Q:

Deduce upper bounds on $a$, $b$, $c$ from bounds on $a^2\sqrt{b}$, $c^2/b^3$, $a/(c\sqrt{b})$

Let's have the set of inequalities:
$$
\begin{cases} a^2\sqrt{b} < C_1, \\ \frac{c^2}{b^3} < C_2, \\ \frac{a}{c}\frac{1}{\sqrt{b}} > C_3 \end{cases} 
$$
Here $C_{i}, i = 1,2,3$ corresponds to a given positive numbers, $a, b, c$ are positive definite variables. 
Is it possible to transform inequalities to the form like $c < D_1, a < D_2, b < D_3$, or not? And how to do it?

A:

No, in general the solution will be a curved set, which cannot be described in this way.  You can see this even more simply in 2D.  For example, let $c=1$ and set $C_2$ so the second constraint is not binding.

