Let T ( f ) and N ( r,c ) denote the usual Nevanlinna characteristic and the counting function for the c -points of a meromorphic function f , respectively. Using a result of Miles and Shea ( Quart. J. Math. Oxford , 24 (2), (1973), 377-383) and two simple estimates for trigonometric functions, we show in connection with a 1929 problem of Nevanlinna for meromorphic functions f of finite order 1 < λ< ∞ $$ \limsup\limits_{r\rightarrow \infty } { N(r, 0)+N(r, \infty ) \over T(r, \,f)}\ge {2\sqrt 2 \over \pi} {|\sin \pi \lambda | \over D(\lambda )}\ge (0.9)\, {{|\sin \pi \lambda | \over {D(\lambda )}, }} $$ with D ( λ) = q +|sin π λ| for $ q\le \lambda \le q + \fraca {1}{2} $ and D ( λ) = q + 1 for $ q + {\fraca {1} {2}} \le \lambda \lt q + 1 $ , where $ q = \lfloor \lambda \rfloor $ .
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