Convolution and quasiconformal extension

We consider the class Σ(p) of univalent meromorphic functions f on D having simple pole at z = p ∈ [0, 1) with residue 1. Let Σ k (p) be the class of functions in Σ(p) which have k-quasiconformal extension to the extended complex plane C where 0 ≤ k < 1. We first give a representation formula for functions in this class and using this formula we derive an asymptotic estimate of the Laurent coefficients for the functions in the class Σ k (p). Thereafter we give a sufficient condition for functions in Σ(p) to belong in the class Σ k (p). Finally we obtain a sharp distortion result for functions in Σ(p) and as a consequence, we get a distortion estimate for functions in Σ k (p).

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“…In 1976, J. G. Krzyż [7], gave a sufficient condition for functions to belong in the class Σ k (p). We state it below.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On some results for a class of meromorphic functions having quasiconformal extension

Bhowmik

Satpati

2018

Proc Math Sci

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“…If g = co = / o r g =f co' = 1 in (13), we obtain conditions f o r / ' -1 a n d / " / / ' , respectively. They generalize the criteria of Krzyz [13] and Becker [6], where G = D, X(z) = 1/z.…”

Section: Suitable Choices Of O(z W) Yield Various Types Of Univalencmentioning

confidence: 94%

Univalence Criteria and Löwner Chains

Betker

1991

Bulletin of the London Mathematical Society

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“…In fact, Krzyz [6] has shown that function in U{λ) admits a Q-quasiconformal extension to the whole complex plane with Q = --γ whenever 0 < Λ < 1. However, until 1 -λ our recent work not much attention has been paid to this class.…”

Section: Convolution Theoremsmentioning

confidence: 97%

Two Parameter Families of Close-to-Convex Functions and Convolution Theorems

Barnard¹,

Naik²,

Obradović³

et al. 2004

Analysis

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Using a recently obtained coefficient condition for functions with positive real part by M. Obradovic and S. Ponnusamy. we construct four different useful examples of two parameter families of close-to-convex functions. As a consequence of another well-known coefficient condition due to M. Obradovic and S. Ponnusamy, we obtain a number of results concerning the convolution of two univalent functions. Using these results, we obtain criteria for combinations of hypergeometric functions to be univalent.

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Convolution and quasiconformal extension

Cited by 26 publications

References 5 publications

On some results for a class of meromorphic functions having quasiconformal extension

On some results for a class of meromorphic functions having quasiconformal extension

Univalence Criteria and Löwner Chains

Two Parameter Families of Close-to-Convex Functions and Convolution Theorems

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