The third logarithmic coefficient for the class S

Obradović, Milutin; Tuneski, Nikola

doi:10.3906/mat-2002-122

Cited by 5 publications

(4 citation statements)

References 4 publications

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“…We shall estimate W 2 using Lemma 2. Namely, Hðx; y; zÞ ; ð2:5Þ where Hðx; y; zÞ The function H as a quadratic function of a variable y achieves its greatest value when y ¼ 5 8 z. Consequently, Hðx; y; zÞ Hðx;…”

Section: ð2:2þmentioning

confidence: 99%

“…is valid for S. A little is known about succeeding logarithmic coefficients of univalent functions. In very recent paper [5], Obradović and Tuneski proved that jc 3 j ffiffiffiffiffi ffi 133 p 15 for all f 2 S. Girela in [3] shown that also in the class C of close-to-convex functions there exist functions such that their logarithmic coefficients are greater than 1 n . The similar fact, but for the class U of univalent functions satisfying the condition z f ðzÞ 2 f 0 ðzÞ À 1 \1 ;…”

Section: Introductionmentioning

confidence: 99%

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Initial logarithmic coefficients for functions starlike with respect to symmetric points

Zaprawa

2021

Bol. Soc. Mat. Mex.

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In this paper, we obtain the bounds of the initial logarithmic coefficients for functions in the classes $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S of functions which are starlike with respect to symmetric points and convex with respect to symmetric points, respectively. In our research, we use a different approach than the usual one in which the coeffcients of f are expressed by the corresponding coeffcients of functions with positive real part. In what follows, we express the coeffcients of f in $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S by the corresponding coeffcients of Schwarz functions. In the proofs, we apply some inequalities for these functions obtained by Prokhorov and Szynal, by Carlson and by Efraimidis. This approach offers a additional benefit. In many cases, it is easily possible to predict the exact result and to select extremal functions. It is the case for $${\mathcal {S}}_S^*$$ S S ∗ and $${\mathcal {K}}_S$$ K S .

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Section: ð2:2þmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Initial logarithmic coefficients for functions starlike with respect to symmetric points

Zaprawa

2021

Bol. Soc. Mat. Mex.

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“…. obtained in [12]. For the subclasses of univalent functions the situation is not a great deal better.…”

Section: Introduction and Definitionsmentioning

confidence: 95%

Some application of Grunsky coefficients in the theory of univalent functions

Obradović¹,

Tuneski²

2020

Preprint

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Let function f be normalized, analytic and univalent in the unit disk D = {z : |z| < 1} and f (z) = z + ∞ n=2 anz n . Using a method based on Grusky coefficients we study several problems over that class of univalent functions: upper bound of the third logarithmic coefficient, upper bound of the coefficient difference |a 4 | − |a 3 |, the special case of the generalised Zalcman conjecture |a 2 a 3 − a 4 | and upper bounds of the second and the third Hankel determinant. Obtained results improve the previous ones.

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“…Note that Obradović and Tuneski [4] obtained an upper bound of |c 3 | for the class S. e problem of estimating the modulus of the rst three logarithmic coe cients is signi cantly studied for the subclasses of S, and in some cases, sharp bounds are obtained. For instance, sharp estimates for the class of starlike functions S * are given by the inequality…”

Section: Introductionmentioning

confidence: 99%

The Third Logarithmic Coefficient for Certain Close‐to‐Convex Functions

Alarifi

2022

Journal of Mathematics

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The logarithmic coefficients γ n of a normalized analytic functions f are defined by log f z / z = 2 ∑ n = 1 ∞ γ n z n . For certain close-to-convex functions f z = z + a 2 z 2 + ⋯ , Cho et al. (on the third logarithmic coefficient in some subclasses of close-to-convex functions) has obtained the upper bound of the third logarithmic coefficient γ 3 when the second coefficient a 2 is real. In the present paper, the upper bound of the third logarithmic coefficient γ 3 is computed with no restriction on the second coefficient a 2 .

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The third logarithmic coefficient for the class S

Abstract: In this paper we give an upper bound of the third logarithmic coefficient for the class S of univalent functions in the unit disc.

Cited by 5 publications

References 4 publications

Initial logarithmic coefficients for functions starlike with respect to symmetric points

Initial logarithmic coefficients for functions starlike with respect to symmetric points

Some application of Grunsky coefficients in the theory of univalent functions

The Third Logarithmic Coefficient for Certain Close‐to‐Convex Functions

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