The sum product phenomenon has received a great deal of attention, since Erd¨os and

Szemer`edi made their well known conjecture that

max(|A + A|, |AA|) ≥ Cǫ

|A|

2−ǫ∀ǫ > 0.

where A is a finite subset of integers and

A + A = {a + b : a ∈ A, b ∈ A},

and

AA = {ab : a ∈ A, b ∈ A}.

In this talk, we will present that if A is a subset in a finite field Fp, p prime, with |A| < p

1

2

then

max(|A + A|, |F(A, A)|) ' |A|

13

12 .

where F : Fp × Fp to Fp , (x, y) → x(f(x) + by), f is any function and b ∈ F

∗

p

. For the

case f=0 and b = 1, it corresponds to the well known sum product theorem by Bourgain,

Katz and Tao. This is joint work with Nets Katz.