# How to reduce/simplify an inequality involving Abs[]?

Can Mathematica reduce this inequality into a domain of z, where {z, r1, r2} are all complex?

Reduce[Abs[z^2 - r1] > Abs[z^3 - r2], z]


The line never returns a result. I also tried just calculating the boundary:

Simplify[Abs[z^2 - r1] == Abs[z^3 - r2]]


But that never returns a result either. I've tried a few other methods that all just crashed or failed. I found a related Question and another related Question but I couldn't figure out how to apply those partial solutions to this problem.

BTW, I realize Solve[] and Reduce[] don't cope with Abs[]. Even this simpler command give a poor result:

Simplify[Abs[z^2] == Abs[z^3]];

(* Abs[z]^2 == Abs[z] *)


And just to be clear, I don't just need the answer to the inequality, I need to know how to make Mathematica do it.

• You should tell Reduce you're working over the reals: Reduce[Abs[z^2 - r1] > Abs[z^3 - r2], z, Reals]. – Chip Hurst Nov 13 '15 at 22:36
• @ChipHurst I'm not. I'll add clarification that the variables are complex. – Jerry Guern Nov 13 '15 at 22:40
• How come my tick marks in the title aren't being turned into nice Font? – Jerry Guern Nov 13 '15 at 23:24
• Titles entry fields don't support all the editing features as the editor pane that accepts the main body of a question. This is a quick of SE that has to be lived with. – m_goldberg Nov 14 '15 at 2:07
• What do you think Simplify[Abs[z^2] == Abs[z^3]] should return? Keep in mind z is considered complex unless otherwise stated. – Chip Hurst Nov 14 '15 at 2:16

In[1]:= Reduce[Abs[z^2 - r1] > Abs[z^3 - r2], {z, r1, r2}, Complexes]

• Thanks for the tip about adding Complexes, but yikes that huge pile result is not a good result. I don't think it should be necessary to explicitly separate Im[z] and Re[z] in the result. – Jerry Guern Nov 13 '15 at 23:08