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Entire functions that share one value with their derivatives. (English) Zbl 0840.30013
Two meromorphic functions \(f\) and \(g\) share the complex value \(a\) if \(f(z) = a\) implies \(g(z) = a\) and vice versa. The value \(a\) is shared CM (counting multiplicities) if, in addition, \(f\) and \(g\) have the same multiplicities at each \(a\)-point. Let \(f\) be a nonconstant entire function. G. Jank, E. Mues and L. Volkmann [Complex Variables, Theory Appl. 6, No. 1, 51-71 (1986; Zbl 0603.30037)] proved the following result: If \(f\) and \(f'\) share the value \(a \neq 0\) and if \(f(z) = a\) implies \(f''(z) = a\) then \(f \equiv f'\). In the paper under review it is shown that \(f''\) cannot be replaced by \(f^{(k)}\), \(k \geq 3\), in this theorem. the author proves the following generalisation (Theorem 1): If \(f\) and \(f'\) share the value \(a \neq 0\) CM and if \(f(z) = a\) implies \(f^{(n)} (z) = f^{(n + 1)} (z) = a\), \(n \geq 1\), then \(f \equiv f^{(n)}\). Two more results on entire functions sharing one value with their derivatives are presented.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
sharing values
Full Text: DOI
[1] L. RUBEL AND C. C. YANG, Values shared by entire functions and their deriva-tives, Complex Analysis, Kentucky 1976, Lecture Notes in Math., 599, Springer-Verlag, Berlin-Heidelberg-New York, 1977, 101-103. · Zbl 0362.30026
[2] G. JANK, E. MUES AND L. VOLKMAN, Meromorphe funktionen, die mit ihre ersten und Zweiten Ableitung einen endlichen Wert teilen, Complex Variables Theory Appl. (1), 6 (1986), 51-71. · Zbl 0603.30037
[3] L. YANG, Value Distribution Theory and New Research on it, Science Press, Bei jing, 1982 (in Chinese). · Zbl 0633.30029
[4] G. FRANK AND W. OHLENROTH, Meromorphe funktionen, die mit einer ihre Ableitungen Werte teilen, Complex Variables Theory Appl. (1), 6 (1986), 23-37. · Zbl 0537.30019
[5] G. GUNDERSEN, Meromorphic functions that share four values, Trans. Amer Math. Soc, 277 (1983) 545-567. correction. 304 (1987), 847-850. · Zbl 0508.30029 · doi:10.2307/1999223
[6] G. GUNDERSEN, Meromorphic functions that share three or four values, J. Londo Math. Soc. (2), 20 (1979), 457-466. · Zbl 0413.30025 · doi:10.1112/jlms/s2-20.3.457
[7] G. GUNDERSEN, Meromorphic functions that share three values IM and a fourt value CM, Complex Variables Theory Appl., 20 (1992), 99-106. · Zbl 0773.30032
[8] W. K. HAYMAN, Meromorphic Functions, Clarendon Press, Oxford, 1964 · Zbl 0115.06203
[9] H. ZHONG, On the extension of F. Nevanlinna Conjecture, Acta Math. Sinic (1), 36 (1993), 90-98. · Zbl 0781.30028
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