Monotonicity and convexity of the function x√Г(x+1)/ (x+α}√Г(x+α+1)

For α>0 a real number, the function x√Γ(x+1)/(x+α)√Γ(x+α+1) is increasing with x∈(x0,∞) and logarithmically concave with x∈[1,∞), wherex0∈(0,1) is a constant. Moreover, some monotonicity and convexity results and inequalities of functions involving the gamma function and polygamma functions are obtained as corollaries and by-products.