Abstract
By looking at the situation when the coefficients Pj(z) (j = 1, 2, ⋯, n − 1) (or most of them) are exponential polynomials, we investigate the fact that all nontrivial solutions to higher order differential equations f(n) + Pn−1(z)f(n−1) + ⋯ + P0(z)f = 0 are of infinite order. An exponential polynomial coefficient plays a key role in these results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chen Z X, Shon K H. Numbers of subnormal solutions for higher order periodic differential equations. Acta Math Sin (Engl Ser), 2011, 27(9): 1753–1768
Cherry W, Ye Z. Nevalinna’s Theory of Value Distribution. The Second Main Theorem and Its Error Terms. Berlin: Springer-Verlag, 2001
Goldberg A A, Ostrovskii I V. Value Distribution of Meromorphic Functions. Providence RI: Amer Math Soc, 2008
Gundersen G G. Finite order solutions of second order linrar differential equations. Trans Amer Math Soc, 1988, 305: 415–429
Gundersen G G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J London Math Soc, 1988, 37(1): 88–104
Heittokangas J, Laine I, Tohge K, Wen Z T. Completely regular growth solutions of second order complex linear differential equations. Ann Acad Sci Fenn Math, 2015, 40(2): 985–1003
Hellerstrin S, Miles J, Rossi J. On the growth of solutions of f″ + gf′ + hf = 0. Trans Amer Math Soc, 1991, 323: 693–706
Kwon K H. On the growth of entire functions satisfying second order linear differential equations. Bull Korean Math Soc, 1996, 33(3): 487–496
Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin: Walter de Gruyter, 1993
Li N, Qi X G, Yang L Z. Some results on the solutions of higher-order linear differential equations. Bull Malays Math Sci Soc, 2019, 42(5): 2771–2794
Steinmetz N. Zur Wertverteilung von Exponentialpolynomen (German). Manuscripta Math, 1978/79, 26(1/2): 155–167
Steinmetz N. Zur Wertverteilung der Quotienten von Exponentialpolynomen. Arch Math (Basel), 1980, 35(5): 461–470
Ozawa M. On a solution of w″ + e−zw′ + (az + b)w = 0. Kodai Math J, 1980, 3: 295–309
Ronkin L I. Functions of completely regular growth//Mathematics and its Applications (Soviet Series), 81. Dordrecht: Kluwer Academic Publishers Group, 1992
Levin B Ja. Distribution of zeros of entire functions. Translations of Mathematical Monographs, 5. Providence, RI: Amer Math Soc, 1980
Miles J, Rossi J. Linear combinations of logarithmic derivatives of entire functions with applications to differential equations. Pacific J Math, 1996, 174(1): 195–214
Wang J, Chen Z X. Limiting direction and Baker wandering domain of entire solutions of differential equations. Acta Math Sci, 2016, 36B(5): 1331–1342
Wang J, Laine I. Growth of solutions of nonhomogeneous linear differential equations. Abstr Appl Anal, 2009, Ar 363927
Wen Z, Gundersen G G, Heittokangas J. Dual exponential polynomials and linear differential equations. J Differential Equations, 2018, 264(1): 98–114
Wittich H. Zur Theorie linearer Differentialgleichungen im Komplexen (German). Ann Acad Sci Fenn Ser A, 1966, 379: 19pp
Wu X B, Long J R, Heittokangas J, Qiu K. Second order complex linear differential equations with special functions or extermal functions as coefficients. Electron J Differ Eq, 2015, 2015(143): 1–15
Yang L. Value Distribution Theory and New Research. Beijing: Science Press, 1982
Yang C C, Yi H X. Uniquenss Theory of Meromorphic Functions. Dordrecht, 2003
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported partly by the National Natural Science Foundation of China (12171050, 11871260) and National Science Foundation of Guangdong Province (2018A030313508)
Rights and permissions
About this article
Cite this article
Huang, Z., Luo, M. & Chen, Z. The Growth of Solutions to Higher Order Differential Equations with Exponential Polynomials as Its Coefficients. Acta Math Sci 43, 439–449 (2023). https://doi.org/10.1007/s10473-023-0124-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-023-0124-5