On Ahlfors' discs theorem and its application

Abstract

Recently J.Dufresnoy [3], M.Tsuji [8] and Z. Yύjόbό [9] have proved a generalization of Ahlfors' discs theorem [1] by use of Ahlfors* theory [2] of covering surfaces. On the other hand, A. Pfluger [6] and Y. Juve [4] have obtained an extension of Koebe's distortion theorem to some univalent pseudo-regular functions.From the results mentioned above, we are motivated to write this paper. First, in 1, we define the functions which are called pseudo-analytic (K) and {K} by following S.Kakutani and A. Pfluger. In 3, by the method due to Z. Yύjόbό and by the aid of lemmas which are stated in 2, we establish a theorem for our function which is pseudo-analytic {K} corresponding to the above Dufresnoy-Tsuji-Yύjόbό's theorem for the analytic function. Further, as its application, an extension of Bloch's theorem is proved in 4.

The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value

“…which is a contradiction. We note here that the estimate in the case when µ 1 = µ 2 = • • • = µ n = 1 was established earlier by Kojima (see [9,10]).…”