trouble with inequality
Inquiry: I intend to address $0<1−an/(mb^2)e^{−r(T−t)}<1$, where $r, a, b, T, t>0$.
The remedy is that either $$an\leq mb^2$$ or $$mb^2\leq an\leq mb^2e^{rT}$$ and also $$t< T − (\ln(an) − \ln(mb^2 ))/r$$.
My Attempt: My ideas are that the first component $0<1−\frac{an}{mb^2}e^{−r(T−t)}$ offers me $an \leq mb^2$ due to the fact that $e^{−r(T−t)}>0$, so I have the first component. The 2nd component $1−\frac{an}{mb^2}e^{−r(T−t)}<1$ does not offer me valuable details given that $\frac{an}{mb^2}e^{−r(T−t)}>0$ constantly.
Just how do I get the various other fifty percent of the remedy ( $mb^2\leq an\leq mb^2e^{rT}$ and also $t< T − (\ln(an) − \ln(mb^2 ))/r$)?
I additionally become aware that the trouble I need to address lowers to addressing $xy<1$ where both $x,y>0$.
Combined from: tricky inequality
How do I deal with addressing $0<1−\frac{an}{mb^2}e^{−r(T−t)}<1$, where a, b, T > t > 0? I have actually been stuck below for time currently.
There are several unneeded variables. Allow
$$\alpha = \frac{an}{mb^2}.\qquad(1)$$
Then the inequality comes to be
$$ 0 < 1 - \alpha e^{-r (T - t)} < 1 $$
The first noticeable action is execute "1 −" on every components,
$$ 1 > \alpha e^{-r (T-t)} > 0 $$
Since the rapid function's array declares and also < 1 (given that r > 0 and also T > t > 0), we can make certain α declares.
If 0 < α ≤ 1, after that every t will certainly please the inequality (the first remedy).
So think α > 1. Currently it's rather noticeable on just how to address t in regards to r , T and also α . Replacement (1) once more to come back a , n , m and also b .
Points you might take into consideration :
- e ^{x } is purely raising.
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