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, Volume 5, Issue 2, pp 155–163 | Cite as

Über den Index der Lösungen Linearer Differentialgieichungen

  • Günter Frank
  • Erwin Mues


In an earliner paper the authors proved the follwoing inequalities for the index (I(r,f) of an entire function f
$$\max (0,\lambda - 1) \leqslant \mathop {\lim }\limits_{r \to \infty } \sup \frac{{l\mathop o\limits^ + g I(r,f)}}{{\log r}} \leqslant \lambda $$
, where λ is the order of f. It is known that the upper bound is sharp. In this paper the authors prove that the lower bound cannot be sharpened if λ≥1 is a rational number. In this direction it is shown that for certain solutions of linear differential equations
$$w^{(n)} + a_{n - 1} w^{(n - 1)} + \ldots + a_o w = 0$$
with polynomial coefficients aj in the left side of the above inequality “≤” and “lim sup” are replaced by “=” and “lim”. Also it is proved that the differential equation has constant coefficients if and only if every solution is of bounded index.


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  1. [1]
    BOAS, R.P.: Entire functions. New York: Academic Press 1954.Google Scholar
  2. [2]
    FRANK, G: Picardsche Ausnahmewerte bei Lösungen linearer Differentialgleichungen. Dissertation an der Universität Karlsruhe (TH), 1969.Google Scholar
  3. [3]
    FRANK, G.: Über den Index einer ganzen Funktion. (Erscheint im Arch. der Math.)Google Scholar
  4. [4]
    FRANK, G., MUES, E.: Über dasWachstum des Index ganzer Funktionen. (Zum Druck eingereitch.)Google Scholar
  5. [5]
    LEPSON, B.: Differential equations of infinite order. Hyperdirichlet series and eintire functions of bounded indes. Entire functions and related parts of analysis. (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Providence, R.I.: Amer. Math. Soc. 1968.Google Scholar
  6. [6]
    PÖSCHL, K.: Über anwachsen und Nullstellenverteilung der ganzen transzendenten Lösungen linearer Differentialgleichungen. I. J. reine und angew. Math. 199, 121–138 (1958).Google Scholar
  7. [7]
    WITTICH, H.. Neuere Untersuchungen über eindeutige analytische Funktionen. Berlin-Göttingen-Heidelberg: Springer 1955.Google Scholar
  8. [8]
    WITTICH, H.: Zur Theorie linearer Differentialgleichungen im Komplexen. Ann. Acad. Sci. Fenn. Ser. AI, Nr. 379 (1966).Google Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • Günter Frank
    • 1
  • Erwin Mues
    • 1
  1. 1.Mathematisches Institut I der UniversitätKarlsruhe

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