×

The filling discs dealing with multiple values of an algebroid function in the unit disc. (English) Zbl 1244.30063

The filling discs of an algebroid function are an interesting part of the value distribution theory. For an algebroid function defined on the complex plane, the existence of its filling discs was proved by D. Sun in [Chin. Ann. Math., Ser. A 16, No. 2, 238–243 (1995; Zbl 0831.30012)]. The corresponding results for algebroid functions of infinite order and zero order were obtained by Z. Gao [Acta Math. Sin. 40, No. 6, 823–830 (1997; Zbl 0948.30042)]. The existence of the sequence of filling discs of algebroid functions dealing with multiple values, of finite or infinite order, was first proved in [Z. Gao, Complex Variables, Theory Appl. 47, No. 3, 203–213 (2002; Zbl 1022.30036); Kodai Math. J. 23, No. 2, 151–163 (2000; Zbl 0963.30019)]. The existence of filling discs in the strong Borel direction of algebroid function with finite order was proved by Y. Huo and Y. Kong in [Bull. Korean Math. Soc. 47, No. 6, 1213–1224 (2010; Zbl 1206.30042)].
Compared with the case of the complex plane, it is interesting to investigate the algebroid functions defined in the unit disc, and there are some essential differences between these two cases. In this paper, the authors continue the work of Y. Kong [J. Math. Anal. Appl. 344, No. 2, 1158–1164 (2008; Zbl 1144.30014)] by considering the case dealing with multiple values, obtaining similar results.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

References:

[1] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. · Zbl 0115.06203
[2] H. Selbreg, Algebroide Funktionen und Umkehrfunktionen Abelscher Integrale, vol. 8 of Avhandlinger-Norske Videnskaps-Akademi i Oslo, 1934. · Zbl 0010.12301
[3] E. Ullrich, “Über den einfluss der verzweigtheit einer algebroide auf ihre wertverteilung,” Journal fur die Reine und Angewandte Mathematik, vol. 169, pp. 198-220, 1931. · Zbl 0003.21202 · doi:10.1515/crll.1932.167.198
[4] G. Valiron, “Sur la dérivée des fonctions algébroïdes,” Bulletin de la Société Mathématique de France, vol. 59, pp. 17-39, 1931. · Zbl 0002.27102
[5] D. C. Sun, “The filling circles of an algebroid function,” Chinese Annals of Mathematics A, vol. 16, no. 2, pp. 238-243, 1995. · Zbl 0831.30012
[6] Z. S. Gao, “A fundamental inequality for quasialgebroid functions and its application,” Acta Mathematica Sinica, vol. 40, no. 6, pp. 823-830, 1997. · Zbl 0948.30042
[7] Z. S. Gao, “On the multiple values of algebroid functions,” Kodai Mathematical Journal, vol. 23, no. 2, pp. 151-163, 2000. · Zbl 0963.30019 · doi:10.2996/kmj/1138044207
[8] Z. S. Gao, “On the value distribution of an algebroid function of infinite order,” Complex Variables. Theory and Application, vol. 47, no. 3, pp. 203-213, 2002. · Zbl 1022.30036 · doi:10.1080/0278107029000143
[9] Y. Y. Huo and Y. Y. Kong, “On filling discs in the strong Borel direction of algebroid function with finite order,” Bulletin of the Korean Mathematical Society, vol. 47, no. 6, pp. 1213-1224, 2010. · Zbl 1206.30042 · doi:10.4134/BKMS.2010.47.6.1213
[10] Z. X. Xuan, “The filling disks of an algebroid function in the unit disk,” Bulletin of the Australian Mathematical Society, vol. 81, no. 3, pp. 455-464, 2010. · Zbl 1189.30070 · doi:10.1017/S0004972709001233
[11] Y. Y. Kong, “On filling discs in Borel radius of meromorphic mapping with finite order in the unit circle,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 1158-1164, 2008. · Zbl 1144.30014 · doi:10.1016/j.jmaa.2008.04.010
[12] N. Wu and Z. X. Xuan, “Common Borel radii of an Algebroid function and its derivative,” Results in Mathematics. In press. · Zbl 1296.30045 · doi:10.1007/s00025-011-0132-y
[13] N. Wu and Z. X. Xuan, “On T points of Algebroid functions,” Mathematica Slovaca, vol. 62, no. 1, pp. 39-48, 2012. · Zbl 1274.30131 · doi:10.2478/s12175-011-0065-7
[14] Y. Y. Kong, “A new singular radius of algebroidal functions in the unit disc,” Acta Mathematica Sinica, vol. 54, no. 2, pp. 257-264, 2011. · Zbl 1240.30156
[15] Z. S. Gao and F. Z. Wang, “Theorems on covering surfaces and multiple values of algebroid functions,” Acta Mathematica Sinica, vol. 44, no. 5, pp. 805-814, 2001. · Zbl 1125.30309
[16] Z. X. Xuan and Z. S. Gao, “The Borel direction of the largest type of algebroid functions dealing with multiple values,” Kodai Mathematical Journal, vol. 30, no. 1, pp. 97-110, 2007. · Zbl 1134.30023 · doi:10.2996/kmj/1175287625
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.