Let $\varphi(x)$ be the PDF of the standard normal distribution and $\Phi(x)$ the respective CDF.
I am looking for estimates of $\frac{\varphi(x)}{\Phi(x)}$ in $x$, i.e., something in the form of $\frac{\varphi(x)}{\Phi(x)}\leq f(x)$ or $\frac{\varphi(x)}{\Phi(x)}\geq g(x)$ with $g,f$ functions of your choice. Anything help.
To provide you with a larger picture: I am trying to find the sign of $\frac{\varphi(z+b)}{\Phi(z+b)}-z$.
Asymptotic expansions for $\Phi$ and the related $\operatorname{erfc}$ are well known: see for instance the handbook entry and its wikipedia equivalent. These expansions cover the case of $\Phi(x)$ for large $|x|$.
Feller's textbook (p166 in the 2nd edition) summarizes the simpler form of this: If $x>0$, then $$ \frac 1 {\sqrt{2\pi}} e^{-x^2/2} \left\{ \frac 1 x - \frac 1 {x^3}\right\} < 1-\Phi(x) < \frac 1 {\sqrt{2\pi}} e^{-x^2/2} \frac 1 x .$$