Slightly improved sum-product estimates in fields of prime order

Let ${\mathbb{F}}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of ${\mathbb{F}}_p$. In this paper we show that if $\mathrm{\mid }A\mathrm{\mid }⪯p^{0.5}$, then $$\max \{\mathrm{\mid }A\pm A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}⪰\mathrm{\mid }A{\mathrm{\mid }}^{13\mathrm{/}12};$$ if $\mathrm{\mid }A\mathrm{\mid }⪰p^{0.5}$, then $$\max \{\mathrm{\mid }A\pm A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}⪸\min \{\mathrm{\mid }A{\mathrm{\mid }}^{13\mathrm{/}12}(\frac{\mathrm{\mid }A\mathrm{\mid }}{p^{0.5}}{)}^{1\mathrm{/}12},\mathrm{\mid }A\mathrm{\mid }(\frac{p}{\mathrm{\mid }A\mathrm{\mid }}{)}^{1\mathrm{/}11}\}\mathrm{.}$$ These results slightly improve the estimates of Bourgain-Garaev and Shen. Sum-product estimates on different sets are also considered.