On the sum product estimates and two variables expanders

Let Fp be the finite field of a prime order p. Let F: Fp x Fp --> Fp be a function defined by F(x, y) = x(f(x) + by), where b ∈ F*/p and f: Fp → Fp is any function. We prove that if A ⊂ Fp and │A│ < p1/2 then │A + A│+│F(A,A) │⪆│A│13/12. Taking f = 0 and b = 1, we get the well-known sum-product theorem by Bourgain, Katz and Tao, and Bourgain, Glibichuk and Konyagin, and also improve the previous known exponent from 14/13 to 13/12