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I need to show

$f(t)=(P-t)\Phi(\frac{t-P}{\sqrt{T(t)}})+\frac{1}{\sqrt{T(t)}}\phi(\frac{t-P}{\sqrt{T(t)}}) <0$ if $t$ is large enough

where $T(t)>0$ and is a function of $t$, with $t>P>0$, and $\Phi, \phi$ are the normal cdf and pdf respectively.

I am totally stumped.

I know that when $t=P$, $f(P)=\frac{1}{\sqrt{2\pi T(P)}}>0$ but I can't think of anything else due to the implicit nature of the problem.

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    May I know why does f(t) has the sign of $\phi(s_t)-T(t)\Phi(s_t)$? T(t) is continuous but that's about it. May I know what does $\Omega(1)$ mean? – 2017-02-21
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    Ah thanks for your reply! I understand what $\Omega(1)$ means now. Its like the opposite of Big O notation $O(1)$. Unfortunately, I'm trying to figure out the behavior of $T(t)$ itself through $f(t)$. The sign of $f(t)$ is actually the sign of $T'(t)$. The rest of the parts of $T'(t)$ are always positive. The implicit form of the function really is causing a lot of problems with the analysis. I do not have an explicit formula for $T(t)$. $T(t)$ is bounded below by $0$. Does that count? – 2017-02-21
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    Also, I think it should be $\phi(s_t)-\sqrt{T(t)}t\Phi(s_t)$ instead. – 2017-02-21
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    For every $s\geqslant0$, $\Phi(s)\geqslant\frac12$ and $s\phi(s)\leqslant\phi(1)$ hence, for every $t\geqslant P$, $$(t-P)f(t)\leqslant\phi(1)-(t-P)^2\tfrac12$$ This implies that $f(t)<0$ for every $$t>P+\sqrt{2\phi(1)}$$ or, more simply, for every $$t\geqslant P+\tfrac12$$ – 2017-02-21
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    Awesome! Thank you so much. – 2017-02-22

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