For α>0\alpha > 0 let Fα{\mathcal {F}_\alpha } denote the class of functions defined for |z|>1|z| > 1 by integrating 1/(1−xz)α1/{(1 - xz)^\alpha } against a complex measure on |x|=1|x|= 1. A function gg holomorphic in |z|>1|z| > 1 is a multiplier of Fα{\mathcal {F}_\alpha } if f∈Fαf \in {\mathcal {F}_\alpha } implies gf∈Fαgf \in {\mathcal {F}_\alpha }. The class of all such multipliers is denoted by Mα{\mathcal {M}_\alpha }. Various properties of Mα{\mathcal {M}_\alpha } are studied in this paper. For example, it is proven that α>β\alpha > \beta implies Mα⊂Mβ{\mathcal {M}_\alpha } \subset {\mathcal {M}_\beta }, and also that Mα⊂H∞{\mathcal {M}_\alpha } \subset {H^\infty }. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if f′f’ is in the Hardy class H1{H^1}, then f∈M1f \in {\mathcal {M}_1}. Also the theorem is improved to f′∈H1f’ \in {H^1} implies f∈Mαf \in {\mathcal {M}_\alpha }, for all α>0\alpha > 0. Finally, let α>0\alpha > 0 and let ff be holomorphic in |z|>1|z| > 1. It is known that ff is bounded if and only if its Cesàro sums are uniformly bounded in |z|≤1|z| \leq 1. This result is generalized using suitable polynomials defined for α>0\alpha > 0.