The Set on Which an Entire Function Is Small
نویسندگان
چکیده
Let f (z) be an entire function and M(r) the maximum of I f (z) on z = r . We give some results on the density of the set of points at which f (z) I is small in comparison with M(r) ; although simple, these results seem not to have been noticed before . If E is a measurable set in the z-plane, we denote by DR (E) the ratio m(z c E, I z < R)/1rRZ and by D(E) and D(E) the upper and lower densities of E, that is the superior and inferior limits of DR (E) as R--> oo . For a fixed function f (z), let Ex be the set of points z for which log I f (z) (1 A) log M ( I z I) . Our results may be stated as follows .
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