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I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.

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Stirling would be of great service... –  J. M. Dec 22 '10 at 16:50
    
Have you tried a proof by induction? Do you know the value of $\Gamma(\pi)$? –  cch Dec 22 '10 at 17:05
    
@cch: Yes, and yes. –  Eugene Dec 22 '10 at 17:23
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On reconsideration: Yes, and no. –  Eugene Dec 22 '10 at 17:31
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@J. M.: Maybe looking at some specific proof of Stirling's formula would help - but to me that is an asymptotic formula. –  AD. Dec 22 '10 at 19:15

5 Answers 5

up vote 6 down vote accepted

This follows directly from Gautschi's inequality, valid for $0 < s < 1$ and $x > 0$:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s}.$$

As a side note, the link is to NIST's new (as of this year) Digital Library of Mathematical Functions. The DLMF is an update of the classic Abramowitz and Stegun text, Handbook of Mathematical Functions.

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Mike, I hope you don't mind if I changed the link to link directly to the formula. In any event, DLMF gives this as the reference for Gautschi's inequality; sadly there seems to be no digital archive for the Journal of Mathematics and Physics. –  J. M. Dec 23 '10 at 8:39
    
@J.M.: That's fine. –  Mike Spivey Dec 23 '10 at 16:44

Here's a direct proof:

The gamma function satisfies $\Gamma(n+1) = n \Gamma(n)$, so the function $g(x) = \ln \Gamma(x)$ satisfies $g(n+1) = \ln n + g(n)$. In other words, the line segment between the points $(n,g(n))$ and $(n+1,g(n+1))$ has slope $\ln n$.

Moreover, $g$ is known to be a convex function for $x>0$ (cf. the Bohr–Mollerup theorem), so the point $(n+1/2, g(n+1/2))$ lies below that line segment: $g(n+1/2) < g(n+1) - \frac12 \ln n$. Exponentiating both sides gives $\Gamma(n+1/2) < \Gamma(n+1) / \sqrt{n}$, as desired.

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I suspect that Gautschi's inequality is proved in a similar way. Incidentally, $n$ doesn't have to be an integer. –  Hans Lundmark Dec 22 '10 at 19:59
    
For a proof that $\Gamma$ is log-convex (using Hölder's inequality), see PlanetMath: planetmath.org/?op=getobj&from=objects&id=3808. –  Hans Lundmark Dec 24 '10 at 15:49
    
Here is a proof of the log-convexity of $\Gamma$ using Young's Inequality. –  robjohn Jan 27 '12 at 11:54

For a completely elementary proof let

$$I_n= \int_0^{ \pi/2} \sin^n x \textrm{ d}x $$

then integration by parts gives

$$I_n = \frac{n-1}{n} I_{n-2} . \quad (1) $$

From which we get

$$I_{2n}= \frac{2n-1}{2n} \frac{2n-3}{2n-2} \cdots \frac{1}{2} \frac{\pi}{2} \quad (2) $$

and

$$I_{2n+1}= \frac{2n}{2n+1} \frac{2n-2}{2n-1} \cdots \frac{2}{3} . \quad (3) $$

Since $ \sin^{2n+1} x < \sin^{2n} x $ for $ x \in (0,\pi/2) $ we have

$$I_{2n+1}< I_{2n} . \quad (4) $$

Also, from $(1)$ we have $(2n+1)I_{2n+1} = 2nI_{2n-1} > 2nI_{2n}$ since $I_{2n-1} >I_{2n}.$

Therefore using this and $(4)$

$$ \frac{2n+1}{2n} > \frac{ I_{2n} }{ I_{2n+1} } > 1 . \quad (5) $$

But from $(2)$ and $(3)$

$$ \frac{ I_{2n} }{ I_{2n+1} } = \frac{2n+1}{2} \frac{ (2n!)^2 }{ 4^{2n} (n!)^4 } \pi .$$

Putting this in $(5)$ gives

$$ \frac{1}{ \sqrt{n} } > \frac{1}{4^n} { 2n \choose n } \sqrt{ \pi } > \frac{1}{ \sqrt{ n + 1/2} }.$$

i.e.

$$ \frac{1}{ \sqrt{n} } > \frac{ \Gamma(n + 1/2) }{ \Gamma(n+1) } > \frac{1}{ \sqrt{ n + 1/2} }.$$

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Tsk, I was expecting that it would be Moron who would post this proof... :D –  J. M. Dec 23 '10 at 1:01
    
@J.M. Yes, one develops a feel for the sort of mathematics that our members enjoy. –  Derek Jennings Dec 23 '10 at 8:29
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@J.M: Wallis! Wallis! Wallis! :-) –  Aryabhata Dec 23 '10 at 16:56

To elaborate on Shai Covo's answer:

$$2n\left[{2n\choose n} 4^{-n}\right]^2 = {1\over2}\,{3\over2}\,{3\over4}\,{5\over4}\cdots {{2n-1}\over{2n-2}}\,{{2n-1}\over{2n}} ={1\over 2}\prod_{j=2}^n\left(1+{1\over4j(j-1)}\right). $$

By Wallis's formula, the middle expression converges to $2\over\pi$. The right hand expression shows that it is strictly increasing. Therefore, multiplying by $\pi/2$ and taking square roots shows that
$$\sqrt{n\pi}{2n\choose n} 4^{-n}\uparrow 1.$$

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Since $$ \Gamma \bigg(n + \frac{1}{2}\bigg) = \frac{{(2n)!}}{{4^n n!}}\sqrt \pi $$ (see e.g. here), we have $$ \frac{{\Gamma (n + \frac{1}{2})}}{{\Gamma (n+1)}} = \frac{{{2n \choose n}}}{{4^n }}\sqrt \pi . $$ This might be very useful, but I have to check.

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Well, at least Byron found this useful... –  Shai Covo Dec 23 '10 at 1:09
    
Yes, so I'll vote it up. –  Byron Schmuland Dec 23 '10 at 3:10
    
Thanks......... –  Shai Covo Dec 23 '10 at 3:12

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