The sum-product estimate for large subsets of prime fields

Let ${\mathbb{F}}_p$ be the field of a prime order $p\mathrm{.}$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset {\mathbb{F}}_p$ with $\mathrm{\mid }A\mathrm{\mid }=N$ such that $$\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}\ll p^{1\mathrm{/}2}\mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{.}$$ In the present paper we prove that if $A\subset {\mathbb{F}}_p$ with $\mathrm{\mid }A\mathrm{\mid }>p^{2\mathrm{/}3},$ then $$\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}\gg p^{1\mathrm{/}2}\mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{.}$$