Some properties of harmonic univalent functions (Division Problem in Douglas Algebras and Related Topics)
俊夫 早味 · Kyoto University Research Information Repository (Kyoto University) · 2016
A sufficient condition on harmonic univalent functions $f_{1}(z)$ and $f_{2}(z)$ in the open unit disk $\mathbb{U}$ for the convex combination $f_{3}(z)=tf_{1}(z)+(1-t)f_{2}(z)$ to be also harmonic univalent in$\mathbb{U}$ and its range $f_{3}(U)$ is convex in the horizontal direction is discussed.Furthermore, several illustrative examples and the images of functions satisfying the obtained condition are enumerated.Remark 1 A function $f(z)=u(x, y)+iv(x, y)$ is analytic in $\mathbb{D}$ if it satisfies the Cauchy- Riemann equations $u_{x}=v_{y}$ and $u_{y}=-v_{x},$ in short, if it has a derivative $f'(z)$ at each point $z\in \mathbb{D}$ .These relations show that every analytic function is harmonic.Now, we consider the following two differential operators 2010 Mathematics Subject Classification: Primary $30C45$ , Secondary $58E20.