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November 4th, 2017, 10:44 AM   #1
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Inequality with fixed parameter

We consider a fixed parameter $a\gg1$ and we note $\cosh(\theta)=\sqrt{4a+1}$ and $\sinh(\theta)=2\sqrt{a}$

for all $t>0$ we note:

$$u(t)=\frac{\sinh\Big(\frac{t}{2}\cosh(\theta)\Bi g)}{\cosh(\theta)}$$

$$A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}-1}{\cosh^2(\theta)}-\Big(\cosh\Big(\frac{t}{2}\Big)-1\Big)$$

$$f(t)=-\ln\Big(1-\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$$

I want to show that $$f(t)\ge \frac{\theta^2}{sinh(\theta)}$$ for all $\frac{2\theta}{cosh(\theta)}\le t\le \frac{2\theta^2}{cosh(\theta)}$

Last edited by mona123; November 4th, 2017 at 11:35 AM.
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