Some Properties of Solutions of Second-Order Linear Differential Equations

where AA and ddjj Ajj = 1r 2A are entire functions of �nite order in the complex plane. It is clear that if ddjj Ajj = 1r 2A are complex numbers or dd1 = ccdd2 where cc is a complex number, then ggff is a solution of (4) or has the same properties of the solutions. It is natural to ask what can be said about the properties of ggff in the case when dd1 ≠ ccdd2 where cc is a complex number and under what conditions ggff keeps the same properties of the solutions of (4). In [6], Chen studied the �xed points and hyper-order of solutions of second-order linear differential equations with entire coefficients and obtained the following results.


Introduction and Main Results
roughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory (see [1][2][3][4]).In addition, we will use  and  to denote, respectively, the exponents of convergence of the zero sequence and distinct zeros of a meromorphic function ,  to denote the order of growth of .
De�nition � (see [4,5]).Let  be a meromorphic function.en the hyperorder of  is de�ned by De�nition 2 (see [4,5]).Let  be a meromorphic function.en the hyper-exponent of convergence of zeros sequence of  is de�ned by where  1 is the counting function of zeros of  in {    .Similarly, the hyperexponent of convergence of the sequence of distinct zeros of  is de�ned by where  1 is the counting function of distinct zeros of  in {    .
Suppose that  1 and  2 are two linearly independent solutions of the complex linear differential equation and the polynomial of solutions where  and    = 1 2 are entire functions of �nite order in the complex plane.It is clear that if    = 1 2 are complex numbers or  1 =  2 where  is a complex number, then   is a solution of (4) or has the same properties of the solutions.
It is natural to ask what can be said about the properties of   in the case when  1 ≠  2 where  is a complex number and under what conditions   keeps the same properties of the solutions of (4).
In [6], Chen studied the �xed points and hyper-order of solutions of second-order linear differential equations with entire coefficients and obtained the following results.eorem A (see [6]).For all nontrivial solutions  of (4) the following hold.
(5) If  is a polynomial with deg     , then one has In fact, we study the growth and oscillation of             where   and   are two linearly independent solutions of (4) and   and   are entire functions of �nite order not all vanishing identically and satisfying   ≠   where  is a complex number, and we obtain the following results.Under the hypotheses of eorem 3, let  ̸ ≡ 0 be an entire function with �nite order such that  ̸ ≡ 0. If   and   are two linearly independent solutions of (4), then the polynomial of solutions(5)satis�es                     ,  ,                         ,  .Let  be a polynomial of deg   .Let      ,  be �nite�order entire functions that are not all vanishing identically such that ℎ ̸ ≡ 0 and max{  ,   }  .If     are two linearly independent solutions of (4), then the polynomial of solutions (5) satis�es If   and   are two linearly independent solutions of (4), then the polynomial of solutions (5) satis�es         ⋯     ′   0    (14) Let  0 ,   , … ,   ,  ̸ ≡ 0 be �nite�order mero� morphic functions.If  is a meromorphic solution of (14) with max       0, , … ,    ,       , eorem 3. Let  be a transcendental entire function of �nite order.Let      ,  be �nite�order entire functions that are not all vanishing identically such that max{  ,   }  .If   and   are two linearly independent solutions of (4), then the polynomial of solutions (5) satis�es              ,  ,                 ,  .(10) eorem 4. Lemma 7 (see [7, 8]).Let  0 ,   , … ,   ,  ̸ ≡ 0 be �nite� order meromorphic functions.If  is a meromorphic solution of the equation with    and     , then  satis�es          ,             .Proof of eorem 3. Suppose that   and   are two linearly independent solutions of (4).en by eorem A, we have           ,               .