## A Generalization of the Tumura-Clunie Theorem and Its Application to the Value Distribution of Meromorphic Functions |

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an(z analogue assume assumption characteristic function Clunie completes our proof complex numbers complex plane conclude condition 2-26 convex function deduce denote differential polynomial entire function f(z exponent of convergence exponential type extended complex plane f and g f(r sin ee1 finite number fixed points fn(z form an A-set formula function of finite function of order functional equation functions meromorphic half-plane Imz Hence inequality infinite measure infinitely many fixed Jensen's formula lemma Let f(z Levin and Ostrovskii log log M(r log T(r,f log+ logarithmic derivative Math Nevanlinna theory number of zeros Ostrovskii 15 pair of entire Picard's theorem poles of f(z polynomial in f proof of theorem real entire function real zeros relation Remark satisfies 1-1 satisfies the condition second fundamental theorem Suppose that f(z Theorem 1-1 transcendental entire functions tt(z Tumura-Clunie theorem upper half-plane values of infinite whole plane zeros and poles zeros of f f