Coefficient characterizations and sections for some univalent functions

Obradović, Milutin; Ponnusamy, Saminathan; Wirths, Karl-Joachim

doi:10.1134/s0037446613040095

Cited by 41 publications

(29 citation statements)

References 17 publications

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“…Finally, we prove the inequality (9). From the formula (12) and the result of Rogosinski (see also [12,Theorem 2.2] and [4, Theorem 6.2]), it follows that for k ∈ N the inequalities k n=1…”

Section: Logarithmic Coefficients Of Functions In U(λ)mentioning

confidence: 74%

Logarithmic coefficients and a coefficient conjecture for univalent functions

2017

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Abstract. Let U(λ) denote the family of analytic functions f (z), f (0) = 0 = f (0) − 1, in the unit disk D, which satisfy the condition z/f (z) 2 f (z) − 1 < λ for some 0 < λ ≤ 1. The logarithmic coefficients γ n of f are defined by the formula log(f (z)/z) = 2 ∞ n=1 γ n z n . In a recent paper, the present authors proposed a conjecture that if f ∈ U(λ) for some 0 < λ ≤ 1, then |a n | ≤ n−1 k=0 λ k for n ≥ 2 and provided a new proof for the case n = 2. One of the aims of this article is to present a proof of this conjecture for n = 3, 4 and an elegant proof of the inequality for n = 2, with equality for f (z) = z/[(1 + z)(1 + λz)]. In addition, the authors prove the following sharp inequality for f ∈ U(λ):where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.

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Section: Logarithmic Coefficients Of Functions In U(λ)mentioning

confidence: 74%

Logarithmic coefficients and a coefficient conjecture for univalent functions

2017

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“…These classes have been studied recently, for example in [7,8]. Functions in C(−1/2) are known to be close-to-convex in D. In [9], Ozaki introduced the class G and proved that functions in G are univalent in D. Later in [10], Umezawa discussed a general version of this class.…”

Section: Conjecturementioning

confidence: 98%

“…In [19,Proposition 2], the authors have proved that if f ∈ S * , then one has for |z| = r < 1 the inequality z f (z) − 1 ≤ 2r + r 2 (7) and the same is not true for the univalent class S. Indeed, for f ∈ S, it is known that (see [19,Proposition 3]) z f (z) − 1 ≤ 2r + 3r 2 for |z| = r < 1.…”

Section: Remarkmentioning

confidence: 99%

Disk of univalence of the ratio of two analytic functions

Obradović

Ponnusamy

2014

Complex Variables and Elliptic Equations

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“…The class G(a) has been studied extensively by Kargar et al [11], Maharana et al [16], Obradović et al [17], and Ponnusamy and Sahoo [20]. It is well-known that the logarithmic coefficients have had great impact in the development of the theory of univalent functions.…”

Section: Introductionmentioning

confidence: 99%

On logarithmic coefficients of certain starlike functions related to the vertical strip

Kargar

2018

J Anal

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In the present paper two certain subclasses of the starlike functions associated with the vertical strip are considered. The main aim of this paper is to investigate some basic properties of these classes such as, subordination relations, sharp inequalities for sums involving logarithmic coefficients and estimate of logarithmic coefficients for functions belonging to these subclasses.2010 Mathematics Subject Classification. 30C50; 30C45.

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Coefficient characterizations and sections for some univalent functions

Cited by 41 publications

References 17 publications

Logarithmic coefficients and a coefficient conjecture for univalent functions

Logarithmic coefficients and a coefficient conjecture for univalent functions

Disk of univalence of the ratio of two analytic functions

On logarithmic coefficients of certain starlike functions related to the vertical strip

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