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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-7,
author = {V. Karunakaran and K. Bhuvaneswari},
title = {On the radius of convexity for a class of conformal maps},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {251-256},
zbl = {1129.30009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-7}
}
V. Karunakaran; K. Bhuvaneswari. On the radius of convexity for a class of conformal maps. Colloquium Mathematicae, Tome 107 (2007) pp. 251-256. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-7/