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Several mathematicians examined classes of meromorphic typically-real functions with a simple pole at the point zero. This article includes results concern class Q' H of complex harmonic typically-real functions with a pole at the point zero. There are determined the relationships between this class and the class Q' r of meromorphic typically-real funtions with a pole at the origin, which was investigated by S. A. Gelfer [4]. We present also coefficient estimates for functions of a subclass of the class Q' H and properties of the Hadamard product with fuctions of the class Q' H.

In this paper we consider a few classes of functions f harmonic in the unit disc ∆ of the form f = h+ overline{g}, where h, g are suitably normalized functions holomorphic in ∆. Our special attention is drawn to some classes generated by respective coefficient conditions and to classes of functions with conditions imposed on coefficient arguments. We examine relationships between these conditions and some analytic conditions of stalikeness or convexity of considered functions.

Well known is the class of typically-real functions holomorphic in the unit disc |z|

In the paper we examine some holomorphic functions and complex harmonic functions, which satisfy certain conditions of a Mocanu kind. We also consider their relations with appropriate coefficient conditions. The paper is a natural supplement to the results published in [1] and [2].