Normality Of Meromorphic Functions With Multiple Zeros And Poles

This thesis is mainly divided into four parts. The first and second part introduce the fundamental knowledge of the normal family theory, the Nevanlinna theory and the Share value theory respectively. The third and the forth part are the main results in this thesis.In the third part of this paper, we obtain some normal criterions which concern the polynomials and shared values of meromorphic functions.Theorem 3.1.1. Let F be a family of meromorphic functions defined in D.Let P(z) be a polynomial with the origin as zero. If for every f∈F, all zeros of f have multiplicity at least k and P(f(z))(f(k)(z))m= a(?)f(k)(z)= b,where k,m≥2 are two positive integers, a≠ 0 and b are two finite complex numbers, and deg(P)≥ 2.Then F is normal in D.Theorem3.1.2. Let F be a family of meromorphic functions defined in D. Let P(z) be a polynomial with the origin as zero. If for every f∈F, all zeros of f have multiplicity at least k and P(f(z))(f(k)(z))m= a(?)|f(k)(z)|≤ A, where A, k,m≥2 are three positive integers, a≠ 0 and b are two finite complex numbers, and deg(P)≥ 2. Then F is normal in D.The forth part concerns the normality of meromorphic functions with multiple zeros and poles. Our results improve the theorems of G. Datt and S. Kumar and get some new normal criterions.Theorem 4.1.1. Let F be a family of meromorphic functions defined in D,for every f∈F, all zeros of f have multiplicity at least k+1, poles of f have multiplicity at least 2; for every f,g∈F, D(f) and D(g) share the value b IM (b is non-zero constant). Then F is normal in D.Theorem 4.1.2. Let F be a family of meromorphic functions defined in D,for every f∈F, all zeros of f have multiplicity at least k+1, poles of f have multiplicity at least 2; for every f∈F,when D(f) is bounded,we have|f(k)(z)|≤ A, A is postive number.Then F is normal in D.Theorem 4.1.3. Leti F be a family of meromorphic functions defined in D, for every f∈ F, all zeros of f have multiplicity at least k+1, poles of f have multiplicity at least 2; for every f∈F, D(f) = b has at most one zero in D.Then.F is normal in D.Theorem 4.1.6. Let F be a family of meromorphic functions defined in D, for every f∈F, all zeros of f have multiplicity at least k+3, poles of f have multiplicity at least k+2(k≥0); for every f,g∈F, D1 (f) and D1 (g) share the value b IM(b is non-zero constant). Then F is normal in D.Theorem 4.1.7. Let F be a family of meromorphic functions defined in D, for every f∈F, all zeros of f have multiplicity at least k+3, poles of f have multiplicity at least k+2(k≥ 0); for every f∈F, D1 (f)-b has at most one zero in D.Then F is normal in D.