Univalency, Starlikeness and Convexity Applied to Certain Classes of Rational Functions

Obradović, Milutin; Ponnusamy, Saminathan; Singh, Vandana; Vasundhra, P.

doi:10.1524/anly.2002.22.3.225

Cited by 29 publications

(28 citation statements)

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“…Theorem 1.6(i) is due to [7] while Theorem 1.6(ii) has been obtained recently by Obradović et al [4]. Our next result improves Theorem 1.6.…”

supporting

confidence: 71%

“…This result was originally stated as a conjecture in [3] and was proved in [4]. In this article, we discuss the relationship between U(λ) and S * (δ), as well as between P(2λ) and K(δ).…”

mentioning

confidence: 82%

“…According to a result due to Ozaki and Nunokawa [6], we have the inclusion U(λ) ⊂ S for 0 < λ ≤ 1, and from [3], we also have the inclusion P(2λ) ⊂ U(λ). In [4], the authors have shown that certain results obtained in [3] also hold if P(2λ) is replaced by U(λ). In this connection, we recall the following result from [4].…”

mentioning

confidence: 96%

“…In [4], the authors have shown that certain results obtained in [3] also hold if P(2λ) is replaced by U(λ). In this connection, we recall the following result from [4].…”

mentioning

confidence: 96%

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Criteria for univalence, starlikeness and convexity

Ponnusamy

Vasundhra

2005

Ann. Polon. Math.

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Abstract. Let A denote the class of all normalized analytic functions f (f (0) = 0 = f (0) − 1) in the open unit disc ∆. For 0 < λ ≤ 1, defineRecently, 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 f ∈ 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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“…Theorem 1.6(i) is due to [7] while Theorem 1.6(ii) has been obtained recently by Obradović et al [4]. Our next result improves Theorem 1.6.…”

supporting

confidence: 71%

mentioning

confidence: 82%

mentioning

confidence: 96%

“…In [4], the authors have shown that certain results obtained in [3] also hold if P(2λ) is replaced by U(λ). In this connection, we recall the following result from [4].…”

mentioning

confidence: 96%

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Criteria for univalence, starlikeness and convexity

Ponnusamy

Vasundhra

2005

Ann. Polon. Math.

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“…Reade-Silverman-Todorov [8], Obradovic [5] and Obradovic-PonnusamySingh-Vasundhra [7] gave coefficient conditions for functions of the form (1.1) to be starlike of order a (0 < a < 1) or convex of order ¡3 (0 < ¡3 < 1) on the unit disc U in C.…”

Section: Introductionmentioning

confidence: 99%