On integral means of star-like functions

Gromova, Larisa; Vasil'ev, Alexander

doi:10.1007/bf02829689

Cited by 5 publications

(10 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this connection one of the classical results of Rogosinski [13] for subordination is useful. Using this, we prove a general result and a particular case solves one of the open problems of Gromova and Vasil'ev [5] on the best estimate for a special integral means for starlike functions of order β. Also, we prove Yamashita's conjecture on area maximum property for α-spirallike functions of order β.…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 71%

“…We shall use this lemma mainly for p = 1, but we have stated it in this form as this will help to extend many results of Gromova and Vasil'ev [5]. However, we would like to point out that Lemma A gives generalizations of the some cases in Gromova and Vasil'ev [5], e.g.…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 97%

“…However, we would like to point out that Lemma A gives generalizations of the some cases in Gromova and Vasil'ev [5], e.g. λ 2 = 0 in [5,Theorems 1,3,4].…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 97%

“…It was also remarked that a sharp estimate is still unknown (see [5, p. 565]). Furthermore, not much is known concerning the estimate for L 1 (r, f ) for many geometric classes of functions from S. One of our aims is to state the following sharp estimate as a corollary to a main result and hence, the open problem of Gromova and Vasil'ev [5] is settled.…”

mentioning

confidence: 99%

See 3 more Smart Citations

On the problem of Gromova and Vasil'ev on integral means, and Yamashita's conjecture for spirallike functions

Ponnusamy¹,

Wirths²

2014

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

Abstract. In this article we consider some integral means problem for certain classes of univalent analytic functions, in particular for the class of the starlike functions of order β and for the class of α-spirallike functions of order β. Our investigation settles one of the open problems of Gromova and Vasil'ev. In addition, we solve another problem concerning area maximum property of α-spirallike functions of order β in the setting of Yamashita and hence, we find the solution to Yamashita's conjecture for certain Dirichlet-finite functions in a general form. Preliminaries and the main resultsIn this paper, we investigate some families of univalent functions for which the range of each of its member function has a specific geometric property. We will be particularly interested in determining estimates for certain integral means for functions with geometric property. In this connection one of the classical results of Rogosinski [13] for subordination is useful. Using this, we prove a general result and a particular case solves one of the open problems of Gromova and Vasil'ev [5] on the best estimate for a special integral means for starlike functions of order β. Also, we prove Yamashita's conjecture on area maximum property for α-spirallike functions of order β.Let D := {z ∈ C : |z| < 1} denote the open unit disk and H denote the class of all functions f analytic in D with the topology of uniform convergence of compact subsets of D and A = {f ∈ H : f (0) = 0 and f

show abstract

Section: Preliminaries and The Main Resultsmentioning

confidence: 71%

Section: Preliminaries and The Main Resultsmentioning

confidence: 97%

“…However, we would like to point out that Lemma A gives generalizations of the some cases in Gromova and Vasil'ev [5], e.g. λ 2 = 0 in [5,Theorems 1,3,4].…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 97%

mentioning

confidence: 99%

See 2 more Smart Citations

On the problem of Gromova and Vasil'ev on integral means, and Yamashita's conjecture for spirallike functions

Ponnusamy¹,

Wirths²

2014

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

show abstract

“…Here

S^{*} (β)

denotes the family of starlike functions of order β, i.e., functions

f \in S

such that

Re (\frac{z f^{'} (z)}{f (z)}) > β, z \in double-struckD,

where

0 \leq β < 1

. For

f \in H

, the integral means

I_{1} (r, f) : = \frac{1}{2 π} \int_{- π}^{π} \frac{d θ}{| f (r e^{i θ}) |^{2}}

and the estimates of I 1 are important in certain problems in fluid dynamics (see , , ). Recently the authors in obtained that if

f \in {scriptS}^{☆} (β)

, then the estimate

L_{1} (r, f) : = r^{2} I_{1} (r, f) \leq \frac{Γ (5 - 4 β)}{Γ^{2} (3 - 2 β)}

holds and the inequality is sharp.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Integral means and Dirichlet integral for analytic functions

Obradović

Ponnusamy

Wirths

2014

Mathematische Nachrichten

View full text Add to dashboard Cite

For normalized analytic functions f in the unit disk, the estimate of the integral meansis important in certain problems in fluid dynamics, especially when the functions f (z) are non-vanishing in the punctured unit disk 0 < |z| < 1. We consider the problem of finding the extremal function f which maximizes the integral means L 1 (r, f ) for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses F of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functionaldenotes the area of the image of |z| < r under z/ f (z). The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990. Preliminaries and main resultsDenote by H the family of all functions f which are analytic in the unit disk D := {z ∈ C : |z| < 1} and by A the subfamily of H with the normalization f (0) = 0 = f (0) − 1. Also, let S = { f ∈ A : f is univalent in D} and S * := S * (0) ⊂ S denote the class of all starlike (univalent) functions in D. Here S * (β) denotes the family of starlike functions of order β, i.e., functions f ∈ S such that [7]

show abstract