We consider classes of harmonic univalent functions $f_k=h_k+\overline{g_k}$, $(k=1,2)$ that are shears of the analytic map $h_k-g_k=1/2\log \left[\left(1+z\right)/\left(1-z\right)\right]$ with dilatation ${\omega }_k={\mathrm{e}}^{\mathrm{i}{\theta }_k}{z}^k$. We prove that if the convolution $f_1\ast f_2$ is locally one-to-one and sense-preserving, then $f_1\ast f_2$ is convex in the direction of the real axis.

中文翻译：

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平面谐波带映射的卷积

我们考虑调和单价函数的类别 $F_{克}=H_{克}+\overline{G_{克}}$, $(克=1,2)$ 是分析图的剪切 $H_{克}-G_{克}=1/2\mathrm{日志}\left[\left(1+z\right)/\left(1-z\right)\right]$ 有扩张 ${\omega }_{克}={\mathrm{电子}}^{\mathrm{一世}{\theta }_{克}}{z}^{克}$. 我们证明如果卷积$F_1＊F_2$ 是局部一对一和意义保持的，那么 $F_1＊F_2$ 在实轴方向是凸的。