If $0 < x < 1$, $y > 1$ and $xy < 1$, $$ \frac{\binom{n}{nx,nx,\cdots,n(1 - yx)}}{\binom{n}{nx,n(1-x)}} \sim \left(\frac{(1-x)^{y(1-x)}}{(1- xy)^{1-xy}}\right)^{\!n} = o(1) $$ only if $$ \frac{(1-x)^{y(1-x)}}{(1- xy)^{1-xy}} < 1. $$ But I cannot find the appropriate way to prove this. Hence I am asking all for your kind help to solve this problem.
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1$\begingroup$ fix $y>1$ and take $f(x)=(1-xy)\log (1-xy)-y(1-x)\log(1-x)$ defined on $[0,1/y]$; simple computation shows that $f' >0 , x >0$ so $f$ increases strictly and $f(0)=0$ hence $f(x) >0$ etc $\endgroup$– ConradFeb 5, 2021 at 22:04
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$\begingroup$ Thanks @Conrad for the proof. I elaborated it further as, $f' = -y (\log (1-xy) +1) + y(\log(1-x) + 1) = y \log\frac{1-x}{1-xy} > 0$. $\endgroup$– Subhankar GhosalFeb 6, 2021 at 0:48
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$\begingroup$ @Conrad kindly add your answer in answer mode. I have to cite this in a paper. $\endgroup$– Subhankar GhosalFeb 8, 2021 at 13:29
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