<s>[INST] Show that $G\left(n,p\right)e^{-\frac{1}{12np\left(1-p\right)}}<{n\choose pn} < G\left(n, p\right)$ The problem:
Using the inequalities
$\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}e^{\left(\frac{1}{12n}-\frac{1}{360n^{3}}\right)} < n! < \sqrt{2\pi n}\left(\frac{n}{e}\right)^ne^{\frac{1}{12n}}$
show that, for a constant $p$ such that $0<p<1$, where $pn$ is an integer,
$G\left(n, p\right)=\frac{2^{nH\left(p\right)}}{\sqrt{2\pi n p\left(1-p\right)}}$, and $H\left(p\right)$ is the binary entropy function, show that
$G\left(n,p\right)e^{-\frac{1}{12np\left(1-p\right)}}<{n\choose pn} < G\left(n, p\right)$
My work so far:
First I have noted that
${n\choose pn} = \frac{n!}{\left(pn\right)!\left(\left(1-p\right)n\right)!}$
$ \approx \frac{\sqrt{2\pi n} \left( \frac{n}{e} \right)^{n}}{\left(pn\right)!\left(\left(1-p\right)n\right)!}$
$\approx \frac{ \sqrt{2\pi \left(1-p\right)n}\left( \frac{\left(1-p\right)n}{e}\right)^{\left(1-p\right)n}}{2\pi \sqrt{\left(1-2p\right)}\left(\frac{pn}{e}\right)^{pn} \left(\frac{\left(1-2p\right)n}{e}\right)^{\left(1-2p\right)n}} e^\left({\frac{1}{12\left(1-p\right)n}}-\frac{1}{12pn+1}-\frac{1}{12\left(1-2p\right)n+1}\right)$
$\leq 2^{nH\left(p\right)} \frac{e^{\frac{1}{12n}}}{\sqrt{2\pi np\left(1-p\right)}}$
and from here I am not sure where to go. Any hints or help would be greatly appreciated.
 [/INST] I put $q=1-p$. First, I shall use the inequalities
$$
\sqrt {2\pi n} \left( {\frac{n}{e}} \right)^n  < n! < \sqrt {2\pi n} \left( {\frac{n}{e}} \right)^n e^{\frac{1}{{12n}}} 
$$
which hold for all positive integer $n$. We obtain
$$
\binom{n}{pn} > \frac{{\sqrt {2\pi n} \left( {\frac{n}{e}} \right)^n }}{{\sqrt {2\pi pn} \left( {\frac{{pn}}{e}} \right)^{pn} e^{\frac{1}{{12pn}}} \sqrt {2\pi qn} \left( {\frac{{qn}}{e}} \right)^{qn} e^{\frac{1}{{12qn}}} }} = \frac{{2^{nH(p)} }}{{\sqrt {2\pi pqn} }}e^{ - \frac{1}{{12npq}}} .
$$
To prove the upper bound, I shall use the inequalities
$$
\sqrt {2\pi n} \left( {\frac{n}{e}} \right)^n e^{\frac{1}{{12n+1}}} < n! < \sqrt {2\pi n} \left( {\frac{n}{e}} \right)^n e^{\frac{1}{{12n}}} 
$$
which hold for all positive integer $n$ (cf. H. Robbins, A remark on Stirling's formula, Amer. Math. Monthly 62 (1955), pp. 26–29.). We deduce
$$
\binom{n}{pn} < \frac{{\sqrt {2\pi n} \left( {\frac{n}{e}} \right)^n e^{\frac{1}{{12n}}} }}{{\sqrt {2\pi pn} \left( {\frac{{pn}}{e}} \right)^{pn} e^{\frac{1}{{12np + 1}}}\sqrt {2\pi qn} \left( {\frac{{qn}}{e}} \right)^{qn} e^{\frac{1}{{12nq + 1}}}}} = \frac{{2^{nH(p)} }}{{\sqrt {2\pi pqn} }}e^{\frac{1}{{12n}} - \frac{1}{{12np + 1}} - \frac{1}{{12nq + 1}}} .
$$
Finally, note that
$$
\frac{1}{{12n}} - \frac{1}{{12pn + 1}} - \frac{1}{{12qn + 1}} =  - \frac{{144n^2 (pq + 1) + 12n - 1}}{{12n(12np + 1)(12nq + 1)}} < 0.
$$
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