# An optimization problem

I have the following two quantities denoted by $$LHS$$ and $$RHS$$:

LHS = Abs[p1 (z12 - z21) + I p2 (z12 - z21) + p3 (z11 - z22) + q (z11 + z22)];
RHS = Sqrt[Abs[z11]^2 + Abs[z21]^2 ] + Sqrt[Abs[z12]^2 + Abs[z22]^2 ];


Here, $$z12$$ and $$z21$$ are complex numbers with $$z12=z21^*$$. Also $$z11$$ and $$z22$$ are reals. I need to find the values of $$p1, p2,p3$$ and $$q$$, such that $$LHS>RHS$$, subjected to the condition that $$|q|+|\vec{p}| \le 1$$, where $$\vec{p}=(p1,p2,p3)$$.

This is my first question on Mathe-SE, apologies if you find the problem trivial or in inappropriate in the description.