Asymptotic inequalities for \(\text{lg}\Gamma(x)\) (Q2785331)

Let \(\psi(x)\) denote the psi-function: NEWLINE\[NEWLINE\psi(x)= \frac{d}{dx} \lg\Gamma(x), \qquad x>0,NEWLINE\]NEWLINE so that \(\psi(x+1)- \psi(x)=\frac1x\), \(x>0\), and let NEWLINE\[NEWLINEg(x)\equiv \psi(x)-\lg x+\frac{1}{2x}+p(x),NEWLINE\]NEWLINE where \(p(x)\) is a suitable function s.t. \(\lim_{x\to+\infty} p'(x)= \lim_{x\to+\infty} p^{n(x)}=0\); \(p'\equiv \frac{dp}{dx}\). If \(V(x)= \frac{2x+1}{2x(x+1)}+ \lg\frac{x}{x+1}+ p'(x+1)-p(x)\) then the author proves that NEWLINENEWLINENEWLINE(i) If \(V'(x)>0\), then \(g'(x)<0\) and \(g(x)>0\); NEWLINENEWLINENEWLINE(ii) If \(V'(x)<0\), then \(g'(x)>0\) and \(g(x)<0\), NEWLINENEWLINENEWLINEand derives some of its consequences.