  Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order
$H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}}
{\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where
${\mathbf{\,e}}_p(z) = \exp(2 \pi i z/p)$, which improves a result of Bourgain
and Garaev (2009). We also obtain new estimates for double exponential sums
with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a
short interval ${\mathcal N}$ of consecutive integers.
