On rational bounds for the gamma function

• Published in: Journal of inequalities and applications Vol. 2017; no. 1; pp. 210 - 17
• Main Authors: Yang, Zhen-Hang, Qian, Wei-Mao, Chu, Yu-Ming, Zhang, Wen
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2017; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; completely monotonic function; Gamma function; Mathematics; Mathematics and Statistics; psi function; rational bound; rational bound; 33B15; psi function; gamma function; 41A60; 26D07; completely monotonic function
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-017-1484-y

In the article, we prove that the double inequality x 2 + p 0 x + p 0 < Γ ( x + 1 ) < x 2 + 9 / 5 x + 9 / 5 holds for all x ∈ ( 0 , 1 ) , we present the best possible constants λ and μ such that λ ( x 2 + 9 / 5 ) x + 9 / 5 ≤ Γ ( x + 1 ) ≤ μ ( x 2 + p 0 ) x + p 0 for all x ∈ ( 0 , 1 ) , and we find the value of x ∗ in the interval ( 0 , 1 ) such that Γ ( x + 1 ) > ( x 2 + 1 / γ ) / ( x + 1 / γ ) for x ∈ ( 0 , x ∗ ) and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) for x ∈ ( x ∗ , 1 ) , where Γ ( x ) is the classical gamma function, γ = lim n → ∞ ( ∑ k = 1 n 1 / k − log n ) = 0.577 … is Euler-Mascheroni constant and p 0 = γ / ( 1 − γ ) = 1.365 …  .