Editing Gamma function (section)

=== Inequalities ===
When restricted to the positive real numbers, the gamma function is a strictly [[logarithmically convex function]].  This property may be stated in any of the following three equivalent ways:
* For any two positive real numbers <math>x_1</math> and <math>x_2</math>, and for any <math>t \in [0, 1]</math>, <math display="block">\Gamma(tx_1 + (1 - t)x_2) \le \Gamma(x_1)^t\Gamma(x_2)^{1 - t}.</math>
* For any two positive real numbers <math>x_1</math> and <math>x_2</math>, and <math>x_2</math> > <math>x_1</math><math display="block"> \left(\frac{\Gamma(x_2)}{\Gamma(x_1)}\right)^{\frac{1}{x_2 - x_1}} > \exp\left(\frac{\Gamma'(x_1)}{\Gamma(x_1)}\right).</math>
* For any positive real number <math>x</math>, <math display="block"> \Gamma''(x) \Gamma(x) > \Gamma'(x)^2.</math>
The last of these statements is, essentially by definition, the same as the statement that <math>\psi^{(1)}(x) > 0</math>, where <math>\psi^{(1)}</math> is the [[polygamma function]] of order 1. To prove the logarithmic convexity of the gamma function, it therefore suffices to observe that <math>\psi^{(1)}</math> has a series representation which, for positive real {{mvar|x}}, consists of only positive terms.

Logarithmic convexity and [[Jensen's inequality]] together imply, for any positive real numbers <math>x_1, \ldots, x_n</math> and <math>a_1, \ldots, a_n</math>,
<math display="block">\Gamma\left(\frac{a_1x_1 + \cdots + a_nx_n}{a_1 + \cdots + a_n}\right) \le \bigl(\Gamma(x_1)^{a_1} \cdots \Gamma(x_n)^{a_n}\bigr)^{\frac{1}{a_1 + \cdots + a_n}}.</math>

There are also bounds on ratios of gamma functions. The best-known is [[Gautschi's inequality]], which says that for any positive real number {{mvar|x}} and any {{math|''s'' ∈ (0, 1)}},
<math display="block">x^{1 - s} < \frac{\Gamma(x + 1)}{\Gamma(x + s)} < \left(x + 1\right)^{1 - s}.</math>