  Using Khinchin's inequality, Ger$\check{\mbox{s}}$gorin's theorem and the
atomic decomposition of Bergman spaces, we estimate the norm and essential norm
of Stevi\'c-Sharma type operators from weighted Bergman spaces $A_\omega^p$ to
$A_\mu^q$ and the sum of weighted differentiation composition operators with
different symbols from weighted Bergman spaces $A_\omega^p$ to $H^\infty$.The
estimates of those between Bergman spaces remove all the restrictions of a
result in [Appl. Math. Comput.,{\bf 217}(2011),8115--8125]. As a by-product, we
also get an interpolation theorem for Bergman spaces induced by doubling
weights.
