Abstract
In this paper, we consider the differential equation f″ + Af′ + Bf = 0, where A(z) and B(z) ≠ 0 are entire functions. Assume that A(z) has a finite deficient value, then we will give some conditions on B(z) which can guarantee that every solution f ≠ 0 of the equation has infinite order.
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References
Bank S B, Laine I. On the oscillation theory of f″ + Af = 0 where A is entire. Trans Amer Math Soc, 1982, 273: 351–363
Barry P D. Some theorems related to the cos πρ theorem. Proc London Math Soc, 1970, 21: 334–360
Chen Z X. The growth of solutions of second order linear differential equations with meromorphic coefficients. Kodai Math J, 1999, 22: 208–221
Chen Z X. On the hyper-order of solution of some second order linear differential equations. Acta Math Sinica English Series, 2002, 18: 79–88
Chen Z X. The growth of f″ + e −z f′ + Q(z)f = 0 where the order (Q) = 1. Sci China Ser A, 1991, 45: 290–300
Frei M. Über die subnormalen Lö sunge der, Differential gleichung w″+e −z w′+(Konst.)w = 0. Comment Math Helv, 1961, 36: 1–8
Fuchs W. Proof of a conjecture of G. Pólya concerning gaps. Illinois J Math, 1963, 7: 661–667
Gundersen G G. On the question of whether f″ + e −z f′ + B(z)f = 0 can admit a solution f ≠ 0 of finite order. Proc Roy Soc Edi Ser A, 1986, 102: 9–17
Gundersen G G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J London Math Soc, 1988, 37: 88–104
Gundersen G G. Finite order solution of second order linear differential equations. Trans Amer Math Soc, 1988, 305: 415–429
Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
Hellenstein S, Miles J, Rossi J. On the growth of solutions of f″ + gf′ + hf = 0. Trans Amer Math Soc, 1991, 324: 693–705
Hellenstein S, Miles J, Rossi J. On the growth of solutions of certain linear differential equations. Ann Acad Sci Fenn Ser A I Math, 1992, 17: 343–365
Hille E. Ordinary Differential Equations in the Complex Domain. New York: Wiley, 1976
Kwon Ki-Ho, Nonexistence of finite order solution of certain second order linear differential equations. Kodai Math J, 1996, 19: 378–387
Kwon K H, Kim J H. Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equations. Kodai Math J, 2001, 24: 344–351
Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin-New York: Walter de Gruyter, 1993
Laine I, Wu P C. Growth of solutions of second order linear differential equations. Proc Amer Math Soc, 2000, 128: 2693–2703
Ozawa M. On a solution of w″ + e −z w′ + (az + b)w = 0. Kodai Math J, 1980, 3: 295–309
Pennycuick K. On a theorem of Besicovitch. J London Math Soc, 1935, 10: 210–212
Yang L. Value Distribution Theory and its New Research (in Chinese). Beiiing: Science Press, 1982
Yang L. Deficient Values and Angular Distibution of Entire Functions. vol. 308. Providence, RI: Amer Math Soc, 1988, 583–601
Zhang G. The Theory of Entire Function and Meromorphic Function (in Chinese). Beiiing: Science Press, 1986
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Wu, P., Zhu, J. On the growth of solutions to the complex differential equation f″ + Af′ + Bf = 0. Sci. China Math. 54, 939–947 (2011). https://doi.org/10.1007/s11425-010-4153-x
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DOI: https://doi.org/10.1007/s11425-010-4153-x