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Rewrite this as $h(\theta)<h(\lambda \theta)$, where $h(\theta)=\Phi(-\theta)e^{\theta^2/2}/\theta$. So, the question is whether the function $h$ decreases on $(0,+\infty)$. We have $$\frac{d\log h}{d\theta}=\frac{-\Phi'(-\theta)}{\Phi(-\theta)}+\theta-\frac1\theta=\frac{-e^{-\theta^2/2}}{\int_{-\infty}^{-\theta} e^{-t^2/2}dt}+\theta-\frac1\theta,$$ this proves that $\frac{d\log h}{d\theta}<0$ for $\theta\geqslant 1$ and for $\theta<1$ this desired inequality may be rewritten as $$\int_{-\infty}^{-\theta} e^{-t^2/2}dt<e^{-\theta^2/2}\frac{\theta}{\theta^2-1}.$$ This becomes equality when $\theta$ goes to $+\infty$, and the difference $$g(\theta)=\int_{-\infty}^{-\theta} e^{-t^2/2}dt-e^{-\theta^2/2}\frac{\theta}{\theta^2-1}$$ increases as is seen from $$\frac{dg}{d\theta}=e^{-\theta^2/2}\left(\frac{\theta^4+1}{(\theta^2-1)^2}-1\right)>0,$$ this finishes the proof.