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 and , and for any , For any two positive real numbers and , and > For any positive real number , The last of these statements is, essentially by definition, the same as the statement that , where is the polygamma function of order 1. To prove the logarithmic convexity of the gamma function, it therefore suffices to observe that has a series representation which, for positive real , consists of only positive terms.
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 and , and for any , For any two positive real numbers and , and > For any positive real number , The last of these statements is, essentially by definition, the same as the statement that  where is the polygamma function of order 1. To prove the logarithmic, convexity of the gamma function, it therefore suffices to observe that has a series representation which, for positive real , consists of only positive terms.