Normality Involving Differential Polynomials And Exceptional Functions

This paper discusses the normality of meromorphic functions involving differential polynomials and exceptional functions based on the knowledge of Nevanlinna value distribution theory and Zaclman-Pang lemma.First,we discuss the normality of exceptional functions involving differential polynomials,and we get:Let f be a family of meromorphic functions in a domain D,and let k be a positive integer,ai(z)(i=0,1,…,k-1),a(z)(≠0),b(z)be holomorphic functions.If for each f∈F,the zeros and the poles of f have multiplicity at least m and n,2-(l+1)/n-1/m>0 and f(k)(z)+ak-1(z)f(k-1)(z)+…+a0(z)f(z)-a(z)f3(z)≠b(z),then Fis normal in D.This result extends and improves a corresponding result of Huaihui Chen and Yongxing Gu.Next,we discuss the normality of fixed points and draw the following conclusions;Let A>1 be a constant and F be afamily of meromorphic functions in a domain D.For each f ∈F,the zeros and the poles of f have multiplicity at least 3 and 2,meanwhile satisfies the following conditions:|f’’’(Z)|≤A|z| whenver f(z)＝0 and f’’’(z)+a2(z)f"(z)+a1(z)f’(z)+a0(z)f(z)＃z,where ai(z)(i=0,1,2)are holomorphic functions in D.If F is not normal in z0,then z0=0,and there exist r>0,{fn} (?)F such that fn(z)=(z-ξn1)3(z-ξn2)3/z-ηn)2-fn(z),whereξni/ρn→Ci(i=1,2),ηn/ρn→(c1+c2)/2,ρn→0+,c1 and C2 are two different constantsy fn(z)is a non-vanishing holomorphic function in △r fn(z)→f(z),with z4f(z)satisfy y’’’+a2(z)y’’+(z)y’(z)+a0(z)y-z=0.Finally,we discuss the normality relationship between the family of original functions and the family of differential polynomial functions,we derive:Let F be a family of meromorphic functions in a domain D,and let k be a positive integer,ai=0,1,…,k-1)are holomorphic functions in D,and for each f∈F,the zeros of f have multiplicity at least k,and there exist M>1 such that |f(k)(z)|<M whenever f(z)=0.If Fk=｛f(k)(z)+ak-1(z)f(k-1)(z)+…+a0(z)f(z):f∈F｝is normal in D,then F is normal in D.The theorem is applied to extend and improve a known result of Mingliang Fang and Jianming Chang.