zbMATH — the first resource for mathematics

On the growth of solutions of some higher order linear differential equations. (English) Zbl 1265.34327
Summary: We discuss the growth of solutions of the higher-order nonhomogeneous linear differential equation $f^{(k)}+A_{k-1}f^{(k-1)}+\dotsb +A_2f''+(D_1(z) +A_1(z) e^{az})f'+( D_0(z)+A_0(z) e^{bz}) f=F (k\geq 2),$ where $$a$$, $$b$$ are complex constants that satisfy $$ab(a-b) \neq 0$$ and $$A_j(z)$$ $$(j=0,1,\dotsc ,k-1)$$, $$D_j(z)$$ $$(j=0,1)$$ and $$F(z)$$ are entire functions with $$\max \{\rho (A_0),\rho (A_1), \dotsc ,\rho (A_{k-1}),\rho (D_0),\rho (D_1)\}<1$$. We also investigate the relationship between small functions and the solutions of the above equation.

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] I. Amemiya, M. Ozawa: Non-existence of finite order solutions of w”+e w’+Q(z)w = 0. Hokkaido Math. J. 10 (1981), 1–17. · Zbl 0554.34003 · doi:10.14492/hokmj/1381758109 [2] B. Belaïdi: Growth and oscillation theory of solutions of some linear differential equations. Mat. Vesn. 60 (2008), 233–246. · Zbl 1274.30112 [3] B. Belaïdi, A. El Farissi: Differential polynomials generated by some complex linear differential equations with meromorphic coefficients. Glas. Mat., Ser. III 43 (2008), 363–373. · Zbl 1166.34054 · doi:10.3336/gm.43.2.09 [4] Z.X. Chen Zeros of meromorphic solutions of higher order linear differential equations. Analysis 14 (1994), 425–438. · Zbl 0815.34003 [5] Z.X. Chen: The growth of solutions of f”+e f’+Q(z)f = 0 where the order (Q) = 1. Sci. China, Ser. A 45 (2002), 290–300. · Zbl 1054.34139 [6] Z.X. Chen, K.H. Shon: On the growth of solutions of a class of higher order differential equations. Acta Math. Sci., Ser. B, Engl. Ed. 24 (2004), 52–60. · Zbl 1056.30029 [7] M. Frei: Über die Lösungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten. Comment. Math. Helv. 35 (1961), 201–222. · Zbl 0115.06903 · doi:10.1007/BF02567016 [8] M. Frei: Über die subnormalen Lösungen der Differentialgleichung w” + e w’ + konst.w = 0. Comment. Math. Helv. 36 (1961), 1–8. · Zbl 0115.06904 · doi:10.1007/BF02566887 [9] G.G. Gundersen: On the question of whether f” + e f’ + B(z)f = 0 can admit a solution f 0 of finite order. Proc. R. Soc. Edinb., Sect. A 102 (1986), 9–17. · Zbl 0598.34002 · doi:10.1017/S0308210500014451 [10] G.G. Gundersen: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc., II. Ser. 37 (1988), 88–104. · Zbl 0638.30030 [11] W.K. Hayman Meromorphic functions. Clarendon Press, Oxford, 1964. [12] G. Jank, L. Volkmann Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen. Birkhäuser, Basel, 1985. · Zbl 0682.30001 [13] J.K. Langley: On complex oscillation and a problem of Ozawa. Kodai Math. J. 9 (1986), 430–439. · Zbl 0609.34041 · doi:10.2996/kmj/1138037272 [14] A. I. Markushevich Theory of functions of a complex variable, Vol. II. Prentice-Hall, Englewood Cliffs, 1965. [15] R. Nevanlinna: Eindeutige analytische Funktionen, Zweite Auflage. Reprint. Die Grundlehren der mathematischenWissenschaften, Band 46. Springer, Berlin-Heidelberg-New York, 1974. (In German.) [16] M. Ozawa On a solution of w” + e w’ + (az + b)w = 0. Kodai Math. J. 3 (1980), 295–309. · Zbl 0463.34028 · doi:10.2996/kmj/1138036197 [17] J. Wang, I. Laine: Growth of solutions of second order linear differential equations. J. Math. Anal. Appl. 342 (2008), 39–51. · Zbl 1151.34069 · doi:10.1016/j.jmaa.2007.11.022 [18] H.Y. Xu, T.B. Cao: Oscillation of solutions of some higher order linear differential equations. Electron. J. Qual. Theory Differ. Equ., paper No. 63, 18 pages (2009).
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.