GROWTH OF \(\varphi\)–ORDER SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS ON THE COMPLEX PLANE

Mohamed Abdelhak Kara     (Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem, Algeria)
Benharrat Belaïdi     (Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem, Algeria)

Abstract


In this paper, we study the growth of solutions of higher order linear differential equations with meromorphic coefficients of \(\varphi\)-order on the complex plane. By considering the concepts of \(\varphi\)-order and \(\varphi \)-type, we will extend and improve many previous results due to Chyzhykov–Semochko, Belaïdi, Cao–Xu–Chen, Kinnunen.


Keywords


Linear differential equations, Entire function, Meromorphic function, \(\varphi \)-order, \(\varphi \)-type.

Full Text:

PDF

References


Belaïdi B. Fast growing solutions to linear differential equations with entire coefficients having the same \(\rho_{\varphi }\)-order. J. Math. Appl., 2019. Vol. 42. P. 63–77. DOI: 10.7862/rf.2019.4

Belaïdi B. Growth of \(\rho_{\varphi }\)-order solutions of linear differential equations with entire coefficients. Pan-American Math. J., 2017. Vol. 27, No. 4. P. 26–42. URL: http://www.internationalpubls.com

Belaïdi B. Growth and oscillation of solutions to linear differential equations with entire coefficients having the same order. Electron. J. Differential Equations, 2009. No. 70. P. 1–10. URL: https://ejde.math.txstate.edu

Bernal L.G. On growth k-order of solutions of a complex homogeneous linear differential equation. Proc. Amer. Math. Soc., 1987. Vol. 101, No. 2. P. 317–322. DOI: 10.1090/S0002-9939-1987-0902549-5

Cao T.-B., Xu J.F., Chen Z.X. On the meromorphic solutions of linear differential equations on the complex plane. J. Math. Anal. Appl., 2010. Vol. 364, No. 1. P. 130–142. DOI: 10.1016/j.jmaa.2009.11.018

Chiang Y.-M., Hayman W.K. Estimates on the growth of meromorphic solutions of linear differential equations. Comment. Math. Helv., 2004. Vol. 79, No. 3. P. 451–470. DOI: 10.1007/s00014-003-0792-7

Chyzhykov I., Semochko N. Fast growing entire solutions of linear differential equations. Math. Bull. Shevchenko Sci. Soc., 2016. Vol. 13. P. 68–83.

Frank G., Hellerstein S. On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients. Proc. London Math. Soc., 1986. Vol. s3–53, No. 3. P. 407–428. DOI: 10.1112/plms/s3-53.3.407

Gundersen G.G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. London Math. Soc., 1988. Vol. s2–37, No. 1. P. 88–104. DOI: 10.1112/jlms/s2-37.121.88

Hayman W.K. Meromorphic Functions. Oxford: Oxford Mathematical Monographs Clarendon Press, 1964. 191 p.

Juneja O.P., Kapoor G.P., Bajpai S. K. On the \((p,q)\)-order and lower \((p,q)\)-order of an entire function. J. Reine Angew. Math., 1976. Vol. 282. P. 53–67. DOI: 10.1515/crll.1976.282.53

Kara M.A., Belaïdi B. Some estimates of the \(\varphi \)-order and the \(\varphi \)-type of entire and meromorphic functions. Int. J. Open Problems Complex Analysis, 2019. Vol. 10, No. 3. P. 42–58. URL: http://www.i-csrs.org

Kinnunen L. Linear differential equations with solutions of finite iterated order. Southeast Asian Bull. Math. 1998. Vol. 22, No. 4. P. 385–405.

Laine I. Nevanlinna Theory and Complex Differential Equations. De Gruyter Studies in Mathematics, Vol. 15. Berlin: Walter de Gruyter & Co., 1993. 341 p. DOI: 10.1515/9783110863147

Li L.M., Cao T.B. Solutions for linear differential equations with meromorphic coefficients of \([p,q]\)-order in the plane. Electron. J. Differential Equations, 2012. No. 195. P. 1–15. URL: https://ejde.math.txstate.edu

Liu J., Tu J., Shi L.Z. Linear differential equations with entire coefficients of \([p,q]\)-order in the complex plane. J. Math. Anal. Appl., 2010. Vol. 372. P. 55–67. DOI: 10.1016/j.jmaa.2010.05.014

Mulyava O.M., Sheremeta M.M., Trukhan Yu.S. Properties of solutions of a heterogeneous differential equation of the second order. Carpathian Math. Publ., 2019. Vol. 11, No. 2. P. 379–398. DOI: 10.15330/cmp.11.2.379-398

Nevanlinna R. Zur theorie der meromorphen funktionen. Acta Math., 1925. Vol. 46, No. 1–2. P. 1–99. (in German). DOI: 10.1007/BF02543858

Seneta E. Regularly Varying Functions. Lecture Notes in Math., Vol. 508. Berlin, Heidelberg: Springer-Verlag, 1976. 116 p. DOI: 10.1007/BFb0079658

Sheremeta M. N. Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion. Izv. Vyssh. Uchebn. Zaved. Mat., 1967. Vol. 2, P. 100–108. (in Russian)

Tu J., Chen Z.-X. Growth of solutions of complex differential equations with meromorphic coefficients of finite iterated order. Southeast Asian Bull. Math., 2009. Vol. 33, No. 1. P. 153–164.

Yang L., Value Distribution Theory. Berlin, Heidelberg: Springer-Verlag, 1993. 269 p. DOI: 10.1007/978-3-662-02915-2




DOI: http://dx.doi.org/10.15826/umj.2020.1.008

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.