I would like to bound $$\cosh^2\left(t\sqrt{1-\gamma^2}\right)$$ for all $t\in\mathbb{R}$, where $\gamma^2\leq1$. How can I do such thing?
This inequality maybe useful cosh x inequality
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This function is unbounded over $\mathbb{R}$. Are you looking for for a dominated function? – Olivier Oloa Apr 26 '16 at 07:04
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Oh yes it's really good if we can find such function – Truong Apr 26 '16 at 07:19
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One may write $$ \cosh^2\left(t\sqrt{1-\gamma^2}\right)=\left(\frac{e^{t\sqrt{1-\gamma^2}}+e^{-t \sqrt{1-\gamma^2}}}2\right)^2\leq \color{blue}{e^{2|t|\sqrt{1-\gamma^2}}},\quad t \in \mathbb{R}, $$ or $$ \cosh^2\left(t\sqrt{1-\gamma^2}\right)=\frac{e^{2t\sqrt{1-\gamma^2}}}4+\frac{e^{-2t\sqrt{1-\gamma^2}}}4+\frac12\leq \color{blue}{\frac{e^{2|t|\sqrt{1-\gamma^2}}}4+\frac34},\quad t \in \mathbb{R}. $$
Truong
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Olivier Oloa
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If $u\leq0$ then $e^u\leq1$. Thus $\frac{e^{-2t\sqrt{1-\gamma^2}}}4+1\leq \frac14+1=\frac34$, if $t\geq0$. – Olivier Oloa Apr 26 '16 at 08:20
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