There are numerous inequalities related to the classical gamma function() ,incomplete gamma function() and the psi function(also called the di-gamma function which is simply the derivative of the logarithm of the gamma function).
One of the simplest bounds for the function where is:
A stronger inequality can be given by
where is the Euler-Mascheroni constant.
with equality on the lower bound iff
Sketch of the proof:
The upper bound is trivial.For the lower bound,we must consider the function and use the the mean-value theorem in the interval to obtain:
where k is some number in between 0 and 1.
Noting that the di-gamma function increases on ,we can choose k to be 1.
can be calculated using the recurrence relation,
can be calculated by the well-known series formula.
So, which means that
In fact,there’s also an interesting inequality related to the ratio of gamma functions.
Another similar inequality is:
Moving on to the di-gamma function,one useful definition of it is given by:
The polygamma functions are usually defined as :
However,using the above definition of ,the polygamma functions can also be written as:
A sharp bound for the di-gamma function is given by:
Another bound is given by the following theorem.
Theorem:Fix any integer ,then
where are the Bernoulli numbers.
The harmonic mean inequality for the gamma function given by
is well-known.Akin to this is the harmonic mean inequality for the di-gamma function;
Many inequalities related to and can be found in the work of Horst Alzer and Graham Jameson.One such inequality is:
There’s also a sharp bound for the absolute value of the polygamma function which is given by:
where admits an infinite series representation:
Note that this formula isn’t valid for as the harmonic series would diverge.In fact,if you wish to find (i.e the digamma function) at ,you can use the series formula(see (2),it has already been done)
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- Andrea Laforgia and Pierpaolo Natalini, On some Inequalities for the Gamma Function, pages 261-267.
- Qi, Feng & Guo, Sen-Lin. (1999). Inequalities for the Incomplete Gamma and Related Functions. Mathematical Inequalities and Applications. 2. 47-. 10.7153/mia-02-05.
- Alzer, Horst & Jameson, Graham. (2017). A harmonic mean inequality for the digamma function and related results. Rendiconti del Seminario Matematico della Università di Padova. 137. 203-209. 10.4171/RSMUP/137-10.