On rational bounds for the gamma function

Abstract

Abstract In the article, we prove that the double inequality x 2 + p 0 x + p 0 ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)$ for x ∈ ( 0 , x ∗ ) $x\in(0, x^{\ast})$ and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)<(x^{2}+1/\gamma)/(x+1/\gamma )$ for x ∈ ( x ∗ , 1 ) $x\in(x^{\ast}, 1)$ , where Γ ( x ) $\Gamma(x)$ is the classical gamma function, γ = lim n → ∞ ( ∑ k = 1 n 1 / k − log n ) = 0.577 … $\gamma=\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}1/k-\log n)=0.577\ldots$ is Euler-Mascheroni constant and p 0 = γ / ( 1 − γ ) = 1.365 … $p_{0}=\gamma/(1-\gamma )=1.365\ldots$ .

Keywords

• gamma function
• psi function
• rational bound
• completely monotonic function