On some new length problem for analytic functions

Authors : Janusz SOKÓL{}, Mamoru NUNOKAWA

Pages : 427-435

View : 30 | Download : 12

Publication Date : 2017-06-01

Article Type : Research Paper

Abstract :Let $\mathcal{H}$ denote the class of analytic functions in the unit disk $|z|<1$. Let $Cinsert ignore into journalissuearticles values(r,f);$ be the closed curve which is the image of the circle $|z|=r<1$ under the mapping $w=finsert ignore into journalissuearticles values(z);\in\mathcal{H}$, $Linsert ignore into journalissuearticles values(r,f);$ the length of $Cinsert ignore into journalissuearticles values(r,f);$ and let $Ainsert ignore into journalissuearticles values(r,f);$ be the area enclosed by $Cinsert ignore into journalissuearticles values(r,f);$. Let $linsert ignore into journalissuearticles values(re^{i\theta},f);$ be the length of the image curve of the line segment joining $re^{i\theta}$ and $re^{iinsert ignore into journalissuearticles values(\theta+\pi);}$ under the mapping $w=finsert ignore into journalissuearticles values(z);$ and let $linsert ignore into journalissuearticles values(r,f);=\max_{0\leq\theta 2 \pi}linsert ignore into journalissuearticles values(re^{i\theta},f);$. We find upper bound for $linsert ignore into journalissuearticles values(r,f);$ for $finsert ignore into journalissuearticles values(z);$ in some known classes of functions. Moreover, we prove that $linsert ignore into journalissuearticles values(r,f);=\mathcal{O}\leftinsert ignore into journalissuearticles values( \log\frac{1}{1-r} \right);$ and that $Linsert ignore into journalissuearticles values(r,f);=\mathcal{O}\left\{ Ainsert ignore into journalissuearticles values(r,f);\log \frac{1}{1-r} \right\}^{1/2}$ as $r\to 1$ under weaker assumptions on $finsert ignore into journalissuearticles values(z);$ than some previous results of this type.
Keywords : length problem, convex functions, starlike functions