Bounding $h^{-1}|e^{hy}-1|$ 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 $\{(h,y):\ y>0\}$.
