| Abstract: | 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 < u < X $$ limsuplimits_{rrightarrow infty } { N(r, 0)+N(r, infty ) over T(r, ,f)}ge {2sqrt 2 over pi} {|sin pi lambda | over D(lambda )}ge (0.9), {{|sin pi lambda | over {D(lambda )}, }} $$ with D ( u ) = q +|sin ~ u | for $ qle lambda le q + fraca {1}{2} $ and D ( u ) = q + 1 for $ q + {fraca {1} {2}} le lambda lt q + 1 $ , where $ q = lfloor lambda rfloor $ . |
