Growth of Solutions of Linear Differential Equations of [p,q]-Order with Meromorphic Coefficients in the Unit Disc

Abstract

In this thesis, we con rmed the usefulness of powerful Nevanlinna theory tools such as the charac- teristic function and the rst fundamental theorem of Nevanlinna. These techniques helped us to improve on several results obtained by other researchers concerning the following linear di⁄erential equations (k) (k 1) 0 f + A (z)f + + A (z)f + A (z)f = 0; 1 0 k 1 where A (z) are analytic or meromorphic in the unit disc = fz : jzj < 1g, i = 0; 1; :::; k 1; k 2. i In the rst instance, when A dominates the other coe¢ cients near a point on the boundary of 0 ; we gave the statement of theorems of Hamouda. Secondly, we investigated the growth of solutions of di⁄erential linear equations of [p; q]-order. In the nal stages of this project, we considered generalizing some of the above-mentioned results by assuming A dominates the other coe¢ cients near a point on the boundary of . s A natural question: Is it possible to generalize the results of previous theorems if the equation is non-homogeneous? If we look at the linear di⁄erential equations of the following form (k) (k 1) 0 f + A (z)f + + A (z)f + A (z)f = K(z); 1 0 k 1 where A (z); K(z) are analytic or meromorphic functions in the unit disc = fz : jzj < 1g, i = 0; 1; :::; k 1; k 2. i Is it possible to use the same approach as in this thesis and Hamouda s paper, namely that one coe¢ cient dominates the other coe¢ cients near a point on the boundary of ? Can we improve the results found in this thesis when the coe¢ cients A (z) are entire functions?