Inequalities related to the Gamma Function and related functions

There are numerous inequalities related to the classical gamma function(\Gamma (x)) ,incomplete gamma function(\Gamma (a;x)) 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 \Gamma (1+a) where 0 \leq a \leq 1 is:

2^{a-1} \leq \Gamma (1+a) \leq 1 .

A stronger inequality can be given by

e^{(x-1)(1- \gamma)} \leq \Gamma(1+a) \leq 1 where \gamma is the Euler-Mascheroni constant.

(1)

with equality on the lower bound iff x=1


Sketch of the proof:

The upper bound is trivial.For the lower bound,we must consider the function log(\Gamma(1+x)) and use the the mean-value theorem in the interval (0,1) to obtain:

log(\Gamma(1+x)=(x-1) \psi(1+k) where k is some number in between 0 and 1.

Noting that the di-gamma function increases on (0,\infty),we can choose k to be 1.

\psi(2)=1-\gamma  can be calculated using the recurrence relation,

\psi(1+x)=\psi(x)+\frac{1}{x}

\Rightarrow \psi(2)=\psi(1)+1

\psi(1) can be calculated by the well-known series formula.

\psi(1+z)=-\gamma+\displaystyle\sum_{n=1}^{\infty} \frac{z}{n(n+z)}

So,\psi(1)=-\gamma which means that \psi(2)=1-\gamma

 

 

(2)


 

In fact,there’s also an interesting inequality related to the ratio of gamma functions.

e^{(1-\lambda) \psi (x+\lambda)} \leq \frac{\Gamma (x+1)}{\Gamma (x+\lambda)} \leq e^{(1-\lambda) \psi (x+1)} for x \geq 0 and \lambda \geq 0

(3)

Another similar inequality is:

(\frac{x}{x+a})^{1-a} \leq \frac{\Gamma (x+a)}{x^{a}\Gamma (x)} \leq 1 where x>0.

(4)

 

Moving on to the di-gamma function,one useful definition of it is given by:

\psi(z) =-\gamma + \displaystyle\sum_{k=0}^{\infty} (\frac{1}{1+k} - \frac{1}{z+k}) .

(5)

The polygamma functions are usually defined as :

\psi^{(k)}(x)=(-1)^{k+1} \displaystyle\int_{0}^{\infty} \frac{t^{k}e^{-xt}}{1-e^{-t}} dt

However,using the above definition of \psi(x),the polygamma functions can also be written as:

\psi^{(n)}(z) = (-1)^{n+1} n! \displaystyle\sum_{k=0}^{\infty} \frac{1}{(z+k)^{n+1}}

(6)

A sharp bound for the di-gamma function is given by:

ln(x+\frac{1}{2})-\frac{1}{x} < \psi(x) <ln(x+e^{-\gamma}) -\frac{1}{x}

(7)

Another bound is given by the following theorem.

Theorem:Fix any integer m \geq 0,then

ln(x) -\frac{1}{2x}-\displaystyle\sum_{j=1}^{2m+1} \frac{B_{2j}}{2jx^{2j}} < \psi(x) <ln(x) -\frac{1}{2x}-\displaystyle\sum_{j=1}^{2m} \frac{B_{2j}}{2jx^{2j}}

where B_{2j} are the Bernoulli numbers.

(8)


 

 

The harmonic mean inequality for the gamma function given by

  1 \leq \frac{2 \Gamma (x) \Gamma (\frac{1}{x})}{ \Gamma (x)+ \Gamma (\frac{1}{x})}   for x>0.

is well-known.Akin to this is the harmonic mean inequality for the di-gamma function;

-\gamma \leq \frac{2\psi (x) \psi (\frac{1}{x})}{\psi (x)+ \psi (\frac{1}{x})} for x>0

(9)

Many inequalities related to \psi(x) and \psi(\frac{1}{x}) can be found in the work of Horst Alzer and Graham Jameson.One such inequality is:

\psi(1+x) \psi(1-x) <\gamma^2 where 0<x<1.

(10)

There’s also a sharp bound for the absolute value of the polygamma function \psi^{(m)}(x) which is given by:

\frac{(k-1)!}{[x+(\frac{(k-1)!}{|\psi^{(k)}(1)|})^{\frac{1}{k}}]^k}+\frac{k!}{x^{k+1}}<|\psi^{(k)}(x)|<\frac{(k-1)!}{(x+\frac{1}{2})^k}+\frac{k!}{x^{k+1}} forx>0 and k \in \mathbb{N}

(11)

where \psi^{(k)}(1) admits an infinite series representation:

\psi^{(k)}(1)=(-1)^{k+1} k! \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{k+1}}

Note that this formula isn’t valid for k=0 as the harmonic series would diverge.In fact,if you wish to find \psi^{(0)}(x)=\psi(x)(i.e the digamma function) at x=1,you can use the series formula(see (2),it has already been done)

References:

  1. Guo, Bai-Ni & Qi, Feng & Zhao, Jiao-Lian & Luo, Qiu-Ming. (2015). Sharp inequalities for polygamma functions. Mathematica Slovaca. 65. 103-. 10.1515/ms-2015-0010.
  2. ] W. Gautschi. Some mean value inequalities for the gamma function. SIAM J. Math. Anal., 5:282–292, 1974.
  3. Andrea Laforgia and Pierpaolo Natalini, On some Inequalities for the Gamma Function, pages 261-267.
  4. 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.
  5. 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.

 

 

 

 

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