Some Results On Distribution Of Entire And Meromorphic Functions

In this thesis, we studied the following four problem and proved several new theorems.1. The first, we studied Picard Sets For Entire Functions and Their Derivatives, and proved the following results.Theorem 1. Let f is a transcendental entire function, and has only zeros of order at least k+2, let be an infinite point set in C with Then f^k assumes all values w ∈C, except possibly zero, infinitely often in C\E.Theorem 2. Suppose that the complex sequence (a_n) and the positive sequence (p_n) satisfy for all n,Then if f is a transcendental entire function, and has only zeros of order at least k+2, the equation f^k = b must have infinitely many solutions outside the union of the discs B(a_n,p_n), for any b ∈ C, b ≠0. Here β > 1 and B(a_n,p_n) = {z : |z-a_n| n}.2. The second, we studied Sharing Values and Normality, and proved the following results.Theorem 3. Let F be a family of holomorphic functions in D = {z : |z| < 1}, and let a be a nonzero finite complex number. If, for any f ∈ F, f and f' share a IM, and for any z₀∈D, N(z₀, r, 1/f) < μT(z-0, r, f), where 0 < μ < 1/2, then F is normal in D.Theorem 4. Let F be a family of meromorphic functions in D = {z :|z| < 1}, and let a, b be two distinct nonzero finite complex numbers. If, for any f G T, f and f' share a IM, and for any Z₀ ∈ D, N(z₀,r, 1/f) + N(z₀,r,1/(f'-b)) < XT(z₀,r,f), where 0 < λ <(1/3), then F is normal in D.