Certain multipliers of univalent harmonic functions (Q812713)

Let \(\overline H\) denote the class of complex-valued harmonic functions which are univalent, orientation preserving, and written in the form \(f=h+\overline g\) where \[ h(z)=z-\sum_{n=2}^{\infty}| a_n| z^n,\;\;\;g(z)=\sum_{n=1}^{\infty}| b_n| z^n,\;\;\;| b_1| <1,\;\;\;| z| <1. \] A function \(f\in\overline H\) is said to be in the multiplier family \(F_{\overline H}(\{c_n\},\{d_n\})\) if there exist sequences \(\{c_n\}\) and \(\{d_n\}\) of positive numbers such that \[ \sum_{n=2}^{\infty}c_n| a_n| +\sum_{n=1}^{\infty}d_n| b_n| \leq1,\;\;\;d_1| b_1| <1. \] For \(c_n\geq n\) and \(d_n\geq n\), the authors determined representation theorem, distortion bounds, invariance under convex combinations, and \(\delta\)-neighborhoods of functions in \(F_{\overline H}(\{c_n\},\{d_n\})\). The representation theorem is generalized to the case when the arguments of the coefficients of \(h\) and \(g\) are unrestricted.