## Coefficient multipliers of Bloch functions

HTML articles powered by AMS MathViewer

- by J. M. Anderson and A. L. Shields PDF
- Trans. Amer. Math. Soc.
**224**(1976), 255-265 Request permission

## Abstract:

The class $\mathcal {B}$ of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by $c/(1 - r)$ on $|z| \leqslant r$. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from ${l^p}$ into $\mathcal {B}$ and from $\mathcal {B}$ into ${l^p}$. We find the second KΓΆthe dual of $\mathcal {B}$ and show its relevance to the multiplier problem. We identify all power series $\sum {a_n}{z^n}$ such that $\sum {w_n}{a_n}{z^n}$ is a Bloch function for every choice of the bounded sequence $\{ {w_n}\}$. Analogous problems for ${H^p}$ spaces are discussed briefly.## References

- J. M. Anderson, J. Clunie, and Ch. Pommerenke,
*On Bloch functions and normal functions*, J. Reine Angew. Math.**270**(1974), 12β37. MR**361090** - Gregory F. Bachelis,
*On the ideal of unconditionally convergent Fourier series in $L_{p}\,(G)$*, Proc. Amer. Math. Soc.**27**(1971), 309β312. MR**271640**, DOI 10.1090/S0002-9939-1971-0271640-X - Gregory F. Bachelis,
*On the upper and lower majorant properties in $L^{p}(G)$*, Quart. J. Math. Oxford Ser. (2)**24**(1973), 119β128. MR**320636**, DOI 10.1093/qmath/24.1.119 - P. L. Duren and A. L. Shields,
*Properties of $H^{p}$ $(0<p<1)$ and its continuing Banach space*, Trans. Amer. Math. Soc.**141**(1969), 255β262. MR**244751**, DOI 10.1090/S0002-9947-1969-0244751-8 - D. J. H. Garling,
*The $\beta$- and $\gamma$-duality of sequence spaces*, Proc. Cambridge Philos. Soc.**63**(1967), 963β981. MR**218881**, DOI 10.1017/s0305004100041992 - D. J. H. Garling,
*On symmetric sequence spaces*, Proc. London Math. Soc. (3)**16**(1966), 85β106. MR**192311**, DOI 10.1112/plms/s3-16.1.85 - Edwin Hewitt,
*The ranges of certain convolution operators*, Math. Scand.**15**(1964), 147β155. MR**187016**, DOI 10.7146/math.scand.a-10738 - Jean-Pierre Kahane,
*Some random series of functions*, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR**0254888** - C. N. Kellogg,
*An extension of the Hausdorff-Young theorem*, Michigan Math. J.**18**(1971), 121β127. MR**280995**
G. KΓΆthe, - Hugo Steinhaus,
*Additive und stetige Funktionaloperationen*, Math. Z.**5**(1919), no.Β 3-4, 186β221 (German). MR**1544384**, DOI 10.1007/BF01203518 - A. Zygmund,
*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

*Topological vector spaces*.I, Springer-Verlag, Berlin, 1960; English transl., Springer-Verlag, New York, 1969. MR

**24**#A411;

**40**#1750.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**224**(1976), 255-265 - MSC: Primary 30A78
- DOI: https://doi.org/10.1090/S0002-9947-1976-0419769-6
- MathSciNet review: 0419769