The monotonicity and convexity of a function involving digamma one and their applications

Abstract

Let L(x, a) be defined on (−1, ∞)×(4/15, ∞) or (0, ∞)×(1/15, ∞) by the formulaWe investigate the monotonicity and convexity of the function x → Fa (x) = ψ (x + 1) − L(x, a), where ψ denotes the Psi function. And, we determine the best parameter a such that the inequality ψ (x + 1) < (>) L(x, a) holds for x ∈ (−1, ∞) or (0, ∞), and then, some new and very high accurate sharp bounds for pis function and harmonic numbers are presented. As applications, we construct a sequence (ln (a)) defined by ln (a) = Hn − L (n, a), which gives extremely accurate values for γ.