Author:
ALI MD FIROZ,VASUDEVARAO A.
Abstract
For a normalized analytic function$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$in the unit disk$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, the estimate of the integral means$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$is an important quantity for certain problems in fluid dynamics, especially when the functions$f(z)$are nonvanishing in the punctured unit disk$\mathbb{D}\setminus \{0\}$. Let${\rm\Delta}(r,f)$denote the area of the image of the subdisk$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$under$f$, where$0<r\leq 1$. In this paper, we solve two extremal problems of finding the maximum value of$L_{1}(r,f)$and${\rm\Delta}(r,z/f)$as a function of$r$when$f$belongs to the class of$m$-fold symmetric starlike functions of complex order defined by a subordination relation. One of the particular cases of the latter problem includes the solution to a conjecture of Yamashita, which was settled recently by Obradovićet al.[‘A proof of Yamashita’s conjecture on area integral’,Comput. Methods Funct. Theory13(2013), 479–492].
Publisher
Cambridge University Press (CUP)
Reference28 articles.
1. On a class of bounded starlike functions;Singh;Indian J. Pure Appl. Math.,1974
2. On maximal area integral problem for analytic functions in the starlike family
3. Area and length maxima for univalent functions
4. On certain classes of starlike functions in the unit disk;Padmanabhan;J. Indian Math. Soc.,1968
5. Univalent Functions in Two-Dimensional Free Boundary Problems
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