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The Order of Certain Classes of Functions Defined in the Unit Disk

Published online by Cambridge University Press:  22 January 2016

D. C. Rung
D. C. Rung*
Affiliation:
Pennsylvania State University, University Park, Pennsylvania
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Let D denote the open unit disk in the complex plane and let C be the boundary of D. If, for a given complex-valued function f(z) defined in D, the existence of a subset M of C is known, with the linear measure of M equal to 2 π, as well as an estimate on the growth of |f/(z)| I on sequences in D which tends to a point of M, then such a result will be called a “statistical” result on order. This terminology is due to Lelong-Ferrand [3].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 26 , February 1966 , pp. 39 - 52
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Gehring, F.. On the radial order of subharmonic functions, J. Math. Soc. Japan, Vol. 9 (1957), pp. 77–79.CrossRefGoogle Scholar
[2] Lehto, O., and Virtanen, K., Boundary behavior and normal meromorphic functions, Acta Math. 97 (1957), pp. 47–65.CrossRefGoogle Scholar
[3] Lelong-Ferrand, J., Representation conforme at transformations à integrale de Dirichlet bornee, Paris 1955.Google Scholar
[4] Rung, D. C., Results on the order of holomorphic functions defined in the unit disk, J. Math. Soc. Japan, Vol. 14, 3 (1962), pp. 322–332.CrossRefGoogle Scholar
[5] Seidel, W., and Walsh, J. L., On the derivatives of functions analytic in the unit circle and their radii of univalence and of p-valence, Trans. Am. Math. Soc., Vol. 52 (1942), pp. 128–216.Google Scholar