The Uniqueness Problem Of Meromorphic Functions Sharing Values With Differential Polynomials

In 1920 s, the uniqueness theory of meromorphic functions is established by R.Nevanlinna. The Nevanlinna theory is the basis of modern meormorphic function theory, and it has great effect on the development of mathematical branches. Moreover, the Nevanlinna theory provides a useful method to the research of differential equations.In this paper, we research the uniqueness problem of meromorphic functions related to differential polynomials sharing values by using the value distribution theory as an important tool. The whole paper divides into four chapters.In the first chapter, we briefly introduce some fundamental results, usual notations and main concepts in Nevanlinna value distribution theory.In the second chapter, by utilizing Nevanlinna’s value distribution theory of meromorphic functions, we first study the growth of functions of exponential type, then as applications, we investigate the growth of transcendental entire solutions of the following type of nonlinear differential equations:( ) ( ) ( ) ( ) ( )’ ’f z +P z f z +Q z f z =0,(0-1)where P(z) and Q(z) are entire functions. When P(z) and Q(z) satisfy some conditions, we prove that every nonzero solution of the above equation has hyper-order 1, which improve the previous results.Theorem 2.1 Let ( ) ( ) ( )1 23()()()1 2 3()b z b zb zF z =a z e +a z e +a z e and F is not a constant,then( ) {( )} ( )1 3, max,,ib i T r F T r e S r F￡ ￡3 +,whereia,ib are entire functions such that ( )31(,),,( 1,2,3)ib iiT r a S r e i=?= =.Remark: if1()inb iiF z a e== ? is not a constant, then for an integer n 31,we have( ) {( )} ( )1, max,,ib i n T r F T r e S r F￡ ￡3 +,whereia,ib are entire functions such that ( )1(,),,( 1,2,,)inb iiT r a S r e i n=?= = ×××.Next, we are going to deal with a more general differential or differential-difference equation of the form:( )1 1( 1) ’1 11() 0n i n n d ndb n i i f c e f c e f a e f---=+ +×××+ + ?=,(0-2)where,i ia b are entire functions with1(,)(,)( 1,2,,)inb iiT r a S r e i n=?= = ××× and,j jc d are entire functions with11(,)(,)( 1,2,, 1)jn d jjT r c S r e j n-=?= = ××× -. By using new methods, we have the following theorem:Theorem 2.2 If( 1,2,,)ib i = ×××n and( 1,2,, 1)jd j = ×××n - are polynomials, if there is{ }111(,) max(,)(,)j i in db b ji n T r e T r e S r e-=￡ ￡?< + then every solution f(o/0) of( 0-2) satisfies( ) {( )}21max degii nrf b￡ ￡=.In the third chapter, we consider the meromorphic solutions of nonlinear differential equations of the form:( )2’ ’ ’ ’0 1 2 3ff -f =k +k f +k f +k f,(0-3)where the ( 0,1,2,3)jk j = are constants. By letting3w =f -k the differential equation(0-3) can be rewritten as( )2’ ’ ’ww -w =aw +bw +g,(0-4)where a,b,g are constants. By using a new method, we give some unified and simplified proofs for these known results.Theorem 3.1 If g 10, consider the solutions of(0-4), we would have( )exp, 0, 0zcz a a z ca gaw aa±±ì ? ?? ? ÷- 1= í è ??+ =?where c is a constant, and242ab b g±- ± -=.Theorem 3.2 If g =0, consider the solutions of(0-4), we would have(1)If ab 10, then ( )z z ceabw-=, here c is a constant;(2) Ifa =0,then( )21211c zc e c z z c cbw bì+???= í- +????where1 2c,c are constants.(3)If b =0, then( )1 221 2, 0, 02 Az Azc e c e A Az z c z c Aawa-ì+ - 1?= í?- + + =??where1 2c,c are constants such that21 2c +2ac =0 if A = 0 or2 21 24 c c A =a if A 10.In the fourth chapter, mainly it is shown that( )1 1()()nk n n np f a f f P f+ -= +,a differential polynomial in f with 0na o/, where f is a transcendental meromorphic function satisfying N(r, f) =S(r, f) and1()nP f-is a differential polynomial in f with small functions as its coefficient and1 deg 1nP n-￡ -, assumes every nonvanishing small function infinitely often when n≥2. In addition, we have extended some known results obtained by others earlier, by different arguments.Theorem 4.1 Let f be a transcendental meromorphic function with N(r, f) =S(r, f),n 32 be an integer. Then( )( )1 0nk n nF a f f P f a-= + + has infinitely many zeros,where ( )n1P f-is a differential polynomial in f with small functions as its coefficientand ( )10 0nP-o,1deg() 1nP f n-￡ -,0,na a are small functions of f such that00 na a o/.Theorem 4.2 Let f be a transcendental meromorphic function with N(r, f) ?S(r, f),p and q be non-vanishing small functions of f, Then?k?pff ?q and?l?pff ?q at least one has infinitely many zeros for integers l and k such that l > k ≥ 2.