More limit cycles for complex differential equations with three monomials

In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, $\dot{z}=A{z}^k{\overline{z}}^l+B{z}^m{\overline{z}}^n+C{z}^p{\overline{z}}^q$, being $k,l,m,n,p,q$ non-negative integers and $A,B,C\in C$. More concretely, if $N=\max (k+l,m+n,p+q)$ and $H_3\left(N\right)\in N\cup \{\infty \}$ denotes the maximum number of limit cycles of the above equations, we show that for $N\ge 4$, $H_3\left(N\right)\ge N-3$ and that for some values of N this new lower bound is $N+1$. We also present examples with many limit cycles and different configurations. Finally, we show that $H_3\left(2\right)\ge 2$ and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles.