On the radius of convexity for a class of conformal maps

@article{V2007,
abstract = { Let 𝓐 denote the class of all analytic functions f in the open unit disc 𝔻 in the complex plane satisfying f(0) = 0, f'(0) = 1. Let U(λ) (0 < λ ≤ 1) denote the class of functions f ∈ 𝓐 for which |(z/f(z))²f'(z) -1| < λ for z ∈ 𝔻. The behaviour of functions in this class has been extensively studied in the literature. In this paper, we shall prove that no member of U₀(λ) = \{f ∈ U(λ): f''(0) = 0\} is convex in 𝔻 for any λ and obtain a lower bound for the radius of convexity for the family U₀(λ). These results settle a conjecture proposed in the literature negatively. We also improve the existing lower bound for the radius of convexity of the family U₀(λ). },
author = {V. Karunakaran, K. Bhuvaneswari},
journal = {Colloquium Mathematicae},
keywords = {convexity; radius of convexity; order of starlikeness},
language = {eng},
number = {2},
pages = {251-256},
title = {On the radius of convexity for a class of conformal maps},
url = {http://eudml.org/doc/283916},
volume = {109},
year = {2007},
}

TY - JOUR
AU - V. Karunakaran
AU - K. Bhuvaneswari
TI - On the radius of convexity for a class of conformal maps
JO - Colloquium Mathematicae
PY - 2007
VL - 109
IS - 2
SP - 251
EP - 256
AB - Let 𝓐 denote the class of all analytic functions f in the open unit disc 𝔻 in the complex plane satisfying f(0) = 0, f'(0) = 1. Let U(λ) (0 < λ ≤ 1) denote the class of functions f ∈ 𝓐 for which |(z/f(z))²f'(z) -1| < λ for z ∈ 𝔻. The behaviour of functions in this class has been extensively studied in the literature. In this paper, we shall prove that no member of U₀(λ) = {f ∈ U(λ): f''(0) = 0} is convex in 𝔻 for any λ and obtain a lower bound for the radius of convexity for the family U₀(λ). These results settle a conjecture proposed in the literature negatively. We also improve the existing lower bound for the radius of convexity of the family U₀(λ).
LA - eng
KW - convexity; radius of convexity; order of starlikeness
UR - http://eudml.org/doc/283916
ER -