Logarithmic coefficients of univalent functions

1979 ◽  
Vol 36 (1) ◽  
pp. 36-43 ◽  
Author(s):  
P. L. Duren ◽  
Y. J. Leung
Keyword(s):  
Univalent Functions ◽  
Logarithmic Coefficients

scholarly journals Logarithmic Coefficients of the Inverse of Univalent Functions

2018 ◽  
Vol 73 (4) ◽  
Author(s):  
Saminathan Ponnusamy ◽  
Navneet Lal Sharma ◽  
Karl-Joachim Wirths
Keyword(s):  
Univalent Functions ◽  
Logarithmic Coefficients

scholarly journals Logarithmic coefficients and a coefficient conjecture for univalent functions

2017 ◽  
Vol 185 (3) ◽  
pp. 489-501 ◽  
Author(s):  
Milutin Obradović ◽  
Saminathan Ponnusamy ◽  
Karl-Joachim Wirths
Keyword(s):  
Univalent Functions ◽  
Logarithmic Coefficients

scholarly journals Erratum to: Logarithmic coefficients and a coefficient conjecture for univalent functions

2017 ◽  
Vol 185 (3) ◽  
pp. 503-506
Author(s):  
Milutin Obradović ◽  
Saminathan Ponnusamy ◽  
Karl-Joachim Wirths
Keyword(s):  
Univalent Functions ◽  
Logarithmic Coefficients

scholarly journals LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 230-249 ◽  
Author(s):  
SAMINATHAN PONNUSAMY ◽  
NAVNEET LAL SHARMA ◽  
KARL-JOACHIM WIRTHS
Keyword(s):  
Upper Bound ◽  
Convex Functions ◽  
Univalent Functions ◽  
List Type ◽  
Starlike And Convex Functions ◽  
The Family ◽  
Logarithmic Coefficients ◽  
Dilogarithm Function ◽  
Image Position ◽  
Analytic And Univalent Functions

Let${\mathcal{S}}$be the family of analytic and univalent functions$f$in the unit disk$\mathbb{D}$with the normalization$f(0)=f^{\prime }(0)-1=0$, and let$\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$denote the logarithmic coefficients of$f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families${\mathcal{F}}(c)$and${\mathcal{G}}(c)$of functions$f\in {\mathcal{S}}$defined by$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)<1+{\displaystyle \frac{c}{2}},\quad z\in \mathbb{D},\end{eqnarray}$$for some$c\in (0,3]$and$c\in (0,1]$, respectively. We obtain the sharp upper bound for$|\unicode[STIX]{x1D6FE}_{n}|$when$n=1,2,3$and$f$belongs to the classes${\mathcal{F}}(c)$and${\mathcal{G}}(c)$, respectively. The paper concludes with the following two conjectures:∙If$f\in {\mathcal{F}}(-1/2)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$for$n\geq 1$, and$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$where$\text{Li}_{2}(x)$denotes the dilogarithm function.∙If$f\in {\mathcal{G}}(c)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$for$n\geq 1$.

SECOND HANKEL DETERMINANT OF LOGARITHMIC COEFFICIENTS OF CONVEX AND STARLIKE FUNCTIONS

2021 ◽  
pp. 1-10
Author(s):  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO
Keyword(s):  
Univalent Functions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Hankel Matrices ◽  
Sharp Bounds ◽  
Second Hankel Determinant ◽  
Logarithmic Coefficients ◽  
Convex And Starlike Functions

Abstract We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.

scholarly journals Logarithmic Coefficients for Univalent Functions Defined by Subordination

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 408 ◽  
Author(s):  
Ebrahim Analouei Adegani ◽  
Nak Eun Cho ◽  
Mostafa Jafari
Keyword(s):  
Univalent Functions ◽  
Logarithmic Coefficients ◽  
The Given

In this work, the bounds for the logarithmic coefficients γ n of the general classes S * ( φ ) and K ( φ ) were estimated. It is worthwhile mentioning that the given bounds would generalize some of the previous papers. Some consequences of the main results are also presented, noting that our method is more general than those used by others.

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