In this paper, we study the normality of a family of analytic functions and prove the following theorem. Let F be a family of analytic functions in a domain D , k be a positive integer and a(z) , a 1(z) , a 2(z) , ... , a k(z) be analytic in D such that a(z)0 . If f(z)≠0 and the zeros of f (k) (z)+a 1(z)f (k-1) (z)+...+a k(z)f(z)-a(z) are of multiplicity at least 2 for each f∈F , then F is normal in D . This result improves Miranda s norm...
The uniqueness of meromorphic functions with one sharing value and an equality on deficiency is studied. We show that if two nonconstant meromorphic functions f(z) and g(z) satisfy δ(0,f)+δ(0,g)+δ(∞,f)+δ(∞,g)=3 or δ 2(0,f)+δ 2(0,g)+δ 2(∞,f)+δ 2(∞,g)=3, and E(1,f)=E(1,g) then f(z),g(z) must be one of five cases.
