On exceptional values of homogeneous differential polynomials
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Let f be a meromorphic function of order , (0 < < 1). Let T(r, f) be the characteristic function of f(z). Let M(r, f) be the maximum of |f(z)| on |z| = r when f(z) is an entire function. For the function g(z) let n(r, 1/f −g) and ¯n(r, 1/f −g) be the number of zeros and the number of distinct zeros respectively of f(z)−g(z) in |z| r, where g is a small function of f i.e. T(r, g) = o(T(r, f)) = S(r, f). Define, N(r, 1 f − g ) = Z r 0 n(t, 1/f − g) t dt and similary ¯N (r, 1/f − g). Let (0.1) P(f) = X a(z)(f0)l1 ...(f(k))lk where a(z) is meromorphic function satisfying T(r, a(z)) = S(r, f) = o(T(r, f)) and f(1), f(2), ... are the successive derivatives of f and l1, l2, l3, ... are non - negative integers is called a homogeneous differential polynomial of degree n not involving f. Let K(r) be an n-slowly changing function [2], as defined below:
Abstract
Let f be a meromorphic function of order , (0 < < 1). Let T(r, f) be the characteristic function of f(z). Let M(r, f) be the maximum of |f(z)| on |z| = r when f(z) is an entire function. For the function g(z) let n(r, 1/f −g) and ¯n(r, 1/f −g) be the number of zeros and the number of distinct zeros respectively of f(z)−g(z) in |z| r, where g is a small function of f i.e. T(r, g) = o(T(r, f)) = S(r, f). Define, N(r, 1 f − g ) = Z r 0 n(t, 1/f − g) t dt and similary ¯N (r, 1/f − g). Let (0.1) P(f) = X a(z)(f0)l1 ...(f(k))lk where a(z) is meromorphic function satisfying T(r, a(z)) = S(r, f) = o(T(r, f)) and f(1), f(2), ... are the successive derivatives of f and l1, l2, l3, ... are non - negative integers is called a homogeneous differential polynomial of degree n not involving f. Let K(r) be an n-slowly changing function [2], as defined below: