| In this work we study growth properties of slowly increasing unbounded harmonic functions on the unit disc from the point of view of the extremal metric. Our results in the first chapter are related to results by Jenkins & Oikawa: On the growth of slowly increasing unbounded harmonic functions, Acta Math. 124 (1970), 37-63. Hamilton, D. H.: A sharp form of the Ahlfors' distortion theorem with applications, Trans. Amer. Math. Soc. 282 (1984), 799-806. and Eke, B. G.: The asymptotic behavior of areally mean valent functions, J. Anal. Math. 20 (1967), 147-212.;In the second chapter we prove some miscellaneous results on the same class of functions concerning the number of directions of maximum growth and the critical points. In this chapter we also include some counter-examples for some natural questions about this class of functions.;Finally in the third chapter we study boundary value problems for functions defined in the unit disc. In particular we prove a Fatou theorem for slowly increasing unbounded harmonic functions. We also construct a function to show that such a theorem is in some sense best possible. In the same chapter we prove a theorem on uniqueness sets for functions meromorphic in the unit disc with finite spherical area which improves a result by Tsuji, M.: Beurling's theorem on exceptional sets, Tohoku Math. J. 2 (1950), 113-125. We give a new, simpler and conceptually different proof of the Fatou theorem, due to Beurling, for the same class of functions. Finally we give a proof of the M. and F. Riesz theorem for areally mean p - valent functions in the unit disc. |
