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On the radius of convexity for a class of conformal maps

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Karunakaran V., Bhuvaneswari K.

Colloquium Mathematicum

2007

tom 109

nr 2

251-256

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).

Ograniczanie wyników

1 Colloquium Mathematicum

1 Bhuvaneswari K.

1 Karunakaran V.

1 2007

Wyniki wyszukiwania

On the radius of convexity for a class of conformal maps