Comments on the proof of the conjecture on logarithmic coefficients

The Bieberbach Conjecture: Proceedings of the Symposium on the Occasion of the Proof - Mathematical Surveys and Monographs ◽

10.1090/surv/021/10 ◽

1986 ◽

pp. 109-112

Author(s):

I. Milin

Keyword(s):

Logarithmic Coefficients

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Initial logarithmic coefficients for functions starlike with respect to symmetric points

Boletín de la Sociedad Matemática Mexicana ◽

10.1007/s40590-021-00370-y ◽

2021 ◽

Vol 27 (3) ◽

Author(s):

Paweł Zaprawa

Keyword(s):

Exact Result ◽

Positive Real Part ◽

Additional Benefit ◽

Extremal Functions ◽

Positive Real ◽

Logarithmic Coefficients ◽

Symmetric Points

AbstractIn 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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On Coefficient Problems for Functions Connected with the Sine Function

Symmetry ◽

10.3390/sym13071179 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1179

Author(s):

Katarzyna Tra̧bka-Wiȩcław

Keyword(s):

Analytic Functions ◽

Sine Function ◽

Hankel Determinant ◽

Coefficient Problems ◽

Logarithmic Coefficients ◽

Symmetric Points

In this paper, some coefficient problems for starlike analytic functions with respect to symmetric points are considered. Bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates for the following: coefficients, logarithmic coefficients, some cases of the generalized Zalcman coefficient functional, and some cases of the Hankel determinant.

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