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:

$\frac{(k-1)!}{[x+(\frac{(k-1)!}{|\psi^{(k)}(1)|})^{\frac{1}{k}}]^k}+\frac{k!}{x^{k+1}}<|\psi^{(k)}(x)|<\frac{(k-1)!}{(x+\frac{1}{2})^k}+\frac{k!}{x^{k+1}}$ for $x>0$ 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:

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.
] W. Gautschi. Some mean value inequalities for the gamma function. SIAM J. Math. Anal., 5:282–292, 1974.
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.

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