The deficiency of entire functions with Fejér gaps
Annales de l'Institut Fourier, Volume 33 (1983) no. 3, p. 39-58
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.
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.
@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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {33},
     number = {3},
     year = {1983},
     pages = {39-58},
     doi = {10.5802/aif.930},
     mrnumber = {723947},
     zbl = {0489.30028},
     mrnumber = {84m:30046},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1983__33_3_39_0}
}
Murai, Takafumi. The deficiency of entire functions with Fejér gaps. Annales de l'Institut Fourier, Volume 33 (1983) no. 3, pp. 39-58. doi : 10.5802/aif.930. http://www.numdam.org/item/AIF_1983__33_3_39_0/

[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 215993 | 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 | MR 1511474

[5] W.H.J. Fuchs, Proof of a conjecture of G. Pólya concerning gap series, Illinois J. Math., 7 (1963), 661-667. | MR 159933 | 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 252644 | MR 40 #5863 | Zbl 0179.11001

[7] W.K. Hayman, Meromorphic functions, Oxford, 1964. | MR 164038 | 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 306497 | 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. | 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