On the radius of convexity for a class of conformal maps

• Autorzy: V. Karunakaran , K. Bhuvaneswari
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
• 30A10: Inequalities in the complex domain

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2007
• Tom: 109
• Numer: 2
• Strony: 251-256

Daty

wydano

2007

Twórcy

autor

V. Karunakaran

• School of Mathematics, Madurai Kamaraj University, Madurai 625021, India

autor

K. Bhuvaneswari

• chool of Mathematics, Madurai Kamaraj University, Madurai 625021, India

Identyfikatory

DOI

10.4064/cm109-2-7