Electron. J. Diff. Eqns., Vol. 2003(2003), No. 17, pp. 1-12.

Order and hyper-order of entire solutions of linear differential equations with entire coefficients

Benharrat Belaidi & Karima Hamani

Abstract:
In this paper, we investigate the growth of solutions of the differential equation
$$ f^{(k)}+A_{k-1}( z)f^{( k-1) }+\dots+A_{1}(z) f'+A_{0}(z) f=0, $$
where $A_{0}( z) ,\dots, A_{k-1}(z)$ are entire functions with $A_{0}(z) \not\equiv 0$. We will show that if the coefficients satisfy certain growth conditions, then every finite order solution of the equation will satisfy certain other growth conditions. We will also find conditions on the coefficients so that every solution $f\not\equiv 0$ will have infinite order and we estimate in one case the lower bounds of the hyper-order.

Submitted November 27, 2002. Published February 20, 2003.
Math Subject Classifications: 30D35, 34M10, 34C11.
Key Words: Linear differential equations, growth of entire functions, hyper-order.

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Benharrat Belaidi
Department of Mathematics
Laboratory of Pure and Applied Mathematics
University of Mostaganem, B. P 227 Mostaganem Algeria
e-mail: belaidi@univ-mosta.dz, belaidi.benharrat@caramail.com
Karima Hamani
Department of Mathematics
Laboratory of Pure and Applied Mathematics
University of Mostaganem, B. P 227 Mostaganem Algeria

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