Area and length maxima for univalent functions

Yamashita, Shinji

doi:10.1017/s0004972700018311

Cited by 17 publications

(16 citation statements)

References 2 publications

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“…We call f a Dirichlet-finite function if ∆(1, f ), the area covered by the mapping z → f (z) for |z| < 1, is finite. Our interest in this paper was originated by the work of Yamashita [21] and Ponnusamy et. al.…”

Section: Introduction Preliminaries and Main Resultsmentioning

confidence: 99%

On maximal area integral problem for analytic functions in the starlike family

Sahoo¹,

Sharma²

2015

Journal of Classical Analysis

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For an analytic function f defined on the unit disk |z| < 1, let ∆(r, f ) denote the area of the image of the subdisk |z| < r under f , where 0 < r ≤ 1. In 1990, Yamashita conjectured that ∆(r, z/f ) ≤ πr 2 for convex functions f and it was finally settled in 2013 by Obradović and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation zf ′ (z)/f (z) ≺ (1 + (1 − 2β)αz)/(1 − αz) for 0 ≤ β < 1 and 0 < α ≤ 1. We prove Yamashita's conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.

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Section: Introduction Preliminaries and Main Resultsmentioning

confidence: 99%

On maximal area integral problem for analytic functions in the starlike family

Sahoo¹,

Sharma²

2015

Journal of Classical Analysis

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“…Thus, a function has finite Dirichlet integral exactly when its image has finite area (counting multiplicities). In 1990, Yamashita proved the following. Theorem We have for the Yamashita functional, (1)

0true max_{f \in scriptS} normalΔ (() r, \frac{f (z)}{z}) = \frac{2 π r^{2} (r^{2} + 2)}{{(1 - r^{2})}^{4}} for 0 < r \leq 1 .

(2)

0true max_{f \in scriptS} normalΔ (() r, \frac{z}{f (z)}) = 2 π r^{2} (r^{2} + 2) for 0 < r \leq 1 .

For each r ,

0 < r \leq 1

, the maximum is attained only by the rotations of the Koebe function

k (z)

.…”

Section: Yamashita's Area Integralmentioning

confidence: 97%

“…For the family

C \subset S

of convex functions, Yamashita [, p. 439] conjectured that

\underset{f \in C}{trueprefixmax} Δ (r, \frac{z}{f false(z false)}) = π r^{2}, for 0 < r \leq 1,

where the maximum is attained only by the rotations of the function

j (z) = z / (1 - z)

. In , the authors proved this conjecture in a more general form and in , , the authors obtained analog result for some other classes of functions and spirallike functions, respectively.…”

Section: Yamashita's Area Integralmentioning

confidence: 99%

Concave univalent functions and Dirichlet finite integral

Ponnusamy

2016

Math. Nachr.

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The article deals with the class Fα consisting of non‐vanishing functions f that are analytic and univalent in double-struckD such that the complement C∖f(D) is a convex set, f(1)=∞, f(0)=1 and the angle at ∞ is less than or equal to απ, for some α∈(1,2]. Related to this class is the class CO(α) of concave univalent mappings in double-struckD, but this differs from Fα with the standard normalization ffalse(0false)=0=f′false(0false)=1. A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for CO(2) settled by Avkhadiev et al. . Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for CO(α) is also presented.

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“…Note that k θ maps the unit disk D onto the complement of a ray. In any case, since the functional A f → L r ( f ) is continuous and the class S is compact, a solution of the extremal problem max f ∈S L r ( f ) (1.1) exists and is in S. We remark that with a clever use of Dirichlet-finite integral and the isoperimetric inequality, Yamashita [16] obtained the upper and lower estimates for the functional (1.1)…”

Section: Introductionmentioning

confidence: 95%

“…In any case, since the functional

A ∋ f \mapsto L_{r} (f)

is continuous and the class

scriptS

is compact, a solution of the extremal problem

\underset{f \in S}{trueprefixmax} L_{r} (f)

exists and is in

scriptS

. We remark that with a clever use of Dirichlet‐finite integral and the isoperimetric inequality, Yamashita obtained the upper and lower estimates for the functional

m (r) \leq L_{r} (k) \leq \underset{f \in S}{trueprefixmax} L_{r} (f) \leq \frac{2 π r}{{(1 - r)}^{2}},

where

m (r) = \frac{2 π r \sqrt{r^{4} + 4 r^{2} + 1}}{{(1 - r^{2})}^{2}} \geq \frac{2 π r (1 + r)}{{(1 - r)}^{2}} \frac{\sqrt{6}}{8} > \frac{π r (1 + r)}{2 {(1 - r)}^{2}} .

This observation provides an improvement over the earlier result of Duren [, Theorem 2] and [, p. 39], and moreover,

m (r) \geq \frac{\sqrt{6}}{2} \frac{π r}{{(1 - r)}^{2}} .

The extremal problem stimulated much research in the theory of univalent functions, and the problem of determining of the maximum value and the extremal functions in

scriptS

remains open. However, the extremal problem

…”

Section: Introductionmentioning

confidence: 96%

Circular symmetrization, subordination and arclength problems on convex functions

et al. 2015

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Abstract. We study the class C(Ω) of univalent analytic functions f in the unit disk D = {z ∈ C : |z| < 1} of the form f (z) = z + ∞ n=2 a n z n satisfyingwhere Ω will be a proper subdomain of C which is starlike with respect to 1(∈ Ω). Let φ Ω be the unique conformal mapping of D onto Ω with φ Ω (0) = 1 anddenote the arclength of the image of the circle {z ∈ C : |z| = r}, r ∈ (0, 1). The first result in this paper is an inequality L r (f ) ≤ L r (k Ω ) for f ∈ C(Ω), which solves the general extremal problem max f ∈C(Ω) L r (f ), and contains many other well-known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class C(Ω).

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Area and length maxima for univalent functions

Cited by 17 publications

References 2 publications

On maximal area integral problem for analytic functions in the starlike family

On maximal area integral problem for analytic functions in the starlike family

Concave univalent functions and Dirichlet finite integral

Circular symmetrization, subordination and arclength problems on convex functions

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