  New lower bounds involving sum, difference, product, and ratio sets for a set
$A\subset \C$ are given. The estimates involving the sum set match, up to
constants, the one obtained by Solymosi for the reals and are obtained by
generalising his approach to the complex plane. The bounds involving the
difference set are slightly weaker. They improve on the best known ones,
including the case $A\subset \R$, which also due to Solymosi, by means of
combining the use of the Szemer\'edi-Trotter theorem with an arithmetic
combinatorics technique.
