Conditional expanding bounds for two-variable functions over prime fields

In this paper we provide in Fp expanding lower bounds for two variables functions f(x,y) in connection with the product set or the sumset. The sum-product problem has been immensely studied in the recent past. A typical result in Fp∗ is the existence of Δ(α)>0 such that if |A|≍pα then max(|A+A|,|A⋅A|)≫|A|1+Δ(α), Our aim is to obtain analogous results for related pairs of two-variable functions f(x,y) and g(x,y): if |A|≍|B|≍pα then max(|f(A,B)|,|g(A,B)|)≫|A|1+Δ(α) for some Δ(α)>0.