# zbMATH — the first resource for mathematics

Complex oscillation of differential polynomials in the unit disc. (English) Zbl 1299.34284
Summary: We consider the complex differential equations $f''+A_1(z)f'+A_0(z)f = F,$ where $$A_0\not\equiv 0$$, $$A_1$$ and $$F$$ are analytic functions in the unit disc $$\Delta =\{z : | z| < 1\}$$. We obtain results on the order and the exponent of convergence of zero-points in $$\Delta$$ of the differential polynomials $$g_f = d_2 f'' + d_1 f' +d_ 0f$$ with non-simultaneously vanishing analytic coefficients $$d_2$$, $$d_1$$, $$d_0$$. We answer a question posed by J. Tu and C. F. Yi [J. Math. Anal. Appl. 340, No. 1, 487–497 (2008; Zbl 1141.34054)] for the case of second-order linear differential equations in the unit disc.

##### 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 34M03 Linear ordinary differential equations and systems in the complex domain
Full Text:
##### References:
 [1] S. Bank, General theorem concerning the growth of solutions of first-order algebraic differential equations, Compos. Math., 25 (1972), 61–70. · Zbl 0246.34006 [2] S. Bank and I. Laine, On the oscillation theory of f” + Af = 0 where A is entire, Trans. Amer. Math. Soc., 273 (1982), 351–363. · Zbl 0505.34026 [3] S. Bank and I. Laine, On the zeros of meromorphic solutions of second order linear differential equations, Comment. Math. Helv., 58 (1983), 656–677. · Zbl 0532.34008 · doi:10.1007/BF02564659 [4] B. Belaïdi, Oscillation of fast growing solutions of linear differential equations in the unit disc, Acta Univ. Sapientiae Mathematica, 2, 1 (2010), 25–38. · Zbl 1215.34112 [5] B. Belaïdi and A. El Farissi, Differential polynomials generated by some complex linear differential equations with meromorphic coefficients, Glas. Mat. Ser. III, 43(63) (2008), 363–373. · Zbl 1166.34054 · doi:10.3336/gm.43.2.09 [6] T. B. Cao, The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl., 352 (2009), 739–748. · Zbl 1160.34357 · doi:10.1016/j.jmaa.2008.11.033 [7] T. B. Cao and H. X. Yi, The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl., 319 (2006), 278–294. · Zbl 1105.34059 · doi:10.1016/j.jmaa.2005.09.050 [8] T. B. Cao and H. X. Yi, On the complex oscillation theory of f” + Af = 0 where A(z) is analytic in the unit disc, Math. Nachr., 282 (2009), 820–831. · Zbl 1193.34174 · doi:10.1002/mana.200610774 [9] T. B. Cao and H. X. Yi, On the complex oscillation theory of linear differential equations with analytic coefficients in the unit disc, Acta Math. Sci., 28A(6) (2008), 1046–1057. · Zbl 1199.34471 [10] Z. X. Chen and K. H. Shon, On the growth and fixed points of solutions of second order differential equations with meromorphic coefficients, Acta Math. Sin. (Engl. Ser.), 21 (2005), 753–764. · Zbl 1100.34067 · doi:10.1007/s10114-004-0434-z [11] A. El Farissi and B. Belaïdi, On oscillation theorems for differential polynomials, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), No. 22, 1–10. · Zbl 1199.34469 [12] A. El Farissi, B. Belaïdi and Z. Latreuch, Growth and oscillation of differential polynomials in the unit disc, Electron. J. Diff. Equ., 2010 (2010), No. 87, 1–7. · Zbl 1213.34112 [13] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964. · Zbl 0115.06203 [14] J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1–54. · Zbl 0965.34075 [15] J. Heittokangas, R. Korhonen and J. Rättyä, Fast growing solutions of linear differential equations in the unit disc, Results Math., 49 (2006), 265–278. · Zbl 1120.34071 · doi:10.1007/s00025-006-0223-3 [16] J. Heittokangas, R. Korhonen and J. Rättyä, Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci. Fenn. Math., 29 (2004), 233–246. · Zbl 1057.34111 [17] I. Laine, Nevanlinna theory and complex differential equations, W. de Gruyter, Berlin, 1993. [18] I. Laine, Complex differential equations, Handbook of Differential Equations: Ordinary Differential Equations, 4 (2008), 269–363. · Zbl 1209.34002 · doi:10.1016/S1874-5725(08)80008-9 [19] I. Laine and J. Rieppo, Differential polynomials generated by linear differential equations, Complex Var. Theory Appl., 49 (2004), 897–911. · Zbl 1080.34076 · doi:10.1080/02781070410001701092 [20] Y. Z. Li, On the growth of the solution of two-order differential equations in the unit disc, Pure Appl. Math., 4 (2002), 295–300. · Zbl 1128.34330 [21] M. Tsuji, Potential theory in modern function theory, New York, Chelsea, 1975. · Zbl 0322.30001 [22] J. Tu and C. F. Yi, On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order, J. Math. Anal. Appl., 340 (2008), 487–497. · Zbl 1141.34054 · doi:10.1016/j.jmaa.2007.08.041 [23] J. Tu and C. F. Yi, Growth of solutions of higher order linear differential equations with the coefficient A 0 being dominant, Acta Math. Sci., 30A(4) (2010), 945–952. · Zbl 1240.30126 [24] J. Wang and H. X. Yi, Fixed points and hyper-order of differential polynomials generated by solutions of differential equation, Complex Var. Theory Appl., 48 (2003), 83–94. · Zbl 1071.30029 · doi:10.1080/0278107021000037048 [25] G. Zhang and A. Chen, Fixed points of the derivative and k-th power of solutions of complex linear differential equations in the unit disc, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), No. 48, 1–9. · Zbl 1188.30047
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.