×

zbMATH — the first resource for mathematics

Growth of solutions of second order linear differential equations. (English) Zbl 1151.34069
The authors study the growth of solutions of the linear differential equation
\[ f''+ A_1(z) e^{az}f'+ A_0(z) e^{bz} f= H(z),\tag{2} \] where \(A_1(z)\), \(A_0(z)\) and \(H(z)\) are entire functions of order less than one, and \(a,b\in\mathbb{C}\).
The authors prove the following theorems
Theorem 1.1. Suppose that \(A_0\not\equiv 0\), \(A_1\not\equiv 0\), \(H\) are entire functions of order less than one, and the complex constants \(a\), \(b\) satisfy \(ab\neq 0\) and \(a\neq b\). Then every nontrivial solution \(f\) of (2) is of infinite order.
Theorem 1.3. Suppose that \(A_0\not\equiv 0\), \(A_1\not\equiv 0\), \(D_0\), \(D_1\), \(H\) are entire functions of order less than one, and the complex constants \(a\), \(b\) satisfy \(ab\neq 0\) and \(b/a< 1\). Then every nontrivial solution \(f\) of equation \[ f''+ (A_1(z) e^{a\cdot z}+ D_1(z)) f'+ (A_0(z) e^{b\cdot z}+ D_0(z)) f= H(z) \] is of infinite order.

MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barry, P.D., On a theorem of Besicovitch, Quart. J. math. Oxford ser. (2), 14, 293-302, (1963) · Zbl 0122.07602
[2] Amemiya, I.; Ozawa, M., Non-existence of finite order solutions of \(w'' + e^{- z} w^\prime + Q(z) w = 0\), Hokkaido math. J., 10, 1-17, (1981) · Zbl 0554.34003
[3] Chen, Z.-X., The growth of solutions of \(f'' + e^{- z} f^\prime + Q(z) f = 0\) where the order \((Q) = 1\), Sci. China ser. A, 45, 290-300, (2002) · Zbl 1054.34139
[4] Frei, M., Über die subnormalen Lösungen der differentialgleichung \(w'' + e^{- z} w^\prime +(\text{konst} .) w = 0\), Comment. math. helv., 36, 1-8, (1962) · Zbl 0115.06904
[5] Fuchs, W., Proof of a conjecture of G. Pólya concerning gap series, Illinois J. math., 7, 661-667, (1963) · Zbl 0113.28702
[6] Gundersen, G., On the question of whether \(f'' + e^{- z} f^\prime + B(z) f = 0\) can admit a solution \(f \not\equiv 0\) of finite order, Proc. roy. soc. Edinburgh sect. A, 102, 9-17, (1986) · Zbl 0598.34002
[7] Gundersen, G., Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London math. soc. (2), 37, 88-104, (1988) · Zbl 0638.30030
[8] Hayman, W., Meromorphic functions, (1964), Clarendon Press Oxford · Zbl 0115.06203
[9] Jank, G.; Volkmann, L., Einführung in die theorie der ganzen und meromorphen funktionen mit anwendungen auf differentialgleichungen, (1985), Birkhäuser Basel · Zbl 0682.30001
[10] Kwon, K.; Kim, J., Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equations, Kodai math. J., 24, 344-351, (2001) · Zbl 1005.34079
[11] Laine, I., Nevanlinna theory and complex differential equations, (1993), W. de Gruyter Berlin
[12] Laine, I.; Wu, P., Growth of solutions of second order linear differential equations, Proc. amer. math. soc., 128, 2693-2703, (2000) · Zbl 0952.34070
[13] Langley, J., On complex oscillation and a problem of ozawa, Kodai math. J., 9, 430-439, (1986) · Zbl 0609.34041
[14] Y. Li, J. Wang, Oscillation of solutions of linear differential equations, Acta Math. Sin. (Engl. Ser.), in press · Zbl 1155.34370
[15] Markushevich, A., Theory of functions of a complex variable, vol. II, (1985), Chelsea Publ. Co. New York
[16] Ozawa, M., On a solution of \(w'' + e^{- z} w^\prime +(a z + b) w = 0\), Kodai math. J., 3, 295-309, (1980) · Zbl 0463.34028
[17] Yang, C.-C.; Yi, H.-X., Uniqueness theory of meromorphic functions, (2003), Science Press/Kluwer Beijing
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.