On the Conversion of Convex Functions to Certain within the Unit Disk

Abstract

A function $g(z)$ is said to be univalent in a domain $D$ if it provides a one-to-one mapping onto its image, $g(D)$. Geometrically , this means that the representation of the image domain can be visualized as a suitable set of points in the complex plane. We are mainly interested in univalent functions that are also regular (analytic, holomorphik) in U . Without lost of generality we assume $D$ to be unit disk $U=\left\{ z:\left\vert z\right\vert <1\right\} $. One of the most important events in the history of complex analysis is Riemann's mapping theorem, that any simply connected domain in the complex plane $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $ which is not the whole complex plane, can be mapped by any analytic function univalently on the unit disk $U$. The investigation of analytic functions which are univalent in a simply connected region with more than one boundary point can be confined to the investigation of analytic functions which are univalent in $U$. The theory of univalent functions owes the modern development the amazing Riemann mapping theorem. In 1916, Bieberbach proved that for every $g(z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}$ in class $S$ , $\left\vert a_{2}\right\vert \leq 2$ with equality only for the rotation of Koebe function $k(z)=\frac{z}{(1-z)^{2}}$ . We give an example of this univalent function with negative coefficients of order $\frac{1}{4}$ and we try to explain $B_{\frac{1}{4}}\left( 1,\frac{\pi }{3},-1\right) $ with convex functions.