On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

Benharrat Belaïdi; Habib Habib

doi:10.2478/auom-2013-0022

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On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 21 (2013): Issue 2 (June 2013)

By: Benharrat Belaïdi and Habib Habib

Open Access

|Sep 2013

Abstract

In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = q_mz^m + ... + q₁z + q₀ is a nonoonstant polynomial and a₁, a₂ are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

References

[1] B. Belaïdi, Growth and oscillation theory of solutions of some linear differential equations. Mat. Vesnik 60 (2008), no. 4, 233-246.
Search in Google Scholar
[2] Z. X. Chen, Zeros of meromorphic solutions of higher order linear differential equations, Analysis 14 (1994), no. 4, 425-438.
Search in Google Scholar
[3] Z. X. Chen, The fixed points and hyper-order of solutions of second order complex differential equations (in Chinese), Acta Math. Sci. Ser. A Chin. Ed. 20 (2000), no. 3, 425-432.
Search in Google Scholar
[4] Z. X. Chen, The growth of solutions of f" -I- e~^z f + Q (z) f = 0 where the order(Q) = 1, Sci. China Ser. A 45 (2002), no. 3, 290-300.
Search in Google Scholar
[5] Z. X. Chen and K. H. Shon, On the growth of solutions of a class of higher older differential equations, Acta Math. Sci. Ser. B Engl. Ed. 24 (2004), no. 1. 52-60.
Search in Google Scholar
[6] F. Gross, On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc. 131(1968), 199-214.10.1090/S0002-9947-1968-0220938-4
Search in Google Scholar
[7] G. G. G undersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88-104.
Search in Google Scholar
[8] W. K. Hayman, Meromorphic functions, Oxford Mathematical Mono- graphs Clarendon Press, Oxford 1964.
Search in Google Scholar
[9] K. H. Kwon, Nonexistence of finite order solutions of certain second order linear differential equations, Kodai Math. J. 19 (1996), no. 3, 378-387.
Search in Google Scholar
[10] M. S. Liu, X. M. Zhang, Fixed points of meromorphic solutions of higher order Linear differential equations, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1. 191-211.
Search in Google Scholar
[11] F. Peng and Z. X. Chen, On the growth of solutions of some second-order linear differential equations, J. Inequal. Appl. 2011, Art. ID 635604, 1-9.
Search in Google Scholar
[12] J. F. Xu, H. X. Yi, The relation between solutions of higher order differential equations with functions of small growth, Acta Math. Sci., Chinese Scries, 53 (2010), 291-296.
Search in Google Scholar
[13] C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557. Kluwer Academic Pub- lishers Group, Dordrecht, 2003. 10.1007/978-94-017-3626-8_10
Search in Google Scholar