 To prove the logarithmic convexity of the gamma function, it therefore suffices to observe that formula_149 has a series representation which, for positive real , consists of only positive terms.Logarithmic convexity and Jensen's inequality together imply, for any positive real numbers formula_151 and formula_152,

There are also bounds on ratios of gamma functions.The best-known is Gautschi's inequality, which says that for any positive real number and any ,

The behavior of formula_65 for an increasing positive variable is simple.