  For analytic functions $g$ on the unit disc with non-negative Maclaurin
coefficients, we describe the boundedness and compactness of the integral
operator $T_g(f)(z)=\int_0^zf(\zeta)g'(\zeta)\,d\zeta$ from a space $X$ of
analytic functions in the unit disc to $H^\infty$, in terms of neat and useful
conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will
be considered contain the Hardy and the Hardy-Littlewood spaces, the
Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA
spaces.
