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Technische Universität Dresden

Herausgeber: Der Rektor

A monotonicity property of the

Γ-function

Hendrik Vogt^{1} and Jürgen Voigt^{1} Institut für Analysis MATH-AN-03-00

A monotonicity property of the Γ-function

Hendrik Vogt^{1} and Jürgen Voigt^{1}

Abstract. It is shown that the function x ↦ 1 + \frac{1}{x} ln Γ(x + 1) − ln(x + 1) is strictly completely monotone on (−1, ∞) and tends to one as x → −1, to zero as x → ∞. This property is derived from a suitable integral representation of ln Γ(x + 1).

MSC 2000: 33B15

Keywords: Gamma function, strictly completely monotone

The starting point of this note was an inequality,

1 ⩽ \frac{Γ(\frac{n}{2} + 1)^{\frac{n−d}{n}}}{Γ(\frac{n−d}{2} + 1)} ⩽ e^{\frac{d}{2}} ,  (1)

for all pairs of integers 0 ⩽ d ⩽ n, in [5; Lemma 2.1]. Note that the left hand side of this inequality is an immediate consequence of the logarithmic convexity of the Γ-function; see [5]. Looking for a stream-lined proof of inequality (1), we first found a proof of the more general inequality

\frac{Γ(p + 1)^{\frac{1}{p}}}{Γ(q + 1)^{\frac{1}{q}}} ⩽ e^{\frac{p}{q} −1},  (2)

valid for all 0 < q ⩽ p, and finally showed

\frac{Γ(p + 1)^{\frac{1}{p}}}{Γ(q + 1)^{\frac{1}{q}}} ⩽ \frac{p + 1}{q + 1},  (3)

for all −1 < q ⩽ p. These inequalities will be immediate consequences of the following result.

Theorem 1. The function f(x) := 1 + \frac{1}{x} ln Γ(x + 1) − ln(x + 1) is strictly completely monotone on (−1, ∞),

lim_{x→−1} f(x) = 1, lim_{x→∞} f(x) = 0,

f(0) = lim_{x→0} f(x) = 1 − γ.

(Here, γ is the Euler-Mascheroni constant, and strictly completely monotone means (−1)^{n}f^{(n)}(x) > 0 for all x ∈ (−1, ∞), n ∈ ℕ_{0}.)

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1partly supported by the Deutsche Forschungsgemeinschaft Address: Technische Universität Dresden, Fachrichtung Mathematik, 01062 Dresden, Germany e-mail: vogt@math.tu-dresden.de, voigt@math.tu-dresden.de

A monotonicity property of the Γ-function 3

Proof. The main ingredient of the proof is the integral representation

ln Γ(x + 1) = x ln(x + 1) − x + ∫_{0}^{∞} (\frac{1}{t} − \frac{1}{e^{t} − 1}) e^{−t} \frac{1}{t}(1 − e^{−xt}) dt,

which is an immediate consequence of [6; formula 1.9 (2) (p. 21)] and [6; formula 1.7.2 (18) (p. 17)]. We obtain

f(x) = ∫_{0}^{∞} (\frac{1}{t} − \frac{1}{e^{t} − 1}) e^{−t} \frac{1}{xt}(1 − e^{−xt}) dt.

The function

g(y) := \frac{1}{y}(1 − e^{−y}) = ∫_{0}^{1} e^{−sy} ds

is strictly completely monotone on ℝ. Since \frac{1}{t} − \frac{1}{e^{t}−1} > 0 for all t > 0, we conclude that f is strictly completely monotone. As y → ∞, g(y) tends to zero, and hence lim_{x→∞} f(x) = 0. The definition of f shows lim_{x→0} f(x) = 1 + ψ(1) = 1 − γ; cf. [6; formula 1.7 (4) (p. 15)]. Finally,

lim_{x→−1} f(x) = 1 − lim_{x→−1} (\frac{1}{x}( ln Γ(x + 2) − ln(x + 1)) − ln(x + 1)) = 1.

Corollary 2. Inequalities (3), (2), (1) are valid for the indicated ranges.

Proof. Inequality (3) is just a reformulation of the monotonicity of the function f from Theorem 1. Continuing (3) to the right,

\frac{p + 1}{q + 1} ⩽ \frac{p}{q} ⩽ e^{\frac{p}{q} −1} (0 < q ⩽ p),

we obtain (2). Setting q = \frac{n−d}{2}, p = \frac{n}{2} we get (1).

Remark 3. (a) In [4] it was shown that the function ξ ↦ ξ(Γ(1 + \frac{1}{ξ}))^{ξ} is increasing on (0, ∞). This fact follows immediately from our Theorem 1, because of

ln (\frac{1}{x}Γ(x + 1)^{\frac{1}{x}} ) + 1 = − ln x + \frac{1}{x}Γ(x + 1) + 1 = ln(x + 1) − ln x + f(x).

(In fact, the latter function even is strictly completely monotone as well.)

(b) For other recent results on (complete) monotonicity properties of the Γ-function we refer to [1], [2], [3].

References

[1] H. Alzer: On some inequalities for the gamma and psi functions. Math. Comp. 66 (1997), no. 217, 373–389.

[2] G. D. Anderson and S.-L. Qiu: A monotoneity property of the gamma function. Proc. Amer. Math. Soc. 125 (1997), no. 11, 3355–3362.

[3] Á. Elbert and A. Laforgia: On some properties of the gamma function. Proc. Amer. Math. Soc. 128 (2000), no. 9, 2667–2673.

[4] D. Kershaw and A. Laforgia: Monotonicity results for the gamma function. Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Natur. 119 (1985), no. 3-4, 127–133.

[5] A. Koldobsky and M. Lifshits: Average volume of sections of star bodies. In: Geometric Aspects of Functional Analysis, V. D. Milman and G. Schechtmann (eds.), Lect. Notes Math. 1745, Springer, Berlin, 2000, 119–146.

[6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi: Higher trancscendental functions. McGraw-Hill Book Company, New York-Toronto-London, 1953.