Mean Value Inequalities for the Digamma Function

Let psi be the digamma function and let L(a, b) = (b - a)/ log(b/a) be the logarithmic mean of a and b. We prove that the inequality(*) (b - a)psi( root ab) < (L(a, b) - a)psi(a) + (b - L(a, b))psi (b)holds for all real numbers a and b with b > a > alpha(0). Here, alpha(0) asymptotic to 0.56155 is the only positive solution of50 psi'(x) + 3x psi''(x) = 0.The constant lower bound alpha 0 is best possible. This refines a result of Chu, Zhang and Tang, who showed that (*) is valid for b > a >= 2. Moreover, we prove that the following companion to (*) holds for all a and b with b > a > 0,(L(a,b)-a) psi (a)+(b-L(a,b))psi(b)< (b-a)psi (a+b/2)