Value distributions of solutions to complex linear differential equations in angular domains

Abstract In this paper we study the iterated order and oscillations of the solutions to some complex linear differential equations in angular domains. Our theorems improve some recent results.


Introduction and main results
In this article, we assume the reader is familiar with standard notations and basic results of Nevanlinna's value distribution theory in the unit disk D fz W jzj < 1g, in an angular region, and in the complex plane C respectively; see [1][2][3][4][5]. The order .f / and lower order .f / of f which is meromorphic in C or are defined as follows: where log OE1 r D log r and log OEnC1 r D log.log OEn r/; n 2 N. The growth and oscillation of solutions to higher-order linear differential equations in C and in have been well studied by many authors. In the paper [6], Cao and Yi studied the properties of solutions to the arbitrary order linear differential equations in of the form where A 0 .6 Á 0/; A 1 ; ; A k are analytic in . In fact, they got the following theorem.
In what follows, we give some notations and definitions of a meromorphic function in an angular domain .˛;ˇ/ D fz W˛< arg z <ˇg. In this paper, usually denotes the angular domain .˛;ˇ/ and " D fz W˛C " < arg z < "g, where 0 < " <ˇ 2 . Let f .z/ be a meromorphic function on .˛;ˇ/ D fz W˛Ä arg z Äˇg. Recall the definition of Ahlfors-Shimizu characteristic in an angular domain; see [5, pp.66 The order and lower order of f on are defined by .f / D lim inf r!1 log T .r; ; f / log r : We remark that the above definitions is reasonable because T .r; C; f / D T .r; f / C O.1/; see [1, pp.20].
Definition 1.5. The iterated n-order n; .f / of a meromorphic function f .z/ in an angular region is defined by where log OE1 r D log r and log OEnC1 r D log.log OEn r/; n 2 N.
Motivated by the definition of a convergent exponent of a-value points of f in in [5, p. 93], we give the following definition.
Recalling the Nevanlinna theory in an angular domain and following the terms in [1], we set For a 2 C [ f1g, the convergence exponent of the sequence of a-point in .˛;ˇ/ of a meromorphic function f is defined by log C˛;ˇ.r; 1 f a / log r : According to the inequality, see [5,Theorem 2.4.7], We consider q pairs of real numbers f˛j ;ˇj g such that and the angular domains X D [ q j D1 fz W˛j Ä arg z Äˇj g. For a function f meromorphic in the complex plane C, we define the order of f on X as In [7], Wu considered the growth of solutions to higher order linear homogeneous differential equations in angular domains. The following theorem was obtained.
where > 0 with Ä Ä . If A j .z/.j D 1; 2; ; n/ are meromorphic functions in C with T .r; A j / D o.T .r; A 0 //, every solution f 6 Á 0 to the equation For the derivatives of the nonzero solutions to the equation in the above theorem, we can get the following result easily.
Theorem 1.11. Let A 0 be a meromorphic function in C with finite lower order < 1 and nonzero order 0 < Ä 1 and ı D ı.1; A 0 / > 0. For q pair of real numbers f˛j ;ˇj g satisfying (8) and The last result relates to the convergence exponent of the sequence of a-point of the solutions of equation (8) in the angular domain X .

G. Zhang
Theorem 1.12. Let A 0 be an entire function in C with finite lower order < 1 and nonzero order 0 < Ä 1 and ı D ı.1; A 0 / > 0. For q pair of real numbers f˛j ;ˇj g satisfying (8) and

Preliminary lemmas
maps the angular domain X D fz W˛< arg z <ˇg; .0 <ˇ ˛< 2 / conformally onto the unit disk f W j j < 1g in the -plane, and maps z D e iÂ 0 to D 0. The image of X " D fz W 1 Ä jzj Ä r;˛C " Ä arg z Äˇ "g; .0 < " On the other hand, the inverse image of the disk h WD f W j j < hg; h < 1 in the z-plane is contained in The inverse transformation of (13) is where the coefficients˛j are the polynomials (with numerical coefficients) in the variables V . /.D such that maxf n; .F /; n; .A j /; j D 0; 1; ; k 1g < n; .f /. Then n; .f / D n; .f / D n; .f /.
where Á is such that 0 < 2Á <ˇ ˛and K is a constant only depending on ; Á;˛andˇ.
It is important and necessary to determine the relations between C˛;ˇ.r; f / and N.r; ; f /, which will be helpful in characterizing meromorphic functions in an angle in terms of the number of points of some values.