Radii of starlikeness and convexity for certain classes of analytic functions

Tuan, P. D.; Anh, Vo

doi:10.1016/0022-247x(78)90027-6

Cited by 13 publications

(8 citation statements)

References 8 publications

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“…, Gupta et al [3] and Kumar [5], (see also [12]) generalized these results. In the present paper, we extend these results and establish some interesting results.…”

Section: Introductionmentioning

confidence: 81%

Radii Problems of Certain Subclasses of Analytic Functions With Fixed Second Coefficients

Porwal¹,

Bulut²

2015

Honam Mathematical Journal

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“…, Gupta et al [3] and Kumar [5], (see also [12]) generalized these results. In the present paper, we extend these results and establish some interesting results.…”

Section: Introductionmentioning

confidence: 81%

Radii Problems of Certain Subclasses of Analytic Functions With Fixed Second Coefficients

Porwal¹,

Bulut²

2015

Honam Mathematical Journal

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“…Remark. (i) By using the expression of f ∈ F 1 (α) in [TA,Theorem 3.1] with α = 0.8417 (by Proposition 7.1), it is quite direct to show that F 3, 1 2 ( 1 z ) maps {|z| > 1.1974} to a starlike set, which is weaker than Proposition 7.2.…”

Section: Self-similar Measures With M =mentioning

confidence: 98%

Cauchy transforms of self-similar measures: Starlikeness and univalence

Dong

Lau

2016

Trans. Amer. Math. Soc.

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For the contractive iterated function system S k z = e 2 π i k / m + ρ ( z − e 2 π i k / m ) S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})} with 0 > ρ > 1 , k = 0 , ⋯ , m − 1 0>\rho >1, k=0,\cdots , m-1 , we let K ⊂ C K\subset \mathbb {C} be the attractor, and let μ \mu be a self-similar measure defined by μ = 1 m ∑ k = 0 m − 1 μ ∘ S k − 1 \mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1} . We consider the Cauchy transform F F of μ \mu . It is known that the image of F F at a small neighborhood of the boundary of K K has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of F F away from K K ; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.

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“…The following lemma is obtained by Tuan and Anh [31] as a particular case of Theorem 3 of [11] for the class of Carathéodory functions P (see [10]). Lemma 3.1.…”

Section: Inclusion and Radius Problemsmentioning

confidence: 99%

“…Tuan and Anh [31] found the radius of starlikeness for the subclass of S, which satisfies the condition Re(f (z)/z) > k, 0 ≤ k < 1. In the following, we extend their result by considering the general class S * (ϕ) in place of S * .…”

Section: Rementioning

confidence: 99%