On the radius of univalence of certain analytic functions

“…Theorem 3.5 follows readily from (29) and (30). Finally, it is easy to see that the bounds in (27) are attained for the function f (z) given by (28).…”

Section: Theorem 35 If a Function F (Z) Defined By (10) Is In The Classmentioning

confidence: 81%

“…In the present paper, we make use of the familiar integral operator I ϑ, p defined by (see, for details, [9,27,30]; see also [55])…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Certain Subclasses of Multivalent Functions Defined by New Multiplier Transformations

Deni̇z

Orhan

2011

Arab J Sci Eng

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The new multiplier transformations J δ p (λ, μ, l) (δ, l ≥ 0, λ ≥ μ ≥ 0; p ∈ N) of multivalent functions are defined. Making use of the operator J δ p (λ, μ, l), two new subclasses P δ λ,μ,l (A, B; σ, p) and P δ λ,μ,l (A, B; σ, p) of multivalent analytic functions are introduced and investigated in the open unit disk. Some interesting relations and characteristics such as inclusion relationships, neighborhoods, and partial sums, and some applications of fractional calculus and quasi-convolution properties of functions belonging to each of these subclasses P δ λ,μ,l (A, B; σ, p) and P δ λ,μ,l (A, B; σ, p) are investigated. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.

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“…This conjecture states that 1 2 (zf (z)) = F 1/2 (f )(z) is univalent in |z| < 1/2 for every f ∈ S. There are a number of papers (see for instance [9,1,2]) dealing with this problem, but Robinson's 1/2-conjecture remains an intriguing open problem. In most of the afore-mentioned papers certain properties of the Robinson operator F 1/2 on geometrically defined subsets of S have been studied and only very little is known about the behaviour of the operator F 1/2 on the set S of all univalent functions.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

On the radius of convexity of linear combinations of univalent functions and their derivatives

Greiner

Roth

2003

Mathematische Nachrichten

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Key words Robinson's conjecture, Hadamard product, univalent functions, radius of convexity MSC (2000) Primary: 30C75 Let S denote the set of normalized univalent functions in the unit disk. We consider the problem of finding the radius of convexity rα of the setfor fixed α ∈ C. Using a linearization method we find the exact value of rα for α ∈ [0, 1] and prove the (sharp) estimate rα ≥ r1 for α ∈ C with |2α − 1| ≤ 1. As an application of these results the sharp lower bound for the radius of convexity of the convolution f * g where f, g ∈ S and g is close-to-convex is found to be 5 − 2 √ 6. The case α = 1/2 is related to an old conjecture of Robinson dating back to 1947.

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On the radius of univalence of certain analytic functions

Abstract: It is possible that these together with the constant unitary matrices generate the whole class of such functions, but we have not been able to prove it.

Cited by 150 publications

References 6 publications

Angular estimations of certain integral operators

Angular estimations of certain integral operators

Certain Subclasses of Multivalent Functions Defined by New Multiplier Transformations

On the radius of convexity of linear combinations of univalent functions and their derivatives

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