Opuscula Math. 37, no. 6 (2017), 853-874
http://dx.doi.org/10.7494/OpMath.2017.37.6.853

Opuscula Mathematica

# On the hyper-order of transcendental meromorphic solutions of certain higher order linear differential equations

Karima Hamani
Benharrat Belaïdi

Abstract. In this paper, we investigate the growth of meromorphic solutions of the linear differential equation $f^{(k)}+h_{k-1}(z)e^{P_{k-1}(z)}f^{(k-1)}+\ldots +h_{0}(z)e^{P_{0}(z)}f=0,$ where $$k\geq 2$$ is an integer, $$P_{j}(z)$$ ($$j=0,1,\ldots ,k-1$$) are nonconstant polynomials and $$h_{j}(z)$$ are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions. We also consider nonhomogeneous linear differential equations.

Keywords: linear differential equation, transcendental meromorphic function, order of growth, hyper-order.

Mathematics Subject Classification: 34M10, 30D35.

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• Karima Hamani
• University of Mostaganem (UMAB), Department of Mathematics, Laboratory of Pure and Applied Mathematics, B.P. 227 Mostaganem, Algeria
• Benharrat Belaïdi
• University of Mostaganem (UMAB), Department of Mathematics, Laboratory of Pure and Applied Mathematics, B.P. 227 Mostaganem, Algeria
• Communicated by Yoshishige Haraoka.
• Revised: 2017-03-03.
• Accepted: 2017-03-06.
• Published online: 2017-09-28.