Average estimate for additive energy in prime field

Assume that $A\subseteq \text{Fp},B\subseteq {\text{Fp}}^{\ast }$, $\text{1}\mathrm{/}4⩽\frac{\mathrm{\mid }B\mathrm{\mid }}{\mathrm{\mid }A\mathrm{\mid }},$ $\mathrm{\mid }A\mathrm{\mid }=p^{\alpha },\mathrm{\mid }B\mathrm{\mid }=p^{\beta }$. We will prove that for $p⩾p_0(\beta )$ one has $$\sum _{b\in B}{E}_{+}(A,bA)⩽15p^{-\frac{\min \{\beta ,1-\alpha \}}{308}}\mathrm{\mid }A{\mathrm{\mid }}^3\mathrm{\mid }B\mathrm{\mid }\mathrm{.}$$ Here $E_{+}(A,bA)$ is an additive energy between subset $A$ and it’s multiplicative shift $bA$. This improves previously known estimates of this type.