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Growth of coefficients of universal Taylor series and comparison of two classes of functions. (English) Zbl 0894.30003
In this paper the authors study the order of growth of the coefficients of universal Taylor series and give several theorems. Let $$U$$ be the class of universal Taylor series (a sub-class of the Taylor series with complex coefficients that converge within the unit disc). As consequences of the results proved, they show that the class $$U$$ is disjoint from the Nevanlinna class and that no universal Taylor series is summable $$(C,k)$$ at any point $$z$$, $$| z|=1$$, for any $$k$$. Relations between some other associated classes of functions are also studied.

MSC:
 30B10 Power series (including lacunary series) in one complex variable
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References:
 [1] N.K. Bary,A Treatise on Trigonometric Series, Vols. I, II, Pergamon Press, Oxford, 1964. · Zbl 0129.28002 [2] C. Chui and M. N. Parnes,Approximation by overconvergence of power series, J. Math. Anal. Appl.36 (1971), 693–696. · Zbl 0224.30006 [3] P. Dienes,The Taylor Series, Dover, New York, 1957. · Zbl 0078.05901 [4] P. Duren,Theory of H p Spaces, Academic Press, New York and London, 1970. · Zbl 0215.20203 [5] J.-P. Kahane,Baire Theory in Fourier and Taylor series, Conference in honor of Donald Newman, Philadelphia, March 1996. [6] J.-P. Kahane,General Properties of Taylor series, 1896–1996, L’Escurial, Spain, June 1996. [7] J.-P. Kahane,A few generic properties of Fourier and Taylor series, Taiwan, November 1996. [8] E. S. Katsoprinakis and M. Papadimitrakis, in preparation. [9] S. Kierst and E. Szpirajn,Sur certaines singularités des fonctions analytiques uniformes, Fund. Math.21 (1933), 267–294. · Zbl 0008.07401 [10] D. Menchoff,Sur les series Trigonometriques Universelles, Comptes Rendus (Doklady) de l’Académie des Sciences de l’URSS, Vol. XLIX, no. 2 (1945), 79–82. · Zbl 0060.18504 [11] V. Nestoridis,Distribution of partial sums of the Taylor development of rational functions, Trans. Amer. Math. Soc.346 (1994), 283–295. · Zbl 0818.30001 [12] V. Nestoridis,Universal Taylor series, Ann. Inst. Fourier (Grenoble)46 (1996), 1293–1306. [13] I. I. Privalov,Boundary Properties of Analytic Functions, Moscow, 1941; 2nd ed. 1950. German transl.: Deutscher Verlag, Berlin, 1956. [14] W. Rudin,Real and Complex Analysis, McGraw-Hill, New York, 1966. · Zbl 0142.01701 [15] A. Zygmund,Trigonometric Series, second edition reprinted, Vols. I, II, Cambridge University Press, 1979.
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