scholarly journals Bergman–Hölder Functions, Area Integral Means and Extremal Problems

2017 ◽  
Vol 87 (4) ◽  
pp. 545-563 ◽  
Author(s):  
Timothy Ferguson
Keyword(s):  
Extremal Problems ◽  
Integral Means ◽  
Hölder Functions ◽  
Area Integral

Area Integral Means of Analytic Functions in the Unit Disk

2018 ◽  
Vol 61 (3) ◽  
pp. 509-517 ◽  
Author(s):  
Xiaohui Cui ◽  
Chunjie Wang ◽  
Kehe Zhu
Keyword(s):  
Analytic Function ◽  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Integral Means ◽  
Area Measure ◽  
Area Integral

AbstractFor an analytic function ऒ on the unit disk , we show that the L2 integral mean of ऒ on c < |z| < r with respect to the weighted area measure (1 − |z|2)αd A(z) is a logarithmically convex function of r on (c, 1), where −3 ≤ ∞ ≤ 0 and c ∈ [0, 1). Moreover, the range [−3, 0] for ∞ is best possible. When c = 0, our arguments here also simplify the proof for several results we obtained in earlier papers.

scholarly journals LOGARITHMIC CONVEXITY OF AREA INTEGRAL MEANS FOR ANALYTIC FUNCTIONS II

2014 ◽  
Vol 98 (1) ◽  
pp. 117-128 ◽  
Author(s):  
CHUNJIE WANG ◽  
JIE XIAO ◽  
KEHE ZHU
Keyword(s):  
Analytic Function ◽  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Integral Means ◽  
Area Measure ◽  
Logarithmic Convexity ◽  
Area Integral

AbstractFor$0<p<\infty$and$-2\leq {\it\alpha}\leq 0$we show that the$L^{p}$integral mean on$r\mathbb{D}$of an analytic function in the unit disk$\mathbb{D}$with respect to the weighted area measure$(1-|z|^{2})^{{\it\alpha}}\,dA(z)$is a logarithmically convex function of$r$on$(0,1)$.

Area integral means over the annuli

2019 ◽  
Vol 473 (1) ◽  
pp. 510-518 ◽  
Author(s):  
Chunjie Wang ◽  
Wenjie Yang
Keyword(s):  
Integral Means ◽  
Area Integral

Convexity for area integral means

2020 ◽  
Vol 491 (2) ◽  
pp. 124345
Author(s):  
Qinxia Hu ◽  
Chunjie Wang
Keyword(s):  
Integral Means ◽  
Area Integral

Hadamard Convolution and Area Integral Means in Bergman Spaces

2020 ◽  
Vol 75 (2) ◽  
Author(s):  
Boban Karapetrović ◽  
Javad Mashreghi
Keyword(s):  
Bergman Spaces ◽  
Integral Means ◽  
Area Integral

INTEGRAL MEANS AND DIRICHLET INTEGRAL FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS

2015 ◽  
Vol 99 (3) ◽  
pp. 315-333
Author(s):  
MD FIROZ ALI ◽  
A. VASUDEVARAO
Keyword(s):  
Fluid Dynamics ◽  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Extremal Problems ◽  
Dirichlet Integral ◽  
Integral Means ◽  
Maximum Value ◽  
Image Position

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].

Convexity of area integral means for analytic functions

2016 ◽  
Vol 62 (3) ◽  
pp. 307-317 ◽  
Author(s):  
Weiqiang Peng ◽  
Chunjie Wang ◽  
Kehe Zhu
Keyword(s):  
Analytic Functions ◽  
Integral Means ◽  
Area Integral

scholarly journals Logarithmic Convexity of Area Integral Means for Analytic Functions

2014 ◽  
Vol 114 (1) ◽  
pp. 149 ◽  
Author(s):  
Chunjie Wang ◽  
Kehe Zhu
Keyword(s):  
Analytic Function ◽  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Integral Means ◽  
Area Measure ◽  
Logarithmic Convexity ◽  
Area Integral

We show that the $L^2$ integral mean on $r\mathsf{D}$ of an analytic function in the unit disk $\mathsf{D}$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$, where $-3\le\alpha\le0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $\alpha$ is best possible.

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