On The Growth Of Solutions Of Second Order Complex Linear Differential Equations

In this thesis,we study the growth of solutions of the following equation f"+A(z)/f'+B(z)f=0,(1)by using Nevanlinna theory and asymptotic method,where A(z),B(z)(?)0 are entire functions.Firstly,we consider a question of Gundersen concerning the growth of solutions of the equation(1),which was raised in the reference[38]:if A(z)is a transcendental entire function of finite order such that ?(A)<?(A),and if B(z)is a nonconstant polynomial,then does every nontrivial solution of the equation(1)satisfy p(f)= ??We give a partial solution about Gundersen's?question by using the precise estimation of logarithmic derivative and Phragmen-Lindelof principle.Secondly,the growth of solutions of the equation(1)is investigated by using the properties of exponential polynomials,the theory of deficient values and singular directions of meromorphic functions.Some sufficient conditions are given to guarantee that every nontrivial solution of the equation(1)has infinite growth order.The main work as follows:1.We consider the growth of solutions of the equation(1)with the exponential polynomials as coefficients.Several sufficient conditions are given to guarantee that every nontrivial solution of the equation(1)has infinite growth order by analyzing deeply the relationship between convex hull and the growth of exponential polynomial in angular domain.2.Let a E C be a finite deficient value of A(z)of lower order ?,and B(z)an exponential polynomial satisfying one of the following conditions:(i)?(B)=?;(ii)?(B)<?,?<?/?-?(B)and ?(a,A)>1-cos?(?-?(B)/2.Then every nontrivial solution of the equation(1)has infinite growth order.3.By using some delicate analysis on the growth in an annulus of A(z)with?(0,A)= 1 and on the spread relation of B(z)which lower order ?<1,we give several sufficient conditions which guarantee the order of any nontrivial solution of the equation(1)is infinite.4.Let A(z)be an entire function extremal for Yang's inequality and have p finite deficient values.Then every nontrivial solution of(1)is of infinite growth order if there exist a sequence {rn} and ??(p/2?(A),1]such that T(rn,B)?? logM(rn,B),(n??).5.We study the value distribution properties of A(z)near its Julia direction and the special growth properties of B(z),such as the extremal function for Denjoy's conjecture and the function with a finite Borel exceptional value,and find several sufficient conditions to guarantee that every nontrivial solution of the equation(1)is infinite growth order.