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$(0,1)$.

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.

scholarly journals Subordination by convex functions

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
Rosihan M. Ali ◽  
V. Ravichandran ◽  
N. Seenivasagan
Keyword(s):  
Analytic Function ◽  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Subordination Result ◽  
Special Cases

For a fixed analytic functiong(z)=z+∑n=2∞gnzndefined on the open unit disk andγ<1, letTg(γ)denote the class of all analytic functionsf(z)=z+∑n=2∞anznsatisfying∑n=2∞|angn|≤1−γ. For functions inTg(γ), a subordination result is derived involving the convolution with a normalized convex function. Our result includes as special cases several earlier works.

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].

scholarly journals Some Connections between Class𝒰- andα-Convex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Edmond Aliaga ◽  
Nikola Tuneski
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk

The class𝒰(λ,μ)of normalized analytic functions that satisfy|(z/f(z))1+μ·f′(z)−1|<λfor allzin the open unit disk is studied and sufficient conditions for anα-convex function to be in𝒰(λ,μ)are given.

scholarly journals Some Sufficient Conditions for Analytic Functions to Belong to Space

2008 ◽  
Vol 2008 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaoge Meng
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions

This paper gives some sufficient conditions for an analytic function to belong to the space consisting of all analytic functions on the unit disk such

scholarly journals Some characteristic properties of analytic functions

2016 ◽  
Vol 24 (1) ◽  
pp. 353-369
Author(s):  
R. K. Raina ◽  
Poonam Sharma ◽  
G. S. Sălăgean
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Integral Representations ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Bilinear Mapping ◽  
Initial Coefficient

AbstractIn this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that,where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

scholarly journals On the second hankel determinant for the kth-root transform of analytic functions

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 227-245 ◽  
Author(s):  
Najla Alarifi ◽  
Rosihan Ali ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
The Given ◽  
Logarithmically Convex Functions

Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of ?.

scholarly journals Radius of Star-Likeness for Certain Subclasses of Analytic Functions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2448
Author(s):  
Caihuan Zhang ◽  
Mirajul Haq ◽  
Nazar Khan ◽  
Muhammad Arif ◽  
Khurshid Ahmad ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Inverse Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sine Function ◽  
Exponential Functions ◽  
Lemniscate Of Bernoulli ◽  
Rotation Function

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.

scholarly journals On area integrals and radial variations of analytic functions in the unit disk

1976 ◽  
Vol 61 ◽  
pp. 135-159
Author(s):  
Takafumi Murai
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Analytic Functions ◽  
Area Integral

We are concerned with the behaviour of analytic functions near the boundary. Let T and D be the unit circle |z| = 1 and the unit disk |z| < 1, respectively. The element of T is denoted by θ (0 ≤ θ < 2π). Let be analytic in D. The area integral A(f, θ) of f at θ is defined by

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