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

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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 Von 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

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