Inequalities for binomial coefficients (Q1304616)

For any real number \(r\) with \(r>1\), let \(c_r= (2\pi(1-{1\over r}))^{-1/2}\) and \(d_r= (r-1)/(1-{1\over r})^r\). Let \(B_{2m}\) \((m= 1,2,\dots)\) be the Bernoulli numbers defined by \[ {z\over e^z-1}=1-{z\over 2}+\sum^\infty_{m=1} B_{2m}{z^{2m}\over (2m)!}. \] The author proves that if \(s\) is a positive number, then \[ c_r d^s_r e^{D_{2k+ 1}(s,r)}/\sqrt s< {rs\choose s}< c_r d^s_r e^{D_{2k}(s,r)}/\sqrt s \] for \(k= 0,1,2,\dots\), where \[ D_n(s,r)= \sum^n_{j=1} B_{2j}((rs)^{1- 2j}- s^{1- 2j}- ((r- 1)s)^{1- 2j})/2j(2j- 1) \] for \(n= 0,1,2,\dots\).