Application of the Second Main Theorem

Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume 162)


For a function meromorphic in the punctured plane z ≠ ∞ the theorem of Picard holds: the function w(z) completely omits at most two values unless it reduces to a constant ; we shall exclude this latter trivial case from future consideration. In connection with the first main theorem we have moreover found that such a function assumes almost all values a with a frequency N(r, a) that corresponds to the characteristic T(r, w). These results were sharpened for the case of a meromorphic function of finite nonintegral order to the extent that at most one value exists which is exceptional in the sense that its counting function is of lower order (class, type) than the characteristic. Even simple examples show that this no longer holds for integral orders; thus, the exponential function has two exceptional values (0, ∞), and in VIII, § 4, we have discussed a case where a lowering of the type or the class of N(r, a) occurs for two values of a. That the number of such exceptional values cannot exceed two, on the other hand, is the essential content of the extension of Picard’s theorem given by Borel [1].


Unit Disk Meromorphic Function Remainder Term Finite Order Counting Function 
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  1. 1.
    G. Valiron [2], R. Nevanlinna [2], [4].Google Scholar
  2. 1.
    If one of these numbers, say b, is equal to ∞, simply replace the quotient by wa.Google Scholar
  3. 1.
    R. Nevanlinna [4].Google Scholar
  4. 1.
    The reader is invited to investigate the above in many ways instructive example more carefully; cf. E. Ullrich [1]. In addition, it should be remarked that because of the relationship between the proximity of the derivative w′ to zero and to infinity, on the one hand, and the proximity of w to the exceptional values of the latter function, on the other, the quantity (math) could serve as a good measure for the total deficiency.Google Scholar
  5. 1.
    For short we speak of the Robin constant of a closed point set; to be more complete one should say: the Robin constant of that region bounded by the point set which contains the point at infinity.Google Scholar
  6. 3.
    The assumption w(0) ≠ ∞ is not an essential restriction.Google Scholar
  7. 3.
    If E n is not closed, then let γ n be the greatest lower bound of the Robin constants of all closed subsets of E n. Then instead of the capacity of E n, one can speak oi the inner capacity eγn of E n.Google Scholar
  8. 1.
    (2.18) and (2.21) were established by L. Ahlfors [3] as generalizations of certain older theorems of G. Valiron [5], J. E. Littlewood [1] and the author [5].Google Scholar
  9. 1.
    These theorems can be looked upon as extensions of certain older theorems of C. Caratheodory [1].Google Scholar
  10. 1.
    E. Picard [2].Google Scholar
  11. 2.
    This proof, which was first given by A. Bloch [2], is completely different from the original proof of Picard. Another proof using the tools of modern value distribution theory and which can be applied directly to any curve of genus p > 1 was given later by H. L. Selberg [2].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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