Growth of Solutions of a Class of Linear Differential Equations Near a Singular Point

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DOI: 10.46793/KgJMat2302.187C


In this paper, we investigate the growth of solutions of the differential equation

                 {            }                  {            }
  ′′                -----a-----    ′                -----b-----
f  +  A  (z) exp   (z  −  z)n   f  +  B  (z) exp   (z  −  z)n   f  =  0,
                      0                               0

where A(z ), B(z ) are analytic functions in the closed complex plane except at z0 and a,b are complex constants such that ab0 and a = cb, c > 1. Another case has been studied for higher order linear differential equations with analytic coefficients having the same order near a finite singular point.


Linear differential equations, growth of solutions, finite singular point.


[1]   I. Amemiya and M. Ozawa, Non-existence of finite order solutions of w′′+ezw+Q(z)w = 0, Hokkaido Math. J. 10 (1981), 1–17.

[2]   L. Bieberbach, Theorie der Gewöhnlichen Differentialgleichungen, Springer-Verlag, Berlin, Heidelberg, New York, 1965.

[3]   Z. X. Chen, The growth of solutions of f′′+ezf+Q(z)f = 0, where the order (Q) = 1, Sci. China Math. 45 (2002), 290–300.

[4]   Z. X. Chen and K. H. Shon, On the growth of solutions of a class of higher order linear differential equations, Acta Math. Sci. Ser. A 24B(1) (2004), 52–60.

[5]   Z. X. Chen and C. C. Yang, Some further results on zeros and growths of entire solutions of second order linear differential equations, Kodai Math. J. 22 (1999), 273–285.

[6]   H. Fettouch and S. Hamouda, Growth of local solutions to linear differential equations around an isolated essential singularity, Electron. J. Differential Equations 2016 (2016), 10 pages.

[7]   G. G. Gundersen, On the question of whether f′′ + ezf + B(z)f = 0 can admit a solution f ⁄≡ 0 of finite order, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 9–17.

[8]   S. Hamouda, Properties of solutions to linear differential equations with analytic coefficients in the unit disc, Electron. J. Differential Equations 2012 (2012), 9 pages.

[9]   S. Hamouda, Iterated order of solutions of linear differential equations in the unit disc, Comput. Methods Funct. Theory 13(4) (2013), 545–555.

[10]   S. Hamouda, The possible orders of growth of solutions to certain linear differential equations near a singular point, J. Math. Anal. Appl. 458 (2018), 992–1008.

[11]   W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

[12]   A. Ya. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli, Mat. Stud. 23(1) (2005), 19–30.

[13]   A. A. Kondratyuk and I. Laine, Meromorphic functions in multiply connected domains, in: Fourier Series Methods in Complex Analysis, Univ. Joensuu Dept. Math. Rep. Ser. 10, Univ. Joensuu, Joensuu, 2006, 9–111.

[14]   R. Korhonen, Nevanlinna theory in an annulus, in: Value Distribution Theory and Related Topics, Adv. Complex Anal. Appl. 3, Kluwer Acad. Publ., Boston, MA, 2004, 167–179.

[15]   I. Laine, Nevanlinna Theory and Complex Differential Equations, W. de Gruyter, Berlin, 1993.

[16]   E. L. Mark and Y. Zhuan, Logarithmic derivatives in annulus, J. Math. Anal. Appl. 356 (2009), 441–452.

[17]   M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, 1975 (reprint of the 1959 edition).

[18]   J. Tu and C-F. Yi, On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order, J. Math. Anal. Appl. 340 (2008), 487–497.

[19]   J. M. Whittaker, The order of the derivative of a meromorphic function, J. Lond. Math. Soc. s1-11 (1936), 82–87.

[20]   L. Yang, Value Distribution Theory, Springer-Verlag Science Press, Berlin, Beijing, 1993.