On the coefficient problem for close-to-convex functions

Trąbka-Więcław, Katarzyna; Zaprawa, Paweł

doi:10.3906/mat-1711-36

Cited by 5 publications

(6 citation statements)

References 19 publications

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“…The proof is divided into six lemmas. The first one is a particular case of the result obtained in [22] (Theorem 3.1 or Theorem 3.3 in [22]), and the second one is obvious.…”

Section: Bounds Of |θ(µ)| For the Class C 0 (K)mentioning

confidence: 70%

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On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

et al. 2019

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In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( μ ) = a 4 − μ a 2 a 3 and Φ f ( μ ) = a 2 a 4 − μ a 3 2 for a real number μ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + … , | z | < 1 . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re f ( z ) / z > α or Re f ′ ( z ) > α , α ∈ [ 0 , 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( μ ) and Φ f ( μ ) for close-to-convex functions such that f ′ ( z ) = h ( z ) / ( 1 − z ) 2 , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of Θ f ( μ ) , Φ f ( μ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions.

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“…The proof is divided into six lemmas. The first one is a particular case of the result obtained in [22] (Theorem 3.1 or Theorem 3.3 in [22]), and the second one is obvious.…”

Section: Bounds Of |θ(µ)| For the Class C 0 (K)mentioning

confidence: 70%

“…For p ∈ (−2, 0], we have ac ≥ 0, and the inequality |b| < 2(1 − |c|) from the last case of Lemma 7 is equivalent to (21). Therefore, Y(a, b, c) is also given by (22).…”

Section: Bounds Of |θ(µ)| For the Class C 0 (K)mentioning

confidence: 97%

On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

et al. 2019

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“…It is worth adding that both estimates can be improved if a more precise inequality than |p 3 − 1 2 p 1 p 2 | ≤ 2 for P ∈ P is applied. In [10] it was proved that…”

Section: Concluding Remarkmentioning

confidence: 99%

Hankel Determinants for Univalent Functions Related to the Exponential Function

Zaprawa

2019

Symmetry

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Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.

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“…equivalently (5) yields wH ′ (w) = q(w)G(w), with G(w) = w + ∞ n=2 h n w n . It is known from [12,39], that a function f is known to be close-to-convex with the argument ϕ if a real number…”

Section: Introductionmentioning

confidence: 99%

Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order

Rehman,,

Mashrafi,,

Salah,

2024

MJMS

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The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the study of bi-close-to-convex functions and its second degree Hankel determinant. At the end of the article, we remark on improvement in the earlier work of some researchers and discover a better value than the one they obtained.

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On the coefficient problem for close-to-convex functions

Cited by 5 publications

References 19 publications

On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

Hankel Determinants for Univalent Functions Related to the Exponential Function

Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order

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