Let L(x,a) be defined on (β1,β)Γ(4/15,β) or (0,β)Γ(1/15,β) by the formula% \begin{equation*}
\mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}%
\right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\allowbreak \tfrac{%
15a-1}{45a}\right) . \end{equation*} We investigate the monotonicity and
convexity of the function xβFaβ(x)=Ο(x+1)βL(x,a), where Ο denotes the Psi function. And, we
determine the best parameter a such that the inequality \psi \left(
x+1\right) \right) \mathcal{L}% (x,a) holds for xβ(β1,β) or (0,β), and then, some new and very
high accurate sharp bounds for pis function and harmonic numbers are presented.
As applications, we construct a sequence (lnβ(a))
defined by lnβ(a)=HnββL(n,a), which
gives extremely accurate values for Ξ³.Comment: 20 page