The Second Main Theorem in the Theory of Meromorphic Functions

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


After discovering the important symmetry property of a function w(z) meromorphic for |z| < R ≦ ∞, which is expressed in the first main theorem through the invariance of the sum m(r, a) + N(r, a), we set up as our main goal the more careful investigation of the relative strength of the two terms in the sum, of the proximity component m(r, a) and of the counting component N(r, a). Individual results in this direction have already been achieved:
  1. 1.

    Picard’s theorem shows that the counting function for a non-constant function meromorphic in the plane z ≠ ∞ can vanish for at most two values of a.

  2. 2.

    For a meromorphic function of finite nonintegral order there is at most one Picard exceptional value, and the sharper theorems of VIII, § 4, hold.

  3. 3.

    That the counting function N(r, a) is in general, i.e., for the great majority of values a, large in comparison with the proximity function m(r, a) follows from the mean value theorems of VI, § 3 and § 4.



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  1. 1.
    R. Nevanlinna [4].Google Scholar
  2. 1.
    In fact, no example is known to the author of a transcendental function where this asymptotic equality would not hold strictly.Google Scholar
  3. 1.
    The idea of introducing the derivative as a comparison function goes back to Borel [1]. This thought is basic to Borel’s proof of that extension of Picard’s theorem which is known as the “Picard-Borel theorem”. E. Ullrich [1] in his version of the proof of the second main theorem has brought out the meaning of the derivative as a collection point for the exceptional values especially clearly.Google Scholar
  4. 1.
    The significance of this general double inequality for investigating the value distribution of the derivative of a meromorphic function has been emphasized by E. Ullrich [1]. Also cf. E. F. Collingwood [1].Google Scholar
  5. 2.
    The author first proved this relation for entire functions in the case p = 3 (1923). The possibility of extending this to the general case p > 3 (for entire functions) was first indicated to me in a letter from J. E. Littlewood in the year 1924. This extension was given independently by E. F. Collingwood [1].Google Scholar
  6. 1.
    Above we have assumed q ≧ 3. But since the mean values m(r, w v) are ≧ 0, the theorem also holds for q = 1, 2.Google Scholar
  7. 1.
    As is clear from the proof that follows, the theorem is also valid when f′(x) is discontinuous only at isolated points, a hypothesis satisfied in the applications that follow.Google Scholar
  8. 1.
    A similar proof has been given by H. L. Selberg [1] and F. Nevanlinna [4].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  1. 1.Academy of FinlandFinland

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