Murai: The deficiency of entire functions with Fejér gaps

Table of contents for this issue | Previous article | Next article

Takafumi Murai
The deficiency of entire functions with Fejér gaps
Annales de l'institut Fourier, 33 no. 3 (1983), p. 39-58, doi: 10.5802/aif.930
Article PDF | Reviews MR 84m:30046 | Zbl 0489.30028

Résumé - Abstract

We say that an entire function $f(z)=\sum _{k=0}a_kz^{n_k}~(0=n_0< n_1< n_2< \ldots )$ has Fejér gaps if $\sum ^\infty _{k=1}1/n_k< \infty .$ The main result of this paper is as follows: An entire function with Fejér gaps has no finite deficient value.

Bibliography

[1] M. BIERNACKI, Sur les équations algébriques contenant des paramètres arbitraires, Bull. Int. Acad. Polon. Sci. Lett., Sér. A (III) (1927), 542-685. JFM 56.0863.02
[2] M. BIERNACKI, Sur les fonctions entières à séries lacunaires, C.R.A.S., Paris, Sér. A-B, 187 (1928), 477-479. JFM 54.0349.02
[3] J. CLUNIE, On integral functions with exceptional values, J. London Math. Soc., 42 (1967), 393-400. MR 35 #6828 | Zbl 0164.08701
[4] L. FEJÉR, Über die Wurzel vom kleinsten absoluten Betrage einer algebraischen Gleichung, Math. Annalen, (1908), 413-423. JFM 39.0123.02
[5] W.H.J. FUCHS, Proof of a conjecture of G. Pólya concerning gap series, Illinois J. Math., 7 (1963), 661-667. Article | MR 28 #3149 | Zbl 0113.28702
[6] W.H.J. FUCHS, Nevanlinna theory and gap series, Symposia on theoretical physics and mathematics, V.9, Plenum Press, New York, 1969. MR 40 #5863 | Zbl 0179.11001
[7] W.K. HAYMAN, Meromorphic functions, Oxford, 1964. MR 29 #1337 | Zbl 0115.06203
[8] W.K. HAYMAN, Angular value distribution of power series with gaps, Proc. London Math. Soc., (3), 24 (1972), 590-624. MR 46 #5623 | Zbl 0239.30035
[9] T. KÖVARI, On the Borel exceptional values of lacunary integral functions, J. Analyse Math., 9 (1961/1962), 71-109. MR 25 #3174 | Zbl 0101.05302
[10] T. KÖVARI, A gap-theorem for entire functions of infinite order, Michigan Math. J., 12 (1965), 133-140. Article | MR 31 #351 | Zbl 0152.06604
[11] T. MURAI, The deficiency of gap series, Analysis (1983) to appear. Zbl 0533.30028
[12] V.S. PETRENKO, Growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR, Ser. Mat., 33 (1969), 414-454. Zbl 0194.11101
[13] G. PÓLYA, Lücken und Singularitäten von Porenzreihen, Math. Z., 29 (1929), 549-640. JFM 55.0186.02
[14] L.R. SONS, An analogue of a theorem of W.H.J. Fuchs on gap series, Proc. London Math. Soc., (3), 21 (1970), 525-539. MR 44 #6962 | Zbl 0206.08801
[15] A. WIMAN, Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Function und dem grössten Gliede der zugehörigen Taylorschen Reihe, Acta Math., 37 (1914), 305-326. JFM 45.0641.02
[16] A. ZYGMUND, Trigonometric series I, Cambridge, 1959. Zbl 0085.05601