Two applications of Grunsky coefficients in the theory of univalent functions
Journal
Acta Universitatis Sapientiae, Mathematica
Date Issued
2023-12-01
Author(s)
Obradović, Milutin
DOI
10.2478/ausm-2023-0017
Abstract
<jats:title>Abstract</jats:title>
<jats:p>Let <jats:italic>S</jats:italic> denote the class of functions f which are analytic and univalent in the unit disk 𝔻 = {z : |z| < 1} and normalized with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0017_eq_001.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
<m:mrow>
<m:mtext>f</m:mtext>
<m:mrow>
<m:mo>(</m:mo>
<m:mtext>z</m:mtext>
<m:mo>)</m:mo>
</m:mrow>
<m:mo>=</m:mo>
<m:mtext>z</m:mtext>
<m:mo>+</m:mo>
<m:msubsup>
<m:mo>∑</m:mo>
<m:mrow>
<m:mtext>n</m:mtext>
<m:mo>=</m:mo>
<m:mn>2</m:mn>
</m:mrow>
<m:mo>∞</m:mo>
</m:msubsup>
<m:mrow>
<m:msub>
<m:mrow>
<m:mi>α</m:mi>
</m:mrow>
<m:mtext>n</m:mtext>
</m:msub>
<m:msup>
<m:mrow>
<m:mtext>z</m:mtext>
</m:mrow>
<m:mtext>n</m:mtext>
</m:msup>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>{\rm{f}}\left( {\rm{z}} \right) = {\rm{z}} + \sum\nolimits_{{\rm{n = 2}}}^\infty {{\alpha _{\rm{n}}}{{\rm{z}}^{\rm{n}}}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. Using a method based on Grusky coefficients we study two problems over the class <jats:italic>S</jats:italic>: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference |α<jats:sub>5</jats:sub>| − |α<jats:sub>4</jats:sub>|.</jats:p>
<jats:p>Let <jats:italic>S</jats:italic> denote the class of functions f which are analytic and univalent in the unit disk 𝔻 = {z : |z| < 1} and normalized with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0017_eq_001.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline">
<m:mrow>
<m:mtext>f</m:mtext>
<m:mrow>
<m:mo>(</m:mo>
<m:mtext>z</m:mtext>
<m:mo>)</m:mo>
</m:mrow>
<m:mo>=</m:mo>
<m:mtext>z</m:mtext>
<m:mo>+</m:mo>
<m:msubsup>
<m:mo>∑</m:mo>
<m:mrow>
<m:mtext>n</m:mtext>
<m:mo>=</m:mo>
<m:mn>2</m:mn>
</m:mrow>
<m:mo>∞</m:mo>
</m:msubsup>
<m:mrow>
<m:msub>
<m:mrow>
<m:mi>α</m:mi>
</m:mrow>
<m:mtext>n</m:mtext>
</m:msub>
<m:msup>
<m:mrow>
<m:mtext>z</m:mtext>
</m:mrow>
<m:mtext>n</m:mtext>
</m:msup>
</m:mrow>
</m:mrow>
</m:math>
<jats:tex-math>{\rm{f}}\left( {\rm{z}} \right) = {\rm{z}} + \sum\nolimits_{{\rm{n = 2}}}^\infty {{\alpha _{\rm{n}}}{{\rm{z}}^{\rm{n}}}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. Using a method based on Grusky coefficients we study two problems over the class <jats:italic>S</jats:italic>: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference |α<jats:sub>5</jats:sub>| − |α<jats:sub>4</jats:sub>|.</jats:p>