# zbMATH — the first resource for mathematics

Complex oscillation theory of differential polynomials. (English) Zbl 1244.34108
Summary: We investigate the relationship between small functions and differential polynomials $$g_{f}(z)=d_{2}f^{\prime \prime }+d_{1}f^{\prime }+d_{0}f$$, where $$d_{0}(z)$$, $$d_{1}(z)$$, $$d_{2}(z)$$ are entire functions that are not all equal to zero with $$\rho (d_j)<1$$ $$(j=0,1,2)$$ generated by solutions of the differential equation $$f^{\prime \prime }+A_{1}(z) e^{az}f^{\prime }+A_{0}(z) e^{bz}f=F$$, where $$a,b$$ are complex numbers that satisfy $$ab( a-b) \neq 0$$ and $$A_{j}( z) \lnot \equiv 0$$ ($$j=0,1$$), $$F(z) \lnot \equiv 0$$ are entire functions such that $$\max \left\{ \rho (A_j),\, j=0,1,\, \rho (F)\right\} <1$$.
##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text:
##### References:
 [1] Amemiya, I., Ozawa, M.: Non-existence of finite order solutions of $$w^{\prime \prime }+e^{-z}w^{\prime }+Q(z) w=0$$. Hokkaido Math. J. 10 (1981), 1-17 · Zbl 0554.34003 [2] Belaïdi, B.: Growth and oscillation theory of solutions of some linear differential equations. Mat. Vesnik 60, 4 (2008), 233-246. · Zbl 1274.30112 [3] Belaïdi, B.: Oscillation of fixed points of solutions of some linear differential equations. Acta Math. Univ. Comenian. (N.S.) 77, 2 (2008), 263-269. · Zbl 1174.34528 · eudml:130852 [4] Belaïdi, B., El Farissi, A.: Relation between differential polynomials and small functions. Kyoto J. Math. Vol. 50, 2 (2010), 453-468. · Zbl 1203.34148 · doi:10.1215/0023608X-2009-019 [5] Chen, Z. X.: Zeros of meromorphic solutions of higher order linear differential equations. Analysis 14 (1994), 425-438. · Zbl 0815.34003 [6] Chen, Z. X.: The fixed points and hyper-order of solutions of second order complex differential equations. (in Chinese), Acta Math. Sci. Ser. A Chin. 20, 3 (2000), 425-432. · Zbl 0980.30022 [7] Chen, Z. X.: The growth of solutions of $$f^{\prime \prime }+e^{-z}f^{\prime }+Q(z)f=0$$ where the order $$(Q) =1$$. Sci. China Ser. A 45, 3 (2002), 290-300. · Zbl 1054.34139 [8] Chen, Z. X., Shon, K. H.: On the growth and fixed points of solutions of second order differential equations with meromorphic coefficients. Acta Math. Sin. (Engl. Ser.) 21, 4 (2005), 753-764. · Zbl 1100.34067 · doi:10.1007/s10114-004-0434-z [9] Frei, M.: Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten. Comment. Math. Helv. 35 (1961), 201-222. · Zbl 0115.06903 · doi:10.1007/BF02567016 · eudml:139218 [10] Frei, M.: Über die Subnormalen Lösungen der Differentialgleichung $$w^{\prime \prime }+e^{-z}w^{\prime }+( Konst.) w=0$$. Comment. Math. Helv. 36 (1961), 1-8. · Zbl 0115.06904 · doi:10.1007/BF02566887 · eudml:139223 [11] Gundersen, G. G.: On the question of whether $$f^{\prime \prime }+e^{-z}f^{\prime }+B(z) f=0$$ can admit a solution $$f\lnot \equiv 0$$ of finite order. Proc. Roy. Soc. Edinburgh Sect. A 102, 1-2 (1986), 9-17. · Zbl 0598.34002 · doi:10.1017/S0308210500014451 [12] Hayman, W. K.: Meromorphic functions. Clarendon Press, Oxford, 1964. · Zbl 0115.06203 [13] Laine, I., Rieppo, J.: Differential polynomials generated by linear differential equations. Complex Var. Theory Appl. 49, 12 (2004), 897-911. · Zbl 1080.34076 · doi:10.1080/02781070410001701092 [14] Langley, J. K.: On complex oscillation and a problem of Ozawa. Kodai Math. J. 9, 3 (1986), 430-439. · Zbl 0609.34041 · doi:10.2996/kmj/1138037272 [15] Levin, B. Ya.: Lectures on entire functions. American Mathematical Society, Providence, RI, 1996 Translations of Mathematical Monographs, Vol. 150. · Zbl 0856.30001 [16] Liu, M. S., Zhang, X. M.: Fixed points of meromorphic solutions of higher order Linear differential equations. Ann. Acad. Sci. Fenn. Math. 31, 1 (2006), 191-211. · Zbl 1094.30036 · eudml:126327 [17] Nevanlinna, R.: Eindeutige analytische Funktionen. Die Grundlehren der mathematischen Wissenschaften Band 46, Zweite Auflage, Reprint, Springer-Verlag, Berlin-New York, 1974. · Zbl 0278.30002 · eudml:203720 [18] Ozawa, M.: On a solution of $$w^{\prime \prime }+e^{-z}w^{\prime }+( az+b) w=0$$. Kodai Math. J. 3, 2 (1980), 295-309. · Zbl 0463.34028 · doi:10.2996/kmj/1138036197 [19] Wang, J., Yi, H. X.: Fixed points and hyper order of differential polynomials generated by solutions of differential equation. Complex Var. Theory Appl. 48, 1 (2003), 83-94. · Zbl 1071.30029 · doi:10.1080/0278107021000037048 [20] Wang, J., Laine, I.: Growth of solutions of second order linear differential equations. J. Math. Anal. Appl. 342, 1 (2008), 39-51. · Zbl 1151.34069 · doi:10.1016/j.jmaa.2007.11.022 [21] Zhang, Q. T., Yang, C. C.: The Fixed Points and Resolution Theory of Meromorphic Functions. Beijing University Press, Beijing, 1988
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.