On entire functions which share one value CM with their first derivative.(English)Zbl 0861.30032

This article is devoted to considering the following conjecture: Let $$f$$ be a nonconstant entire function of finite noninteger iterated order $\rho_1(f) = \lim\sup_{r\to \infty} \log \log T(r,f)/ \log r.$ If $$f$$, $$f'$$ share a value $$a\in \mathbb{C}$$ counting multiplicity, then for a constant $$c\in \mathbb{C} \backslash \{0\}$$, $$(f'-a)/(f-a)=c$$. Now, two theorems related to this conjecture will be proved: (1) The conjecture is true for $$a= 0$$. (2) If $$f$$ is nonconstant entire, $$N(r,{1 \over f'}) = S(r,f)$$ ad $$f$$, $$f'$$ share $$a=1$$ counting multiplicity, then for a constant $$c\in \mathbb{C} \backslash \{0\}$$, $$(f'-1)/(f-1) =c$$. Observe that the growth condition has been dropped in (2).
Reviewer: I.Laine (Joensuu)

MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text:

References:

 [1] G. G. Gundersen, Meromorphic functions that share finite values with their derivative, J. Math. Anal. Appl. 75 (1980), 441–446. · Zbl 0447.30018 [2] W. K. Hayman, M eromorphic Functions, Clarendon Press, Oxford, 1964. [3] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser Verlag, Basel, 1985. · Zbl 0682.30001 [4] E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), 195–206. · Zbl 0416.30028
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.