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I am reading a book on the probabilistic method, and the following claim was made:

$\dbinom{y}{n}$ is convex.

Why is this the case?

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  • $\begingroup$ I think it's even convex if you take the log of it $\endgroup$
    – user66081
    Feb 5, 2014 at 21:18
  • $\begingroup$ can you prove it for me? $\endgroup$
    – pyrrhic
    Feb 5, 2014 at 21:24
  • $\begingroup$ @Henry Concavity with respect to n is out of the question, by inspection for y=6 or by comparison with a normal density when y is large. $\endgroup$
    – Did
    Feb 5, 2014 at 21:42
  • $\begingroup$ @Did: Fair enough - I will delete that comment $\endgroup$
    – Henry
    Feb 5, 2014 at 21:44

1 Answer 1

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Hint:

Compare $\dfrac{\dbinom{y+1}{n} }{\dbinom{y}{n}}$ with $\dfrac{\dbinom{y}{n} }{\dbinom{y-1}{n}}$ for $y \gt n$

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  • $\begingroup$ I admit that I don't see how this leads to a proof that the sequence is convex in y. But I think that considering the differences instead of the ratios, will work. $\endgroup$
    – Simon
    Jan 5 at 17:52

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