Coefficient conditions for harmonic close-to-convex functions (Q448687)

Assume that NEWLINE\[NEWLINEf(z)= z+\sum^\infty_{n=2} a_nz^n+\overline{\sum^\infty_{n=1} b_nz^n}NEWLINE\]NEWLINE is a sense preserving harmonic mapping on the unit disk \(\Delta\). Some years ago, \textit{Y. Avci} and \textit{E. J. Złotkiewicz} [Ann. Univ. Mariae Curie-Sklodowska, Sect. A 44, 1--7 (1990; Zbl 0780.30013)] proved that the inequality NEWLINE\[NEWLINE\sum^\infty_{n=2} n(|a_n|+|b_n|)\leq 1-|b_1|\tag{1}NEWLINE\]NEWLINE implies univalence and starlikeness of \(j(z)\).NEWLINENEWLINE The author of this article observes that if \(F(z)= z+\sum^\infty_{n=2} A_nz^n\), \(z\in\Delta\), satisfies for a given \(\varphi\), \(0\leq\varphi<2\pi\), the condition \(\sum^\infty_{n=2} |nA_n- e^{i\varphi}(n- 1)A_n|\leq 1\), then \(F(z)\) is univalent and close-to-convex. Then he shows, by making use of a result due to \textit{J. Clunie} and \textit{T. Sheil-Small} [Ann. Acad. Sci. Fenn., Ser. A I 9, 3--25 (1984; Zbl 0506.30007)], that the condition NEWLINE\[NEWLINE\sum^\infty_{n=2} \{|na_n- e^{i\varphi}(n- 1)a_{n-1}|+|nb_n- e^{i\varphi}(n- 1)b_{n-1}|\}\leq 1-|b_1|\tag{2}NEWLINE\]NEWLINE guarantees univalence and close-to-convexity of \(f(z)\). It is observed that the mapping \(f(z)=-\overline z- 2\log|1-z|\) satisfies (2) but it does not satisfy (1).