The deficiency of entire functions with Fejér gaps
Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 39-58.

On dit qu’une fonction entière f(z)=∑ k=0 a k z n k (0=n 0 <n 1 <n 2 <...) a des lacunes de Fejér si ∑ k=1 ∞ 1/n k <∞. Le résultat principal de cet article est le suivant : Une fonction entière avec des lacunes de Fejér n’a pas de valeur déficiente finie.

We say that an entire function f(z)=∑ k=0 a k z n k (0=n 0 <n 1 <n 2 <...) has Fejér gaps if ∑ k=1 ∞ 1/n k <∞. The main result of this paper is as follows: An entire function with Fejér gaps has no finite deficient value.

@article{AIF_1983__33_3_39_0,
     author = {Murai, Takafumi},
     title = {The deficiency of entire functions with {Fej\'er} gaps},
     journal = {Annales de l'Institut Fourier},
     pages = {39--58},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     number = {3},
     year = {1983},
     doi = {10.5802/aif.930},
     zbl = {0489.30028},
     mrnumber = {723947},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.930/}
}
TY  - JOUR
AU  - Murai, Takafumi
TI  - The deficiency of entire functions with Fejér gaps
JO  - Annales de l'Institut Fourier
PY  - 1983
SP  - 39
EP  - 58
VL  - 33
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.930/
DO  - 10.5802/aif.930
LA  - en
ID  - AIF_1983__33_3_39_0
ER  - 
%0 Journal Article
%A Murai, Takafumi
%T The deficiency of entire functions with Fejér gaps
%J Annales de l'Institut Fourier
%D 1983
%P 39-58
%V 33
%N 3
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.930/
%R 10.5802/aif.930
%G en
%F AIF_1983__33_3_39_0
Murai, Takafumi. The deficiency of entire functions with Fejér gaps. Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 39-58. doi : 10.5802/aif.930. http://www.numdam.org/articles/10.5802/aif.930/

[1] M. Biernacki, Sur les équations algébriques contenant des paramètres arbitraires, Bull. Int. Acad. Polon. Sci. Lett., Sér. A (III) (1927), 542-685. | JFM

[2] M. Biernacki, Sur les fonctions entières à séries lacunaires, C.R.A.S., Paris, Sér. A-B, 187 (1928), 477-479. | JFM

[3] J. Clunie, On integral functions with exceptional values, J. London Math. Soc., 42 (1967), 393-400. | MR | Zbl

[4] L. Fejér, Über die Wurzel vom kleinsten absoluten Betrage einer algebraischen Gleichung, Math. Annalen, (1908), 413-423. | EuDML | JFM | MR

[5] W.H.J. Fuchs, Proof of a conjecture of G. Pólya concerning gap series, Illinois J. Math., 7 (1963), 661-667. | MR | Zbl

[6] W.H.J. Fuchs, Nevanlinna theory and gap series, Symposia on theoretical physics and mathematics, V.9, Plenum Press, New York, 1969. | MR | Zbl

[7] W.K. Hayman, Meromorphic functions, Oxford, 1964. | MR | Zbl

[8] W.K. Hayman, Angular value distribution of power series with gaps, Proc. London Math. Soc., (3), 24 (1972), 590-624. | MR | Zbl

[9] T. Kövari, On the Borel exceptional values of lacunary integral functions, J. Analyse Math., 9 (1961/1962), 71-109. | MR | Zbl

[10] T. Kövari, A gap-theorem for entire functions of infinite order, Michigan Math. J., 12 (1965), 133-140. | MR | Zbl

[11] T. Murai, The deficiency of gap series, Analysis (1983) to appear. | Zbl

[12] V.S. Petrenko, Growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR, Ser. Mat., 33 (1969), 414-454. | Zbl

[13] G. Pólya, Lücken und Singularitäten von Porenzreihen, Math. Z., 29 (1929), 549-640. | JFM

[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 | Zbl

[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

[16] A. Zygmund, Trigonometric series I, Cambridge, 1959. | Zbl

Cité par Sources :