Rational Deficient Functions of Meromorphic Functions

The following theorem is shown: Suppose that f is a transcendental meromorphic function. Then for ε > 0 and distinct rational functions a 1 , a 2 …, a q we havem ( r , f ) + ∑ j = 1 q m ( r , 1 f ‐ a i) ⩾ ( 2 + ɛ ) T ( r , f ) + S ( r , f ) .As a corollary we have a theorem on deficiencies for rational deficient functions: δ ( ∞ , f ) + Σ δ ( a , f ) ⩽ 2 ,where δ ( a , f ) = lim   inf r → ∞m ( r , 1 f ‐ a)T ( r , f )and the sum is taken over an arbitrary set of rational functions. The corollary is an extension of the classical theorem on the deficiency sum of Nevanlinna.