In real analysis, a branch of mathematics, Gautschi's inequality is an inequality for ratios of gamma functions.
It is named after Walter Gautschi.
be a positive real number, and let
Then,[1] In 1948, Wendel proved the inequalities for
[2] He used this to determine the asymptotic behavior of a ratio of gamma functions.
The upper bound in this inequality is stronger than the one given above.
In 1959, Gautschi independently proved two inequalities for ratios of gamma functions.
His lower bounds were identical to Wendel's.
One of his upper bounds was the one given in the statement above, while the other one was sometimes stronger and sometimes weaker than Wendel's.
An immediate consequence is the following description of the asymptotic behavior of ratios of gamma functions: There are several known proofs of Gautschi's inequality.
One simple proof is based on the strict logarithmic convexity of Euler's gamma function.
The resulting inequalities are: Rearranging the first of these gives the lower bound, while rearranging the second and applying the trivial estimate
A survey of inequalities for ratios of gamma functions was written by Qi.
[3] The proof by logarithmic convexity gives the stronger upper bound Gautschi's original paper proved a different, stronger upper bound, where
[4] Kershaw proved two tighter inequalities.
,[5] Gautschi's inequality is specific to a quotient of gamma functions evaluated at two real numbers having a small difference.
are positive real numbers, then the convexity of
, this leads to the estimates A related but weaker inequality can be easily derived from the mean value theorem and the monotonicity of
[7] A more explicit inequality valid for a wider class of arguments is due to Kečkić and Vasić, who proved that if
, we have: Guo, Qi, and Srivastava proved a similar-looking inequality, valid for all