gives q−1 0 −fp,q,r (x) = r qψ(x) − pψ(px) + + (p − q) ln x + p ln p , (3.1) 2x where ψ(x) = Γ0 (x)/Γ(x). In [2, p. 893], the following formula is given for x > 0: ∞ x 1 X , (3.2) ψ(x) = −γ − + x n=1 n(n + x) where γ is Euler-Mascheroni’s constant. Substituting (3.2) into (3.1) leads to 1−q 0 −fp,q,r (x) = r p ln p + (p − q)γ + + (p − q) ln x + ω(x) , (3.3) 2x where ω(x) = qx
∞ X n=1
∞ X 1 1 2 −p x n(n + x) n(n + px) n=1
(3.4)
0 (x) is with limx→0+ ω(x) = 0. If fp,q,r (x) is completely monotonic, then −fp,q,r nonnegative in (0, ∞). Hence 1 r(1 − q) 0 0 ≤ lim+ [−fp,q,r (x)] = r[p ln p + (p − q)γ] + lim+ + r(p − q)x ln x . x→0 x→0 x 2
From this, it is concluded that either r(1 − q) > 0 or r(1 − q) = 0 and r(1 − p) ≥ 0. The proof of of Theorem 1 is complete. 2 Proof of Theorem 2. Direct calculation yields Z ∞ Z ∞ δ(u) −pxu δ(t) −xt fp,q,r (x) = r e du − q e dt u t 0 0 Z ∞ Z ∞ δ(t/p) −xt δ(t) −xt =r e dt − q e dt t t 0 0 Z ∞ e−xt , k(p, q, r; t) dt. t 0 By (2.2), it is reasoned that k(p, q, r; t) > 0 if p, q and r satisfy one of the following conditions:
A Class of Completely Monotonic Functions
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1. r > 0, 0 < p < 1 and q ≤ 1; 2. r > 0, p > 1 and q ≤ 1/p; 3. r < 0, 0 < p < 1 and q ≥ 1/p; 4. r < 0, p > 1 and q ≥ 1. For p = 1, then k(p, q, r; t) = r(1 − q)δ(t) ≥ 0 if either r > 0, p = 1 and q ≤ 1 or r < 0, p = 1 and q ≥ 1. In conclusion, if r > 0 and q ≤ min{1, 1/p} or r < 0 and q ≥ max{1, 1/p}, then k(p, q, r; t) ≥ 0, which implies that fp,q,r (x) is completely monotonic. 2 Proof.[Proof of Theorem 3] Applying r = 1 and 0 < p = q < 1 in Theorem 2 yields fp,q,r (x) = θ(px) − pθx ≥ 0. Hence −θ(x) is star-shaped in (0, ∞), which implies that −θ(x) is superadditive. Consequently, θ(x) is subadditive. 2
References [1] A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203–1215. [2] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, New York, 2000. [3] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, 1966. [4] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [5] F. Qi, A monotonicity result of a function involving the exponential function and an application, RGMIA Res. Rep. Coll. 7 (2004), no.3, Art. 16, 507–509; Available online at http://rgmia.vu.edu.au/v7n3.html.
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[6] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, New Jersey, 1941.
