  We establish several sum-product estimates over finite fields that involve
polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially
larger than |A| for an arbitrary polynomial f over F_p. Second, a
characterization is given for the rational functions f and g for which
|f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that
under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided
|A| is large. We also present a conjecture on what the general sum-product
result should be.
