Algebroid Solutions of Second Order Complex Differential Equations

Using value distribution theory and maximum modulus principle, the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that our results are sharp. 1. Introduction and Main Results We use the standard notations and results of the Nevanlinna theory of meromorphic or algebroid functions; see, for example, [1, 2]. In this paper we suppose that second order algebraic differential equation (3) admit at least one nonconstant -valued algebroid solution in the complex plane. We denote by a subset of for which and by a positive constant, where denotes the linear measure of . or does not always mean the same one when they appear in the following. Let ？？ be entire functions without common zeroes such that . We put Some authors had investigated the problem of the existence of algebroid solutions of complex differential equations, and they obtained many results ([2–10], etc.). In 1989, Toda [4] considered the existence of algebroid solutions of algebraic differential equation of the form He obtained the following. Theorem A (see [4]). Let be a nonconstant -valued algebroid solution of the above differential equation and all are polynomials. If , then is algebraic. The purpose of this paper is to investigate algebroid solutions of the following second order differential equation in the complex plane with the aid of the Nevanlinna theory and maximum modulus principle of meromorphic or algebroid functions: where , . We will prove the following two results. Theorem 1. Let be a nonconstant -valued algebroid solution of differential equation (3) and all are polynomials. If , then is algebraic, . Theorem 2. Let be a nonconstant -valued algebroid solution of differential equation (3) and the orders of all are finite. If , then the following statements are equivalent:(a) ;(b) ;(c) is a Picard exceptional value of . 2. Some Lemmas Lemma 3 (see [2]). Suppose that , , are meromorphic functions, and . Then one has Examining proof of Lemma？？4.5 presented in [2, pp. 192-193], we can verify Lemma 4. Lemma 4. Let be a transcendental algebroid function such that has only finite number of poles, and let , , and have no poles in . Then, for some constants , , and it holds: where . Lemma 5 (see [11]). The absolute values of roots of equation are bounded by Lemma 6. Let be a nonconstant -valued algebroid solution of the differential equation (3) and let be a polynomial. If , then where , is a positive constant. Proof. We first prove that the poles of are contained in the zeroes of . Suppose that is a pole of of order and is not