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.


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