## A note on the Brück conjecture.(English)Zbl 1213.30058

Arch. Math. 95, No. 3, 257-268 (2010); erratum ibid. 99, No. 3, 255-259 (2012).
This paper is an interesting contribution to the recent literature around the R. Brück conjecture, originally formulated in [R. Brück, Results Math., 30, No.1–2, 21–24 (1996; Zbl 0861.30032)]. The key results here are related to the derivative version, the shift version and the difference operator version of the conjecture. As to the last two of these versions, see [J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang, J. Math. Anal. Appl., 355, No. 1, 352–363 (2009; Zbl 1180.30039)] and [K. Liu, L.-Z. Yang, Arch. Math., 92, No. 3, 270–278 (2009; Zbl 1173.30018)], respectively. The main result for the difference operator version reads as follows: Let $$f(z)$$ be a non-periodic transcendental entire function of finite order $$\rho$$, and suppose that $$f(z)$$ and the iterated difference operator $$\Delta_{\eta}^{n}f(z)$$ share a nonzero finite complex value $$a$$ CM. Then $$1\leq\rho\leq \lambda (f-a)+1$$, where $$\lambda (f-a)$$ stands for the exponent of convergence for the sequence of $$a$$-points of $$f$$. The corresponding result for the shift version is more complicated. The reviewer remarks that the abstract of the paper is somewhat misleading. Indeed, what is actually observed in the paper, is that the Brück conjecture (in its original derivative version) is equivalent to the following conjecture: Let $$f(z)$$ be a non-constant entire function such that the hyper-order $$\rho_{2}(f)<\infty$$ is not a natural number, and suppose that $$f(z),f'(z)$$ share a finite value $$a$$ CM. Then there exist nonzero constants $$A,c$$ such that $$f(z)=\frac{1}{c}(Ae^{cz}-a)+a$$.

### MSC:

 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 39A05 General theory of difference equations

### Keywords:

Brück conjecture; derivative; shift; difference; shared value

### Citations:

Zbl 0861.30032; Zbl 1180.30039; Zbl 1173.30018
Full Text:

### References:

 [1] Bergweiler W., Langley J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Phil. Soc. 142, 133–147 (2007) · Zbl 1114.30028 [2] Brück R.: On entire functions which share one value CM with their first derivative. Results Math. 30, 21–24 (1996) · Zbl 0861.30032 [3] Chen Z.X.: The growth of solutions of f” + e f’ + Q(z)f = 0 where the order of Q = 1. Science in China (Ser.A) 45, 290–300 (2002) · Zbl 1054.34139 [4] Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic f(z + {$$\eta$$}) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008) · Zbl 1152.30024 [5] Chiang Y.M., Feng S.J.: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Amer. Math. Soc. 361, 3767–3791 (2009) · Zbl 1172.30009 [6] Gundersen G.G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc. 37, 88–104 (1988) · Zbl 0638.30030 [7] Gundersen G.G., Yang L.Z.: Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl. 223, 88–95 (1998) · Zbl 0911.30022 [8] Halburd R.G., Korhonen R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006) · Zbl 1085.30026 [9] Hayman W.: Meromorphic Functions. Clarendon Press, Oxford (1964) [10] Heittokangas J. et al.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355, 352–363 (2009) · Zbl 1180.30039 [11] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Boston, 1985. · Zbl 0682.30001 [12] Laine I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin (1993) · Zbl 0784.30002 [13] Liu K., Yang L.Z.: Value distribution of the difference operator. Arch. Math. 92, 270–278 (2009) · Zbl 1173.30018 [14] Markushevich A.I.: Theory of functions of a complex variable, Vol. II. Prentice-Hall, Englewood Cliffs, New Jersey (1965) · Zbl 0135.12002 [15] Ozawa M.: On the existence of prime periodic entire functions. Kodai Math. Sem. Rep. 29, 308–321 (1978) · Zbl 0402.30025 [16] L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Lecture Notes in Math. 599, Berlin, Springer-Verlag, 101–103 (1977). · Zbl 0362.30026 [17] Yang L.: Value Distribution Theory and New Research. Science Press, Beijing (1982) (in Chinese) · Zbl 0633.30029 [18] C. C. Yang and H. X. Yi, The Uniqueness Theory of Meromorphic Functions, Math. Appl., 557, Kluwer Academic Publishers Group, Dordrecht, 2003. · Zbl 1070.30011
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.