zbMATH — the first resource for mathematics

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.

MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
sharing values
Full Text:
References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.