## 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 $x=1$

## Connection between a symmetric linear transformation and the unit sphere

There is an interesting correspondence among the quadratic form of a symmetric linear transformation $T:V \mapsto V$on a real Euclidean space,the extreme values of the sphere and the eigenvectors of $T$

Let $Q(x)=(T(x),x)$ be the quadratic form associated with a symmetric linar transformation which maps $V$ into itself,then the set of elements $u$ in V satisfying $\langle u,u \rangle=1$ is called the unit sphere of $V$