Inequalities And Complete Monotonicity Properties For Some Special Functions

In this paper, the inequalities and the complete monotonicity properties for some special functions are proved. The concrete result is as follows:1,The inequalities of the type are named Turan-type inequalities. We prove that the inequality hold for all integers n≥0, where is the normalized binomial mid-coefficient.2,LetΓdenotes gamma function, the logarithmic derivative of the gamma functionψ=Γ'/Γis side to be the psi (or digamma) function. In 2005 D. Kershaw proved that for real x≥0 and 0 0 and a∈R, let We prove that the function x→fa(x) is strictly decreasing on (0,∞) if and only if a≥1/2, and the function f1/2 is logarithmically completely monotonic on (0,∞). From the monotonicity of the function f1/2, we obtain the inequality where A(x,x+1),G(x,x+1)and I(x,x+1)are the arithmetic,geometric and the exponential means of two positive numbers x and x+1,respectively.