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Prove x > 2y

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,690
Hi MHB,

Problem:

Assume $x>0$, and $y$ satisfy that $e^y=\dfrac{x}{1-e^{-x}}$, prove that $x>2y$.

Attempt:

I first tried to express $y$ in terms of $x$ and get:

$y=x+\ln x- \ln (e^x-1)$

and I am aware that one of the method to prove the intended result is to rewrite the equation above as

$y-\dfrac{x}{2}=\dfrac{x}{2}+\ln x- \ln (e^x-1)$

and if I can prove $\dfrac{x}{2}+\ln x- \ln (e^x-1)<0$, which also implies $y-\dfrac{x}{2}<0$, then the result is proved.

Now, I see it that no inequality theorems that I know of could be applied to the expression $\dfrac{x}{2}+\ln x- \ln (e^x-1)$ and hence I get stuck, so stuck that I wish to drop this problem behind my mind!

But, I hope to solve it nonetheless and that's why I posted it here and hope someone can chime in to help me out.

Thanks in advance.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Prove x>2y

Can you show that the given definition of $y$ and the inequality implies:

\(\displaystyle \sinh\left(\frac{x}{2} \right)>\frac{x}{2}\) ?

After that, a bit of calculus is all you need to demonstrate this is true. :D
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,690
Re: Prove x>2y

Can you show that the given definition of $y$ and the inequality implies:

\(\displaystyle \sinh\left(\frac{x}{2} \right)>\frac{x}{2}\) ?
We can simplify the terms of natural logarithm in $\dfrac{x}{2}+\ln x- \ln (e^x-1)$ and get:

$\dfrac{x}{2}+\ln x- \ln (e^x-1)=\dfrac{x}{2}+\ln \dfrac{x}{e^x-1}$

and to be honest with you, I don't understand how to get what you obtained.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Prove x>2y

I began with the given inequality:

\(\displaystyle x>2y\)

and this implies:

\(\displaystyle e^x>e^{2y}=\left(e^y \right)^2=\left(\frac{x}{1-e^{-x}} \right)^2=\left(\frac{xe^x}{e^x-1} \right)^2\)

Hence:

\(\displaystyle e^x>\frac{x^2e^{2x}}{\left(e^x-1 \right)^2}\)

Multiplying through by \(\displaystyle \frac{\left(e^x-1 \right)^2}{e^{2x}}>0\) we obtain:

\(\displaystyle e^{-x}\left(e^x-1 \right)^2>x^2\)

Now this implies:

\(\displaystyle e^{-\frac{x}{2}}\left(e^x-1 \right)>x\)

\(\displaystyle e^{\frac{x}{2}}-e^{-\frac{x}{2}}>x\)

Dividing through by 2:

\(\displaystyle \frac{e^{\frac{x}{2}}-e^{-\frac{x}{2}}}{2}>\frac{x}{2}\)

Using the definition of the hyperbolic sine function, we may now write:

\(\displaystyle \sinh\left(\frac{x}{2} \right)>\frac{x}{2}\)
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,690
Re: Prove x>2y

Ah, I see...thanks!
 

chisigma

Well-known member
Feb 13, 2012
1,704
Re: Prove x>2y

Hi MHB,

Problem:

Assume $x>0$, and $y$ satisfy that $e^y=\dfrac{x}{1-e^{-x}}$, prove that $x>2y$.

Attempt:

I first tried to express $y$ in terms of $x$ and get:

$y=x+\ln x- \ln (e^x-1)$

and I am aware that one of the method to prove the intended result is to rewrite the equation above as

$y-\dfrac{x}{2}=\dfrac{x}{2}+\ln x- \ln (e^x-1)$

and if I can prove $\dfrac{x}{2}+\ln x- \ln (e^x-1)<0$, which also implies $y-\dfrac{x}{2}<0$, then the result is proved.

Now, I see it that no inequality theorems that I know of could be applied to the expression $\dfrac{x}{2}+\ln x- \ln (e^x-1)$ and hence I get stuck, so stuck that I wish to drop this problem behind my mind!

But, I hope to solve it nonetheless and that's why I posted it here and hope someone can chime in to help me out.

Thanks in advance.
I'm afraid that You have to try the McLaurin expansion around x = 0 of the logarhitm of the function...


$\displaystyle y = \ln x - \ln (1 - e^{-x})\ (1)$


Starting from f(0) and using l'Hopital rule You have...


$\displaystyle \lim_{x \rightarrow 0} \frac{x}{1 - e^{-x}} = \lim_{x \rightarrow 0} e^{x} = 1 \implies y(0) = 0\ (2)$

The procedure for the derivatives however is much more complex and 'Monster Wolfram' can supply a significative aid...

ln x - ln [1 - e^(- x)] - Wolfram|Alpha

The McLaurin expansion we are interersted for is...

$\displaystyle y(x) = \frac{x}{2} - \frac{x^{2}}{24} + \frac{x^{4}}{2880} + \mathcal {O} (x^{6})\ (3)$

As curiosity y(x) is defined for all value of x [positive as well as negatives...] and is $y(x) \le \frac{x}{2}$. May be that a simpler approach to the problem exists and, if yes, it must be found...


Kind regards


$\chi$ $\sigma$
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,690
Re: Prove x>2y

I'm afraid that You have to try the McLaurin expansion around x = 0 of the logarhitm of the function...


$\displaystyle y = \ln x - \ln (1 - e^{-x})\ (1)$


Starting from f(0) and using l'Hopital rule You have...


$\displaystyle \lim_{x \rightarrow 0} \frac{x}{1 - e^{-x}} = \lim_{x \rightarrow 0} e^{x} = 1 \implies y(0) = 0\ (2)$

The procedure for the derivatives however is much more complex and 'Monster Wolfram' can supply a significative aid...

ln x - ln [1 - e^(- x)] - Wolfram|Alpha

The McLaurin expansion we are interersted for is...

$\displaystyle y(x) = \frac{x}{2} - \frac{x^{2}}{24} + \frac{x^{4}}{2880} + \mathcal {O} (x^{6})\ (3)$

As curiosity y(x) is defined for all value of x [positive as well as negatives...] and is $y(x) \le \frac{x}{2}$.
Hey chisigma, thank you for your reply...and I am happy to learn these two methods to solve the problem!

May be that a simpler approach to the problem exists and, if yes, it must be found...
I think MarkFL's approach is very straightforward and smart! :):eek:
 

johng

Well-known member
MHB Math Helper
Jan 25, 2013
236
Re: Prove x>2y

Here's another solution which just uses a careful application of the mean value theorem.

MHBcalculus3.png