Sharp starlikeness conditions for analytic functions with bounded derivative

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]

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“…The numbers

2 / \sqrt{5}

and

3 / \sqrt{10}

in Lemma A were proved to be sharp (see for instance , ). We refer to for many other interesting properties of the class

R (α, λ)

.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 97%

Integral means and Dirichlet integral for analytic functions

Obradović

Ponnusamy

Wirths

2014

Mathematische Nachrichten

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“…For example, Ponnusamy and Singh [15] have shown that Uðl; mÞ J S Ã if m < 0 and 0 a l a 1 À m ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð1 À mÞ 2 þ m 2 q ¼: l Ã ðmÞ and in [8], Obradovic ´proved that the above inclusion continues to hold for 0 < m a 1 and with the same bound for l. The sharpness part of these results may be obtained as a consequence of results from [21]. However, it is not known whether each function f in Uð1; mÞ (or more generally, Uðl; mÞ with l Ã ðmÞ < l a 1) is univalent in D for certain values of m in the open interval ð0; 1Þ.…”

Section: Introductionmentioning

confidence: 86%

Study of some subclasses of univalent functions and their radius properties

Ponnusamy

Sahoo

2006

Kodai Math. J.

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An analytic functionfor some l b 0 and m > À1. For À1 a a a 1, we introduce a geometrically motivated S p ðaÞ-class defined by;where S represents the class of all normalized univalent functions in D. In this paper, the authors determine necessary and su‰cient coe‰cient conditions for certain class of functions to be in S p ðaÞ. Also, radius properties are considered for S p ðaÞ-class in the class S. In addition, we also find disks jzj < r :¼ rðl; mÞ for which 1 r f ðrzÞ A Uðl; mÞ whenever f A S. In addition to a number of new results, we also present several new su‰cient conditions for f to be in the class Uðl; mÞ.

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A Survey On Some Special Classes of Bazilevič Functions and Related Function Classes

Sahoo

Mohapatra

2014

Trends in Mathematics

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Sharp starlikeness conditions for analytic functions with bounded derivative

Cited by 8 publications

References 5 publications

Integral means and Dirichlet integral for analytic functions

Integral means and Dirichlet integral for analytic functions

Study of some subclasses of univalent functions and their radius properties

A Survey On Some Special Classes of Bazilevič Functions and Related Function Classes

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