tailieunhanh - Báo cáo hóa học: " Research Article On the Exponent of Convergence for the Zeros of the Solutions of y Ay By 0"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Exponent of Convergence for the Zeros of the Solutions of y Ay By 0 | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 428936 8 pages doi 2010 428936 Research Article On the Exponent of Convergence for the Zeros of the Solutions of y Ay By 0 Abdullah Alotaibi Department of Mathematics King Abdulaziz University . Box 80203 Jeddah 21589 Saudi Arabia Correspondence should be addressed to Abdullah Alotaibi mathker11@ Received 1 July 2010 Accepted 12 September 2010 Academic Editor P. J. Y. Wong Copyright 2010 Abdullah Alotaibi. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Let B and C be entire functions of order less than 1 with C 0 and B transcendental. We prove that every solution f 0 of the equation y Ay By 0 A z C z eaz a e C 0 being has zeros with infinite exponent of convergence. 1. Introduction It is assumed that the reader of this paper is familiar with the basic concepts of Nevanlinna theory 1 2 such as T r f mfr f Nfr f and Sfr f . Suppose that f is a meromorphic function then the order of growth of the function f and the exponent of convergence of the zeros of f are defined respectively as log T r f log j 1 f PƠ limsup o. Vf limsup V1 rCtt iog r rCtt iog r Let E be a measurable subset of 1 x . The lower logarithmic density and the upper logarithmic density of E are defined respectively by . -ÍĨ x .t dt t ---- c x t dt t _ log dens E liminf log dens E limsup- f - c log r r cv log r 2 Journal of Inequalities and Applications where f is the characteristic function of E defined as xt 1 if t e E 0 if teE. Now let us recall some of the previous results on the linear differential equation y e -y B z y 0 where B z is an entire function of finite order When B z is polynomial many authors 3-6 have studied the properties of the solutions of . If B z is a transcendental entire function with p B