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Criteria for univalence, starlikeness and convexity

100%

Ponnusamy S., Vasundhra P.

Annales Polonici Mathematici

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2005

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tom 85

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nr 2

121-133

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.

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Sufficient conditions for starlike and convex functions

100%

Ponnusamy S., Vasundhra P.

Annales Polonici Mathematici

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2007

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tom 90

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nr 3

277-288

EN

For n ≥ 1, let 𝓐 denote the class of all analytic functions f in the unit disk Δ of the form $f(z) = z + ∑_{k=2}^∞ a_kz^k$. For Re α < 2 and γ > 0 given, let 𝓟(γ,α) denote the class of all functions f ∈ 𝓐 satisfying the condition |f'(z) - α f(z)/z + α - 1| ≤ γ, z ∈ Δ. We find sufficient conditions for functions in 𝓟(γ,α) to be starlike of order β. A generalization of this result along with some convolution results is also obtained.

3

Univalence, strong starlikeness and integral transforms

81%

Obradović M., Ponnusamy S., Vasundhra P.

Annales Polonici Mathematici

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2005

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tom 86

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nr 1

1-13

EN

Let 𝓐 represent the class of all normalized analytic functions f in the unit disc Δ. In the present work, we first obtain a necessary condition for convex functions in Δ. Conditions are established for a certain combination of functions to be starlike or convex in Δ. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on f and μ so that the non-linear integral transform $∫_0^z (ζ/f(ζ))^{μ} dζ$ is univalent in Δ. Finally, we give sufficient conditions for functions to be strongly starlike of order α.

Ograniczanie wyników

3 Annales Polonici Mathematici

3 Ponnusamy S.

3 Vasundhra P.

1 Obradović M.

1 2007

2 2005

Criteria for univalence, starlikeness and convexity

Sufficient conditions for starlike and convex functions

Univalence, strong starlikeness and integral transforms