Let $\theta>0$, $0<\lambda<1$, and $\Phi$ be the standard normal cumulative distribution function. Is it true that

$$\frac{\lambda \Phi(-\theta)}{\Phi(-\lambda \theta) }< e^{\frac{\theta^2(-1+\lambda^2)}{2}} $$

Preliminary numerical analysis appears to agree with my assertion. Here are plots of $\lambda=0.2,0.3,\ldots,0.9$.