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1. By the method of Dolzhenko's paper, we studied relations between non-tangetial (angular) boundary behaviors and horocyclic boundary behaviors of arbitrary functions defined in the open unit disc of the complex plane in . Vessey ,  investigated the behavior of arbitrary functions on paths which are "more tangential" than horocycles. The purpose of the present paper is to prove the fact that is sharper than the results in Vessey , , and generalize the results in  to obtaindoi:10.1017/s002776300001480x fatcat:ex77ddj4jvgkpasbqpuim575ki