# Bounds for the Gamma Function

Bounds for the Gamma Function We improve the upper bound of the following inequalities for the gamma function $$\Gamma$$ Γ due to H. Alzer and the author. \begin{aligned} \exp \left( -\frac{1}{2}\psi (x+1/3)\right)<\frac{\Gamma (x)}{x^xe^{-x}{\sqrt{2\pi }}} <\exp \left( -\frac{1}{2}\psi (x)\right) . \end{aligned} exp - 1 2 ψ ( x + 1 / 3 ) < Γ ( x ) x x e - x 2 π < exp - 1 2 ψ ( x ) . We also prove the following new inequalities: for $$x\ge 1$$ x ≥ 1 \begin{aligned} {\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a_*\right) ^{\frac{1}{4}}<\Gamma (x+1)<{\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a^*\right) ^{\frac{1}{4}} \end{aligned} 2 π x x e - x x 2 + x 3 + a ∗ 1 4 < Γ ( x + 1 ) < 2 π x x e - x x 2 + x 3 + a ∗ 1 4 with the best possible constants $$a_*=\frac{e^4}{4\pi ^2}-\frac{4}{3}=0.049653963176\ldots$$ a ∗ = e 4 4 π 2 - 4 3 = 0.049653963176 … , and $$a^*=1/18=0.055555\ldots$$ a ∗ = 1 / 18 = 0.055555 … , and for $$x\ge 0$$ x ≥ 0 \begin{aligned} \exp \left[ x\psi \left( \frac{x}{\log (x+1)}\right) \right] \le \Gamma (x+1)\le \exp \left[ x\psi \left( \frac{x}{2}+1\right) \right] , \end{aligned} exp x ψ x log ( x + 1 ) ≤ Γ ( x + 1 ) ≤ exp x ψ x 2 + 1 , where $$\psi$$ ψ is the digamma function. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# Bounds for the Gamma Function

, Volume 72 (2) – May 27, 2017
10 pages

/lp/springer_journal/bounds-for-the-gamma-function-raQHlnHSi6
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0698-0
Publisher site
See Article on Publisher Site

### Abstract

We improve the upper bound of the following inequalities for the gamma function $$\Gamma$$ Γ due to H. Alzer and the author. \begin{aligned} \exp \left( -\frac{1}{2}\psi (x+1/3)\right)<\frac{\Gamma (x)}{x^xe^{-x}{\sqrt{2\pi }}} <\exp \left( -\frac{1}{2}\psi (x)\right) . \end{aligned} exp - 1 2 ψ ( x + 1 / 3 ) < Γ ( x ) x x e - x 2 π < exp - 1 2 ψ ( x ) . We also prove the following new inequalities: for $$x\ge 1$$ x ≥ 1 \begin{aligned} {\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a_*\right) ^{\frac{1}{4}}<\Gamma (x+1)<{\sqrt{2\pi }}x^xe^{-x}\left( x^2+\frac{x}{3}+a^*\right) ^{\frac{1}{4}} \end{aligned} 2 π x x e - x x 2 + x 3 + a ∗ 1 4 < Γ ( x + 1 ) < 2 π x x e - x x 2 + x 3 + a ∗ 1 4 with the best possible constants $$a_*=\frac{e^4}{4\pi ^2}-\frac{4}{3}=0.049653963176\ldots$$ a ∗ = e 4 4 π 2 - 4 3 = 0.049653963176 … , and $$a^*=1/18=0.055555\ldots$$ a ∗ = 1 / 18 = 0.055555 … , and for $$x\ge 0$$ x ≥ 0 \begin{aligned} \exp \left[ x\psi \left( \frac{x}{\log (x+1)}\right) \right] \le \Gamma (x+1)\le \exp \left[ x\psi \left( \frac{x}{2}+1\right) \right] , \end{aligned} exp x ψ x log ( x + 1 ) ≤ Γ ( x + 1 ) ≤ exp x ψ x 2 + 1 , where $$\psi$$ ψ is the digamma function.

### Journal

Results in MathematicsSpringer Journals

Published: May 27, 2017

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