  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 $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if
$|A|\succeq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succapprox
\min\{|A|^{13/12}(\frac{|A|}{p^{0.5}})^{1/12},|A|(\frac{p}{|A|})^{1/11}\}.\]
These results slightly improve the estimates of Bourgain-Garaev and Shen.
Sum-product estimates on different sets are also considered.
