I would like to find an upper and/or lower bound for the parametric function:

$\displaystyle (x,y)=\left\{ \left\( \frac{\Phi(t)}{\phi(t)}, \frac{\Phi(t)}{\phi(t)}\Phi(-t) + t\Phi(t) +\phi(t) \right\) : -\infty\leq t \leq \infty\right\}$

where $\displaystyle \phi(t)$ is the standard normal density and $\displaystyle \Phi(t)$ the corresponding distribution function. The bound should be in terms of $\displaystyle x$ and $\displaystyle y$, and not in $\displaystyle t$.

To get a rough idea of the function, the attached figure shows the function for $\displaystyle t\leq 2$.

Can anybody help?