Criteria for univalence, starlikeness and convexity

• Autorzy: S. Ponnusamy , P. Vasundhra
• Języki publikacji: EN

Abstrakty

EN

Let 𝓐 denote the class of all normalized analytic functions f (f(0) = 0 = f'(0)-1) in the open unit disc Δ. For 0 < λ ≤ 1, define
𝓤(λ) = {f ∈ 𝓐 : |(z/f(z))²f'(z) - 1| < λ, z ∈ Δ}
and
𝓟(2λ) = f ∈ 𝓐 : |(z/f(z))''| < 2λ, z ∈ Δ.cr Recently, the problem of finding the starlikeness of these classes has been considered by Obradović and Ponnusamy, and later by Obradović et al. In this paper, the authors consider the problem of finding the order of starlikeness and of convexity of 𝓤(λ) and 𝓟(2λ), respectively. In particular, for fi ∈ 𝓐 with f''(0) = 0, we find conditions on λ, β*(λ) and β(λ) so that 𝓤(λ) ⊊ 𝓢*(β*(λ)) and 𝓟(2λ) ⊊ 𝒦(β(λ)). Here, 𝓢*(β) and 𝒦(β) (β < 1) denote the classes of functions in 𝓐 that are starlike of order β and convex of order β, respectively. In addition to these results, we also provide a coefficient condition for functions to be in 𝒦(β). Finally, we propose a conjecture that each function f ∈ 𝓤(λ) with f''(0) = 0 is convex at least when 0 < λ ≤ 3 - 2√2.

Kategorie tematyczne

• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
• 30A10: Inequalities in the complex domain
• 30C55: General theory of univalent and multivalent functions
• 30C80: Maximum principle; Schwarz's lemma, Lindel\"of principle, analogues and generalizations; subordination

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2005
• Tom: 85
• Numer: 2
• Strony: 121-133

Daty

wydano

2005

Twórcy

autor

S. Ponnusamy

• Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai 600 036, India

autor

P. Vasundhra

• Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai- 600 036, India

Identyfikatory

DOI

10.4064/ap85-2-2