Do we have an upper bound for $\frac{ax+by+cz}{a_{2} x +b_{2} y +c_2 z}$. What about when $x,y,z<1$ ?
I don't know how to go about it? any tips?
EDIT: Someone kindly presented a counterexample. But are there nice restrictions to put so that it is upper bounded by some parameter. Like can it somehow be upper bounded by say $\frac{a}{a_2}+\frac{b}{b_2}+\frac{c}{c_2}$. (This looks familiar and I've seen it somewhere :) ) Thanks!
Numerator and denominator represent two planes , when each is put equal to $d_1, \, d_2$ .
So either they are parallel and there are no common points, or they are coincident, and the ratio is constant everywhere, or they have a common intersection line where the ratio is $d_1 / d_2$.
Then if you increase $d_1$ the upper plane will move .. and the intersection line will move ..