On Two Extremal Problems Related to Univalent Functions

Fournier, Richard; Ruscheweyh, Stephan

doi:10.1216/rmjm/1181072416

Cited by 59 publications

(44 citation statements)

References 4 publications

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“…This proves the required result by observing that |I(w)| ≤ 1/|2 − α|. Indeed, we have equality in the last inequality, which is proved exactly as in the proof of Theorem 1 in [7].…”

Section: Proof Of Main Resultssupporting

confidence: 75%

“…The above question is motivated by a general result due to Fournier and Ruscheweyh in [7] which has already been extended in a number of ways (see [10,2,1,11]). For example, one has Corollary 1.1 ( [7]).…”

mentioning

confidence: 99%

“…For example, one has Corollary 1.1 ( [7]). Let β = 1 − 1/2(1 − log 2) ≈ −0.6294 and let g ∈ A satisfy the condition Re{e iφ (g ′ (z) − β)} > 0 in ∆ for some φ ∈ R. Then the Alexander transform Ag is starlike, and the value of β is sharp.…”

mentioning

confidence: 99%

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Sufficient conditions for starlike and convex functions

Ponnusamy

Vasundhra

2007

Ann. Polon. Math.

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Abstract. For n ≥ 1, let A denote the class of all analytic functions f in the unit disk ∆ of the form f (z) = z + ∞ k=2 a k z k . For Re α < 2 and γ > 0 given, let P(γ, α) denote the class of all functions f ∈ A satisfying the conditionWe 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.

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“…This proves the required result by observing that |I(w)| ≤ 1/|2 − α|. Indeed, we have equality in the last inequality, which is proved exactly as in the proof of Theorem 1 in [7].…”

Section: Proof Of Main Resultssupporting

confidence: 75%

mentioning

confidence: 99%

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Sufficient conditions for starlike and convex functions

Ponnusamy

Vasundhra

2007

Ann. Polon. Math.

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“…This operator introduced by Fournier and Ruscheweyh [5] has been studied by a number of authors by various choices of λ(t). For 0 γ 1 and β < 1, let P γ (β) denote the class of all functions in A such that…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Duality techniques for certain integral transforms to be starlike

Balasubramanian

Ponnusamy

2004

Journal of Mathematical Analysis and Applications

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For β < 1, let P γ (β) denote the class of all normalized analytic functions f such thatfor some ϕ ∈ R. Let S * (µ), 0 µ < 1, denote the usual class of starlike functions of order µ,In this paper we find conditions so that the integral transform V λ (f )(z) = 1 0 λ(t) f (tz) t dt carries P γ (β) into S * (µ). We provide a number of applications for certain special choices of λ(t). These results extend previously known results by a number of authors.

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“…In 2002, for η = 0 in (1.2), Liu [14] gave a univalence criterion for functions in the class P γ (α, β). For α = γ = 1, Fournier and Ruscheweyh [12] and Ali and Singh [3] used the Duality Principle [19,20] to prove starlikeness and convexity of the linear integral transform V λ,α ( f ), when f varies in the class P γ (α, β). For α = 1, Kim and Rønning [13] and Choi et al [9] studied starlikeness and convexity of the linear transform V λ,α ( f ), f ∈ P γ (α, β).…”

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confidence: 99%

Geometric Properties of an Integral Operator

Verma

Gupta

Singh

2016

Bull. Malays. Math. Sci. Soc.

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Let P γ (α, β), denote the class of all normalized analytic functions f defined in the unit disc E = {z : |z| < 1} such thatfor all z ∈ E and η ∈ R, where β < 1, α ≥ 0 and 0 ≤ γ ≤ 1. For a real-valued nonnegative function λ with the normalization 1 0 λ(t)dt = 1, we consider the integral operatorsand a class S δ (ν) of normalized analytic functions f which satisfy the conditionfor δ < ν ≤ 1+δ and 0 ≤ δ < 1. The aim of this paper is to find the sharp value of β so that the operator V λ,α ( f ) carries P γ (α, β) into S δ (α). Some interesting applications for different choices of λ are discussed.

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On Two Extremal Problems Related to Univalent Functions

Cited by 59 publications

References 4 publications

Sufficient conditions for starlike and convex functions

Sufficient conditions for starlike and convex functions

Duality techniques for certain integral transforms to be starlike

Geometric Properties of an Integral Operator

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