For \(j=1,2,\) let the sense-preserving locally univalent harmonic mappings \({\mathcal {F}}_j={\mathcal {H}}_j+\overline{{\mathcal {G}}_j}\) on \({\mathcal {D}}:=\left\{ z\in {\mathbb {C}}: |z|<1\right\} \) be such that \(\overline{{\mathcal {F}}_j({\overline{z}})}={\mathcal {F}}_j(z)\) and the mappings \(z({\mathcal {H}}_j+{\mathcal {G}}_j)'\) are either odd starlike or starlike of order 1/2. It is shown that the convolution \({\mathcal {F}}_1*{\mathcal {F}}_2\) of \({\mathcal {F}}_1\) and \({\mathcal {F}}_2\) is directional convex univalent mapping if it is locally univalent sense-preserving. Also, some examples are given where \({\mathcal {F}}_1*{\mathcal {F}}_2\) is locally univalent sense-preserving.

中文翻译：

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关于调和映射的卷积

对于\（j = 1,2，\），让保留感知的局部单价谐波映射\（{\ mathcal {F}} _ j = {\ mathcal {H}} _ j + \ overline {{\ mathcal {G}} _ j } \）在\（{\ mathcal {D}}：= \ left \ {z \ in {\ mathbb {C}}：| z | <1 \ right \} \）中的\（\ overline {{ \ mathcal {F}} _ j（{\ overline {z}}）} = {\ mathcal {F}} _ j（z）\）和映射\（z（{\ mathcal {H}} _ j + {\ mathcal { G}} _ j）'\）是奇数星状或1/2级星状。显示了\（{\ mathcal {F}} _ 1 \）和\（{\ mathcal {F}}的卷积\（{\ mathcal {F}} _ 1 * {\ mathcal {F}} _ 2 \）_2 \）是方向性凸单价映射（如果它是局部单价保留的）。此外，给出了一些示例，其中\（{\ mathcal {F}} _ 1 * {\ mathcal {F}} _ 2 \）是局部单义的。