The Bergman space \(A^ p\) is the space of analytic \(L^ p\) functions in the unit disk with respect to the area measure \((0< p<\infty)\). A sequence \(\{\lambda_ n\}\) is a coefficient multiplier from \(A^ p\) into \(A^ q\) if \(\sum a_ n z^ n\in A^ p\Rightarrow \sum \lambda_ n a_ n z^ n\in A^ q\), and the set of all such \(\{\lambda_ n\}\) is denoted by \((A^ p,A^ q)\). A recent result of \textit{P. Wojtaszczyk} [Can. Math. Bull. 33, 151-161 (1991; Zbl 0737.46006)]\ completely describes \((A^ p,A^ q)\) for \(0\leq q\leq 2\leq p< \infty\). This paper complements his work by giving several new conditions, necessary or sufficient, for multipliers in terms of summability of the series \(\sum n^ \alpha|\lambda_ n|^ q\), \(\alpha= \alpha(p,q)\), mostly in the case \(0\leq p\leq 2\leq q< \infty\). In particular, an exact characterization of \((A^ 1,A^ 2)\) is obtained, which is an analogue of a known theorem of Hardy-Littlewood-Duren-Shields for the Hardy spaces. An analogue of a theorem of Duren for \(H^ p\) spaces is also obtained, showing that any sequence \(O\left(n^{{2\over q}-{2\over p}}\right)\in (A^ p,A^ q)\), for \(0\leq p\leq 2\leq q< \infty\) or \(0< p\leq 1\), \(q= \infty\), and the exponent is best possible.