I am trying to find a tight upper bound on $\sum_{i=1}^nf(s_i)\left(\frac{1 - \exp(-s_it)}{s_it} - \frac{1}{1 + s_it}\right)$ where the maximum value of function is $a$ and minimum is $b$, and both of them are positive, also, $s_i, t>0$. I am able to find the bound on $\left(\frac{1 - \exp(-s_it)}{s_it} - \frac{1}{1 + s_it}\right)$ (see below), but how do I incorporate the information of $f(s_i)$ in $g(t)$?
\begin{align*} &\frac{(1 - \exp(-s_it))(1 + s_it)}{s_it} \leq g(t) \\ &\Rightarrow \frac{1 - \exp(-s_it)}{s_it} \leq g(t) \frac{1}{1 + s_it}\\ &\Rightarrow \frac{1 - \exp(-s_it)}{s_it} - \frac{1}{1 + s_it} \leq (g(t) - 1) \frac{1}{1 + s_it}, \\ \end{align*} where $g(t)$ is given as: \begin{align*} g(t) = \left\{ \begin{array}{ll} \frac{(1-\exp(-vt))(1+vt)}{vt} & \mbox{if } t \leq 1.7933/s_{\max} \\ \frac{(1-\exp(-wt))(1+wt)}{wt} & \mbox{if } t \geq 1.7933/s_{\min} \\ 1.2985 & \mbox{if } 1.7933/s_{\max} < t < 1.7933/s_{\min} \\ \end{array} \right. \end{align*}
Edit 1: I am not sure if $g(t)$ is tight enough or not, if the difference between maximum value and minimum value of $s$ is very large, then it is quite loose. Should I use some other function? Also, at $t \rightarrow 0$ and $t \rightarrow \infty$, the function should go to 0.