Some Questions of the Nevanlinna Theory for the Complex Half-Plane

In this paper we give variants of the logarithmic derivative lemma and the second main theorem for the complex half-plane. Here the second main theorem is a special case of a theorem concerning polynomial functions of the coordinates of a holomorphic curve f : C₊ Pⁿ. This theorem is a variant for C₊ of a theorem of A. Eremenko and M. Sodin, who considered an entire curve f : C₊ Pⁿ. The proofs are realized because of the introduction of natural classes of subharmonic, -subharmonic, holomorphic, meromorphic functions in C₊ . Such functions we shall call just functions.