Ask for a proof of an inequality bounding the ratio of two consecutive geometric means of the first finitely many natural numbers

Question

A colleague asked me the first one of the following problems:

• For $n\in\mathbb{N}=\{1,2,3,\dotsc\}$, is the inequality \begin{equation} \frac{\sqrt[n+1]{(n+1)!}}{\sqrt[n]{n!}}>1+\frac{1}{n}+\frac{\frac{1}{2}-\ln\sqrt{2n\pi}\,}{n^2}-\frac{5}{12n^3}\label{1}\tag{1} \end{equation} valid?
• If the inequality \eqref{1} is valid, can one find the best constants $\alpha>5$ and $\beta\le5$ such that the double inequality \begin{equation} 1+\frac{1}{n}+\frac{\frac{1}{2}-\ln\sqrt{2n\pi}\,}{n^2}-\frac{\alpha}{12n^3}>\frac{\sqrt[n+1]{(n+1)!}}{\sqrt[n]{n!}}>1+\frac{1}{n}+\frac{\frac{1}{2}-\ln\sqrt{2n\pi}\,}{n^2}-\frac{\beta}{12n^3}, \quad n\in\mathbb{N}\label{2}\tag{2} \end{equation} is valid?

Notice that $n!=\Gamma(n+1)$, where $$ \Gamma(z)= \int_{0}^{\infty}t^{z-1}e^{-t} \textrm{d}t, \quad \Re(z)>0 $$ is the classical Euler gamma function.

To the best of my knowledge, in the papers [1, 2, 3, 4, 5] below, some properties, including increasing property, inequalities, and (logarithmicaly) complete monotonicity, of the functions \begin{align} &\frac{[{\Gamma(x+\alpha+1)}]^{1/(x+\alpha)}}{[{\Gamma(x+1)}]^{1/x}}, & &\frac{[\Gamma(x+1)]^{1/x}}{[\Gamma(x+1+\beta)]^{1/(x+\beta)}}\biggl(1+\frac{\beta}{x}\biggr)^\alpha, \\ &\frac{[\Gamma(x+1)]^{1/x}}{[\Gamma(x+1+\beta)]^{1/(x+\beta)}}\biggl(1+\frac{\beta}{x+1}\biggr)^\alpha,& &\frac{[\Gamma(x+a+1)]^{1/(x+a)}}{[\Gamma(x+b+1)]^{1/(x+b)}} \end{align} were investigated. These results may be useful for answering the above two questions.

References

1. H. Alzer and C. Berg, Some classes of completely monotonic functions, II, Ramanujan J. 11 (2006), no. 2, 225--248; available online at https://doi.org/10.1007/s11139-006-6510-5.
2. Chao-Ping Chen and Feng Qi, Monotonicity results for the gamma function, Journal of Inequalities in Pure and Applied Mathematics 4 (2003), no. 2, Article 44; available online at http://www.emis.de/journals/JIPAM/article282.html.
3. Feng Qi and Chao-Ping Chen, Monotonicity and convexity results for functions involving the gamma function, International Journal of Applied Mathematical Sciences 1 (2004), no. 1, 27--36.
4. Feng Qi and Bai-Ni Guo, Some logarithmically completely monotonic functions related to the gamma function, Journal of the Korean Mathematical Society 47 (2010), no. 6, 1283--1297; available online at https://doi.org/10.4134/JKMS.2010.47.6.1283.
5. J. Sandor, Sur la fonction gamma, Publ. C.R.M.P. Neuchatel, Serie I, 21 (1989), 4--7.