The Zeros Of F~nf'-a And Normal Families Related To Shared Values

Normality is important subject in complex analysis, many scholars have made a large number of outstanding contributions to this and developed a number of important results.In this paper we study the zeros of fⁿf'-a and the normality of meromorphic functions. Many mathematicians in this area has done a lot of discussion and research, and got a number of important conclusions.W.K.Hayman[41]proposed a conjecture in 1959:If / is a transcendental meromorphicfunction in C, then for any n, fⁿf' assumes every finite non-zero complex number infinitely often.lt set off an animated discussion among many world famous scientists in maths.Fristly Hayman[41] himself confirmed it for n≥3;Further, it was proved by E.Mues[42] when n≥2.Moreover, Hayman proved for entire functions, this conjecture is true when n≥2 in [46], J.Clunie[43] proved for entire function, this conjecture is true when n≥1. At last in 1995, Huaihui Chen and Mingliang Fang[31] Considered Hayman's conjecture In the case of n = 1, and proved that if f is a transcendental meromorphic function in C, then ff' assumes every finite non-zero complex number infinitely often. Y.F.Wang[30]proved the conjecture On the other hand in 1992, and get that if f is a transcendental meromorphic function in C, n≥3, k≥0 are two integer numbers, then (fⁿ(z))^k assumes every finite non-zero complex number infinitely often. Mingliang Fang[32][33] proved that if f is a meromorphic functionin C, n≥2 is a positive integer, 6 is a non-zero constant, if fⁿf'≠b, then f is a constant;if f is a transcendental meromorphic function, then fⁿf' assumes every finite non-zero complex number infinitely often. Obviously, we get:if f is a non-constant meromorphic function, then fⁿf'-a(a≠0,∞) inevitably has zeros.According to Bloch principle, we could apply above theorems to mormal families and then obtain a series of normal criterion. In fact, some normal criteriaof such a course have been got by senior scientists in maths. Influenced from above results on value distribution, Hayman proposed the conjecture:F is a normal family in D, n is a finite non-zero complex number, if for any f∈F, fⁿf'≠a, then F is normal. It was been confirmed on some additional conditions respectively by some famous mathematicians. Recently, Q.C,Zhang obtained a theorem by discussing the problem related to value distribution:Given F is a family of meromorphic functions, n is a positive integer, a, b is two constantnumber, (a≠0,b≠∞).If n≥4, and for any f and g in F, f'-afⁿ and g'-agⁿ share b, then F is normal.Influenced from above theorems, we discuss the zeros of fⁿf'-a in this article, and furthermore, we extend our discussion to normal families, obtaining some normal criteria from it according to Bloch principle.In the preface, we give a review about the historical background of normal families and research achievements in these field.The present paper is divided into five parts.In chapter one, we state a few fundamental knowledge, definitions and notationsabout the theory of normal families of meromorphic function and basic results in the theory of normal families.In chapter two, we state the primary results and their theory backgrounds, and get theorem 1: Let f be a nonconstant meromorphic function in C, n be a positiveinteger and a be a non-zero constant, then fⁿf'-a(a≠0,∞) has more than two zeros when n≥2; If n = 1 and f only has multiply zeros, fⁿf'-a(a≠0,∞) has more than two zeros .Theorem 2:Let f be a nonconstant meromorphic functionin C, k is a integer, n≥k+1 be a positive integer, if f^(k)(?)0, then fⁿf^(k)-a(a≠0,∞) has at least one zero.In chapter three, we discuss the application of theorem 1,2 in the prove of normal criterion, improve and get four normal criterion:(1.)Let F be a family of meromorphic functions in D, n be a positive integer and a, b be two constants such that a≠0,b≠0,∞.when n≥4, or n = 3 and for each every function f in F only has multiple pole number, if for each pair of functions f and g in F, f'-afⁿ and g'-agⁿ share the value b, then F is normal in D.(2.)Let F be a family of meromorphic functions in D, n be a positive integer and a, b be two constants such that a≠0,b≠0,∞, when n≥2 or n = 1 and for each every function f in F only has multiple pole number, if for each pair of functions f and g in F, fⁿf'-a and gⁿg'-a share the value 0, then F is normalin D. (3.)Let F be a family of meromorphic functions in D, k is a integer, f^(k)(?) 0, if for n≥k+1, fⁿf^(k)-a≠0, then F is normal in D.(4.)Let F be a family of meromorphic functions in D, k is a integer, f^(k)(?) 0, n be a positive integer and a, b be two constants such that a≠0,b≠0,∞.If for each pair of functions f and g in F, fⁿf^(k)-a and gⁿg^(k)-a share the value b, then F is normal in D.In chapter four, we prove theorem 1,2.In chapter five, we apply theorem 1,2 in proving theorem 3,4 and theorem 5,6, respectively.