On rational bounds for the gamma function

In the article, we prove that the double inequality (Formula presented.) holds for all x∈ (0 , 1) , we present the best possible constants λ and μ such that (Formula presented.) for all x∈ (0 , 1) , and we find the value of x^∗ in the interval (0 , 1) such that Γ (x+ 1) > (x²+ 1 / γ) / (x+ 1 / γ) for x∈ (0 , x^∗) and Γ (x+ 1) < (x²+ 1 / γ) / (x+ 1 / γ) for x∈ (x^∗, 1) , where Γ (x) is the classical gamma function, γ=limn→∞(∑k=1n1/k−logn)=0.577… is Euler-Mascheroni constant and p₀= γ/ (1 − γ) = 1.365 … .