I want to bound $f_h(y) = h^{-1}|e^{hy}-1|$ where $h\in(0,1)$ and $y\in\mathbb R$ with something independent on $h$ and growing as slow as possible with $y\to \pm\infty$.
Can I do better than $f_h(y) \ \leq e^{|y|}$?
I want to bound $f_h(y) = h^{-1}|e^{hy}-1|$ where $h\in(0,1)$ and $y\in\mathbb R$ with something independent on $h$ and growing as slow as possible with $y\to \pm\infty$.
Can I do better than $f_h(y) \ \leq e^{|y|}$?
For $y\rightarrow -\infty$ its possible, but for $y\rightarrow \infty$ it is not possible, because
\begin{eqnarray} \frac{|e^{hy}-1|}{h} &\geq& \frac{e^{hy}}{h}-\frac{1}{h} \nonumber \end{eqnarray}
From the last inequality, we see that for fixed $h$, your function grows equal or even more than an exponential in the set $\{(h,y):\ y>0\}$.