Question
All variables belong to R. Prove (A) and (B) and provide reasoning for (C) and (D) Prop A Suppose that xy > 0. Then exactly one of the following two possibilities must occur: (i)x, y ? R>0 (ii) ?x,?y ? R>0 Prop B Suppose that y2 > x2. Then exactly one of the following three possibilities must occur: (i)y > x and y > ?x (ii) y < x and y < ?x Project C Find conditions on a, b, c so that abc > 0. (No Proof is required) Project D Find conditions on x and y so that xy2 > x2y. You do not have to give a formal proof but you do need to explain your reasoning using project C.
Explanation / Answer
|1-zr|^2 - |z-r|^2 = (1-|z|^2)(1-r^2) > 0 ...yes? This looks right, and it shows that ?r(z) is in the unit disk. A similar inequality holds when r is complex. But this should really be proven prior to this last step we are discussing. If nothing else, z in this last step is really w?z2(z1), and we would like to know that |z| < 1. I had a friend who worked on some very complicated research, but his papers were always much shorter than mine. I asked him why he thought that was. His answer was that much of his work involved long strings of equalities. According to him, most of these could be omitted and left to the reader, because there are only a limited number of ways to get from one end of a string of equalities to the other. My work, on the other hand, involves long strings of inequalities. And, again according to him, these need to be explicitly described to the reader, because the reader would not be able to easily reproduce them. I am not sure how much truth is in my friend's comments. I do, however, feel there is a lot of art in doing inequalities. Whatever the case may be, there is no denying that inequalities are at the heart of analysis.