Logarithmic Coefficients Problems in Families Related to Starlike and Convex Functions

Ponnusamy, Saminathan; Sharma, Navneet Lal; Wirths, Karl-Joachim

doi:10.1017/s1446788719000065

Cited by 31 publications

(19 citation statements)

References 16 publications

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“…The sharp upper bounds for modulus of logarithmic coefficients are known for functions in very few subclasses of S. For functions in the class S * , it is easy to prove that |γ n | ≤ 1/n for n ≥ 1 and equality holds for the Koebe function. For another subclasses of S, see also [8,16,17,30]. Following, we estimate the logarithmic coefficients of f ∈ S * t (α 1 , α 2 ).…”

Section: On Logarithmic Coefficients and Coefficientsmentioning

confidence: 99%

On a certain subclass of strongly starlike functions

Kargar,

Sokół,

Mahzoon

2018

Preprint

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Let S * t (α 1 , α 2 ) denote the class of functions f analytic in the open unit disc ∆, normalized by the condition f (0) = 0 = f ′ (0) − 1 and satisfying the following two-sided inequality:) is a subclass of strongly starlike functions of order β where β = max{α 1 , α 2 }. The object of the present paper is to derive some certain inequalities including (for example), upper and lower bounds for Re{zf ′ (z)/f (z)}, growth theorem, logarithmic coefficient estimates and coefficient estimates for functions f belonging to the class S * t (α 1 , α 2 ).2010 Mathematics Subject Classification. 30C45.

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Section: On Logarithmic Coefficients and Coefficientsmentioning

confidence: 99%

On a certain subclass of strongly starlike functions

Kargar,

Sokół,

Mahzoon

2018

Preprint

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“…play as extremal functions for some issues of the families ST hpl ðsÞ and CV hpl ðsÞ, respectively. Lately, several researchers have subsequently investigated same problems regarding the logarithmic coefficients and the coefficient problems [9,[13][14][15][16][17][18][19][20][21][22][23], to mention a few of them. For instance, the rotation of the Koebe function kðzÞ = z ð1 − e iθ zÞ −2 for each θ ∈ ℝ has the logarithmic coefficients…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

Alimohammadi

Adegani

Bulboacă

et al. 2021

Journal of Function Spaces

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It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f z = z + ∑ n = 2 ∞ a n z n analytic and univalent in the open unit disk U , then the logarithmic coefficients γ n f of the function f ∈ S are defined by log f z / z = 2 ∑ n = 1 ∞ γ n f z n . In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like C 1 + α z for α ∈ 0 , 1 and C V hpl 1 / 2 were estimated. Further, conjectures for the logarithmic coefficients γ n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f ∈ C 1 + α z , then the logarithmic coefficients of f satisfy the inequalities γ n ≤ α / 2 n n + 1 , n ∈ ℕ . Equality is attained for the function L α , n , that is, log L α , n z / z = 2 ∑ n = 1 ∞ γ n L α , n z n = α / n n + 1 z n + ⋯ , z ∈ U .

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“…In 2017, Ali and Allu [1] obtained the initial logarithmic coefficients bounds for close-to-convex functions. In 2020, Ponnusamy et al [17] computed the sharp estimates for the initial three logarithmic coefficients for a subclass of S * . The problem of computing the bound of the logarithmic coefficients is also considered in [6,18,21] for several subclasses of close to convex functions.…”

Section: Introductionmentioning

confidence: 99%