# Is it possible to show this expression is positive and bounded between 0 and 1?

#### chamilka

##### New member
Hi all,
I am working with some beta functions. I want to show that the following is positive and bounded between 0 and 1. Is it possible to show this?

$$\frac{ \frac{B( a + b , \frac{2}{ c} )}{B(a, \frac{2}{c}) } - \big\{\frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) }\big\}^{2} }{ \frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) } - \big\{\frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) }\big\}^{2} }$$
where a, b and c are positive real numbers.

Thanks for your help.

Here is a image of the above expression.

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#### Sudharaka

##### Well-known member
MHB Math Helper
Hi all,
I am working with some beta functions. I want to show that the following is positive and bounded between 0 and 1. Is it possible to show this?

$$\frac{ \frac{B( a + b , \frac{2}{ c} )}{B(a, \frac{2}{c}) } - \big\{\frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) }\big\}^{2} }{ \frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) } - \big\{\frac{B( a + b , \frac{1}{ c} )}{B(a, \frac{1}{c}) }\big\}^{2} }$$
where a, b and c are positive real numbers.

Thanks for your help.

Here is a image of the above expression.

Hi chamilka,

This certainly seem to be bounded between 0 and 1 although I can't think of a way to prove it. Where did you encounter this fraction? You may be interested by the following journal article which includes lots of inequalities for Beta and Gamma functions.

http://rgmia.org/papers/v2n3/begaostr.pdf

Kind Regards,
Sudharaka.

#### chamilka

##### New member
Hi chamilka,

This certainly seem to be bounded between 0 and 1 although I can't think of a way to prove it. Where did you encounter this fraction? You may be interested by the following journal article which includes lots of inequalities for Beta and Gamma functions.

http://rgmia.org/papers/v2n3/begaostr.pdf

Kind Regards,
Sudharaka.
Thank you sudharaka. I am checking the reference you suggested. I hope that by considering the Beta integrals it can be proved. Let me to try it!!

#### Sudharaka

##### Well-known member
MHB Math Helper
Hi chamilka,

$0<x^b<1\mbox{ for }0<x<1$

$\therefore x^{a+b}(1-x)^{\frac{1}{c}}<x^{a}(1-x)^{\frac{1}{c}}\mbox{ for }0<x<1$

$\Rightarrow \int_0^1 x^{a+b}(1-x)^{\frac{1}{c}}\,dx<\int_0^1 x^{b}(1-x)^{\frac{1}{c}}\,dx$

$\Rightarrow \beta\left(a+b,\frac{1}{c}\right)<\beta\left(a, \frac{1}{c}\right)$

The value of the Beta function is always positive. Therefore,

$0<\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}<1~~~~~~~~~~(1)$

Using the relationship between Beta and Gamma function it could be shown that,

$\frac{ \beta\left(a+b,\frac{2}{c} \right)}{\beta \left(a, \frac{2}{c}\right)}\div\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}=\frac{\beta\left(a+\frac{2}{c},a+b+\frac{1}{c} \right)}{\beta\left(a+\frac{1}{c},a+b+\frac{2}{c} \right)}$

Using Theorem 2 on page 7 in the article that I linked in my previous post we can get,

$\frac{\beta\left(a+\frac{2}{c},a+b+\frac{1}{c} \right)}{\beta\left(a+\frac{1}{c},a+b+\frac{2}{c} \right)}<1$

$\therefore \frac{ \beta\left(a+b,\frac{2}{c} \right)}{\beta \left(a, \frac{2}{c}\right)}<\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}$

$\Rightarrow \frac{ \beta\left(a+b,\frac{2}{c} \right)}{\beta \left(a, \frac{2}{c}\right)}-\left(\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}\right)^2<\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}-\left(\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}\right)^2$

By (1) it is clear that the right hand side of the inequality is positive. Therefore,

$\frac{\frac{ \beta\left(a+b,\frac{2}{c} \right)}{\beta \left(a, \frac{2}{c}\right)}-\left(\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}\right)^2}{\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}-\left(\frac{ \beta\left(a+b,\frac{1}{c} \right)}{\beta \left(a, \frac{1}{c}\right)}\right)^2}<1$

By (1) the denominator is positive, so it remains to show that the numerator is positive. I'll have to think how to do that....

Kind Regards,
Sudharaka.