-behavior of power series with positive coefficients and Hardy spaces

Authors:
Miodrag Mateljević and Miroslav Pavlović

Journal:
Proc. Amer. Math. Soc. **87** (1983), 309-316

MSC:
Primary 30D55

DOI:
https://doi.org/10.1090/S0002-9939-1983-0681840-0

MathSciNet review:
681840

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For the power series with , certain weighted -norms of on are estimated from above and below in terms of the coefficients . Some consequences of this are obtained. For example, some known results concerning Hardy spaces may be extended to a wider class of spaces.

**[1]**Richard Askey,*𝐿^{𝑝} behavior of power series with positive coefficients*, Proc. Amer. Math. Soc.**19**(1968), 303–305. MR**0223786**, https://doi.org/10.1090/S0002-9939-1968-0223786-X**[2]**R. Askey and R. P. Boas Jr.,*Some integrability theorems for power series with positive coefficients*, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 23–32. MR**0277956****[3]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[4]**G. H. Hardy and J. E. Littlewood,*Some properties of fractional integrals. II*, Math. Z.**34**(1932), no. 1, 403–439. MR**1545260**, https://doi.org/10.1007/BF01180596**[5]**-,*Elementary theorems concerning power series with positive coefficients and moment constants of positive functions*, J. Reine Angew. Math.**157**(1927), 141-158.**[6]**-,*Some new properties of Fourier constants*, Math. Ann.**97**(1926), 159-209.**[7]**W. K. Hayman,*Multivalent functions*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge University Press, Cambridge, 1958. MR**0108586****[8]**F. Holland and J. B. Twomey,*On Hardy classes and the area function*, J. London Math. Soc. (2)**17**(1978), no. 2, 275–283. MR**0486532**, https://doi.org/10.1112/jlms/s2-17.2.275**[9]**F. Holland and J. B. Twomey,*Conditions for membership of Hardy spaces*, Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979) Academic Press, London-New York, 1980, pp. 425–433. MR**623486****[10]**J. E. Littlewood and R. E. A. C. Paley,*Theorems on Fourier series and power series*. II, Proc. London Math. Soc.**42**(1936), 52-89.**[11]**M. Mateljević and M. Pavlović,*On the integral means of derivatives of the atomic function*(to appear).**[12]**J. W. Noonan and D. K. Thomas,*The integral means of regular functions*, J. London Math. Soc. (2)**9**(1974/75), 557–560. MR**0364625**, https://doi.org/10.1112/jlms/s2-9.4.557

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30D55

Retrieve articles in all journals with MSC: 30D55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0681840-0

Keywords:
Hardy spaces,
Hardy-Stein identity

Article copyright:
© Copyright 1983
American Mathematical Society