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 November 3rd, 2017, 04:43 AM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 Inequality with $sinh$ and $cosh$ We consider a parameter $a>>1$ and we note $cosh(\theta)=\sqrt{4a+1}$ and $sinh(\theta)=2\sqrt{a}$ for all $t>0$ we note: $u(t)=\frac{\sinh(\frac{t}{2}\cosh(\theta))}{\cosh (\theta)}$ $A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}-1}{\cosh^2(\theta)}-\Big(\cosh(\frac{t}{2})-1\Big)$ $f(t)=-ln\Big(1-\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$ I want to find the minimal power $\alpha\ge 0$ such that $$\frac{\cosh^2(\theta)}{f(t)}e^{-argsh(u(t))}\le \frac{\theta^{\alpha}}{t^3}$$ for all $t>0$
November 3rd, 2017, 07:24 AM   #2
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Quote:
 Originally Posted by mona123 We consider a parameter $a>>1$ and we note $cosh(\theta)=\sqrt{4a+1}$ and $sinh(\theta)=2\sqrt{a}$ for all $t>0$ we note: $u(t)=\frac{\sinh(\frac{t}{2}\cosh(\theta))}{\cosh (\theta)}$ $A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}-1}{\cosh^2(\theta)}-\Big(\cosh(\frac{t}{2})-1\Big)$ $f(t)=-ln\Big(1-\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$ I want to find the minimal power $\alpha\ge 0$ such that $$\frac{\cosh^2(\theta)}{f(t)}e^{-argsh(u(t))}\le \frac{\theta^{\alpha}}{t^3}$$ for all $t>0$
What is the difference between t and $\displaystyle \theta$? u(t) is defined as a function of t, not $\displaystyle \theta$??

-Dan

 November 3rd, 2017, 09:12 AM #3 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 $\theta$ is fixed from the begining
 November 4th, 2017, 07:32 AM #4 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 1 @topsquark, can you help me to prove the iniquality by taking $\alpha=4$?

 Tags $cosh$, $sinh$, inequality

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