The Bound of the Hankel Determinant of the Third Kind for Starlike Functions

Kwon, Oh Sang; Lecko, Adam; Sim, Young Jae

doi:10.1007/s40840-018-0683-0

Cited by 101 publications

(51 citation statements)

References 18 publications

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“…Further for the sharpness, he examined the subfamilies of S * and C consisting of functions with m-fold symmetry and obtained the sharp bounds. Moreover this determinant was further improved by Kwon et al [38] and proved |H 3,1 ( f )| ≤ 8/9 for f ∈ S * , yet not best possible. The authors in [39][40][41] contributed in similar direction by generalizing different classes of univalent functions with respect to symmetric points.…”

Section: The Family S *mentioning

confidence: 99%

An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function

et al. 2019

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In the current article, we consider certain subfamilies S e ∗ and C e of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family S e ∗ in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions.

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Section: The Family S *mentioning

confidence: 99%

An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function

et al. 2019

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“…The sharp results for H 3,1 are difficult to obtain. It is worth citing the sharp bound |H 3,1 | ≤ 4/135 for K and the non-sharp estimate |H 3,1 | ≤ 8/9 for S * obtained by Kowalczyk et al and Kwon et al, respectively (see, [2,3]).…”

Section: Introductionmentioning

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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“…The classes of Janowski type starlike and Janowski type convex functions are defined, respectively, by [14]). For more details, we refer the reader to [6,15].…”

Section: Introductionmentioning

confidence: 99%