On some subclasses of typically real harmonic functions (Q2757063)

Denote by \(T_H(\beta)\), \(0\leq\beta< 1\), the class of all functions which are harmonic, complex valued, sense-preserving, locally univalent of the form \(f(z)= h(z)+\overline{g(z)}\), \(z\in U= \{z:|z|< 1\}\), where NEWLINE\[NEWLINEh(z)= z+ \sum^\infty_{n=2} a_n z^n,\quad g(z)= \sum^\infty_{n=1} b_n z^n,\quad a_n= \overline{a_n},\quad b_n=\overline{b_n}\tag{1}NEWLINE\]NEWLINE are holomorphic in \(U\), satisfying the condition NEWLINE\[NEWLINE\text{Re}\Biggl\{{1- z^2\over z} (h(z)- g(z))\Biggr\}> (1- b_1)\beta.\tag{2}NEWLINE\]NEWLINE Let \(T^0_H(\beta)\) denote a subclass of \(T_H(\beta)\) with \(b_1= 0\). In this paper the authors obtain some theorems concerning the classes \(T_H(\beta)\) and \(T^0_H(\beta)\). In particular, they obtain: Theorem 2.1. Let \(h\) and \(g\) be given as in (1), \(b_1= 0\). Then \(f= h+\overline g\in T^0_H(\beta)\) if and only if \(h- g\in T(\beta)\) (here \(T(\beta)\) denotes the classes of all holomorphic functions in \(U\) typically-real of order \(\beta\)). Theorem 2.3. If a function \(f= h+\overline g\) belongs to the class \(T^0_H(\beta)\), then \(||a_{2n}|-|b_{2n}||\leq 2(1- \beta)n\), \(n= 1,2,\dots\)\ . Some different class of harmonic typically-real functions in \(U\) was studied by \textit{Z. J. Jakubowski}, \textit{W. Majchrzak} and \textit{K. Skalska} [J. Ramanujan Math. Soc. 9, No. 1, 35-48 (1994; Zbl 0820.30009)].