Abstract. The relation between the Jacobian and the orders of a linear invariant family of locally univalent harmonic mapping in the plane is studied. The new order (called the strong order) of a linear invariant family is defined and the relations between order and strong order are established.

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In this paper we determine some examples of the domains of n-valence for functions which are analytic with classical normalization and locally univalent. This problem has not been studied yet. It seems to be rather difficult. We formulate a few conjectures concerning some subclasses of typically real functions and odd typically real functions.

Abstract. In this paper we discuss the class Tp,e consisting of typically real functions which do not admit values wo = pe lB and wo-We estimate the second and the third coefficients of a function / 6 TPi$ and we determine the Koebe domain for the class of typically real functions with fixed second coefficient.Let T denote the class of analytic functions / in the unit disk A = {z G C : \z\ < 1} and normalized by /(0) = /'(0) -1 = 0 for which the condition