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

Let $\mathcal{L}(x,a)$ be defined on $\left( -1,\infty \right) \times \left( 4/15,\infty \right) $ or $\left( 0,\infty \right) \times \left( 1/15,\infty \right) $ 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 $x\rightarrow F_{a}\left( x\right) =ψ\left( x+1\right) -\mathcal{L}(x,a)$, where $ψ$ denotes the Psi function. And, we determine the best parameter $a$ such that the inequality $ψ\left( x+1\right) <\left( >\right) \mathcal{L}% (x,a) $ holds for $x\in \left( -1,\infty \right) $ or $\left( 0,\infty \right) $, and then, some new and very high accurate sharp bounds for pis function and harmonic numbers are presented. As applications, we construct a sequence $\left( l_{n}\left( a\right) \right) $ defined by $l_{n}\left( a\right) =H_{n}-\mathcal{L}\left( n,a\right) $, which gives extremely accurate values for $γ$.