Linear Differential Equations With Analytic Coefficients of ${[P,Q]}$-Order in the Unit Disc

Authors

  • Zinelaâbidine Latreuch Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), Mostaganem, Algeria
  • Benharrat Belaïdi Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), Mostaganem, Algeria

DOI:

https://doi.org/10.5644/SJM.09.1.06

Keywords:

Linear differential equations, $[p,q]$-order, analytic function, hyper-order, exponent of convergence of the sequence of distinct zeros

Abstract

In this paper, we investigate the complex higher order linear differential equations in which the coefficients are analytic functions in the unit disc of $[p,q]$-order. We obtain several theorems about the growth and oscillation of solutions of differential equations.

 

2010 Mathematics Subject Classification. 34M10, 30D35

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References

S. Bank, General theorem concerning the growth of solutions of first-order algebraic differential equations, Compositio Math., 25 (1972), 61–70.

B. Bela¨ıdi, Oscillation of fast growing solutions of linear differential equations in the unit disc, Acta Univ. Sapientiae, Mathematica, 2 (1) (2010), 25–38.

B. Bela¨ıdi, Growth of soluitions to linear differential equations with analytic coefficients of $[P,Q]$-order in the unit disc, Electron. J. Diff. Equ., Vol. 2011 (156) (2011), 1–11.

L. G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equation, Proc. Amer. Math. Soc., 101 (2) (1987), 317–322.

T. B. Cao and H. X. Yi, On the complex oscillation theory of $f^{prime prime }+Af=0$ where $A(z)$ is analytic in the unit disc, Math. Nachr., 282 (6) (2009), 820–831.

T. B. Cao and H. X. Yi, On the complex oscillation theory of of lineare differential equations in the unit disc, Acta Math. Sci., 2008, 28A (6) :1046–1057.

T. B. Cao, The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl., 352 (2) (2009), 739–748.

T. B. Cao and H. X. Yi, The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl., 319 (2006), 278–294.

Z. X. Chen and C. C. Yang, Quantitative estimations on the zeros and growths of entire solutions of linear differential equations, Complex Variables, 42 (2000), 119–133.

C. T. Chuang, Sur la comparaison de la croissance d’une fonction m´eromorphe et de celle de sa d´eriv´ee, Bull. Sci. Math., 75 (2) (1951), 171–190.

A. El Farissi, B. Bela¨ıdi and Z. Latreuch, Growth and oscillation of differential polynomials in the unit disc, Electron. J. Diff. Equ., 2010 (87) (2010), 1–7.

A. Goldberg and I. Ostrovskii, Value Distribution of Meromorphic functions, Transl. Math. Monogr., Vol. 236, Amer. Math. Soc., Providence RI, 2008.

G. Gundersen, E. M. Steinbart and S. P. Wang, The possible order of solutions of linear differential equations with polynomial coefficients, Trans. Amer. Math. Soc., 350 (3) (1998), 1225–1247.

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

J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1–54.

J. Heittokangas, R. Korhonen and J. R¨atty¨a, Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci. Fenn. Math., 29 (2004), 233–246.

J. Heittokangas, R. Korhonen and J. R¨atty¨a, Fast growing solutions of linear differential equations in the unit disc, Result. Math., 49 (2006), 265–278.

S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of $f^{prime prime }+gf^{prime}+hf=0$}, Trans. Amer. Math. Soc., 324 (1991), 693–706.

O. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the $(p,q)$-order and lower $(p,q)$-order of an entire function, J. Reine Angew. Math. 282 (1976), 53–67.

O. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the $(p,q)$-type and lower $(p,q)$-type of an entire function, J. Reine Angew. Math., 290 (1977), 385–405.

L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math., 22 (4) (1998), 4, 385–405.

I. Laine, Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, 15. Walter de Gruyter & Co., Berlin, 1993.

Y. Z. Li, On the growth of the solution of two-order differential equations in the unit disc, Pure Appl. Math., 4 (2002), 295–300.

J. Liu, J. Tu and L. Z. Shi, Linear differential equations with entire coefficients of (p, q)-order in the complex plane, J. Math. Anal. Appl., 372 (2010), 55–67.

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

J. Tu and T. Long, Oscillation of complex high order linear differential equations with coefficients of finite iterated order, Electron. J. Qual. Theory Differ. Equ., 66 (2009), 1–13.

H. Wittich, Neuere Untersuchungen über eindeutige analytishe Funktionen, 2nd Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1968.

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Published

07.06.2024

How to Cite

Latreuch, Z., & Belaïdi, B. (2024). Linear Differential Equations With Analytic Coefficients of ${[P,Q]}$-Order in the Unit Disc. Sarajevo Journal of Mathematics, 9(1), 71–84. https://doi.org/10.5644/SJM.09.1.06

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