<abstract xmlns="http://www.w3.org/1999/xhtml">

This paper deals with the class <italic>S</italic> containing functions which are analytic and univalent in the open unit disc <italic>U</italic> = {<italic>z</italic> ∈ ℂ : |<italic>z</italic>| &lt; 1}. Functions <italic>f</italic> in <italic>S</italic> are normalized by <italic>f</italic>(0) = 0 and <italic>f</italic>′(0) = 1 and has the Taylor series expansion of the form <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2021-0019_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mo>∞</mo></munderover><mrow><msub><mrow><mi>a</mi></mrow><mi>n</mi></msub><msup><mrow><mi>z</mi></mrow><mi>n</mi></msup></mrow></mrow></math>
<tex-math>f\left( z \right) = z + \sum\limits_{n = 2}^\infty {{a_n}{z^n}}</tex-math>
</alternatives>
</inline-formula>. In this paper we investigate on the subclass of <italic>S</italic> of close-to-convex functions denoted as <italic>C</italic><italic>gα</italic>(<italic>λ, δ</italic>) where function <italic>f</italic> ∈ <italic>C</italic><italic>gα</italic>(<italic>λ, δ</italic>) satisfies <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2021-0019_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>Re</mo><mrow><mo>{</mo> <mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>λ</mi></mrow></msup><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mrow><mi>g</mi><mi>α</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></mfrac></mrow> <mo>}</mo></mrow></mrow></math>
<tex-math>{\mathop{\rm Re}\nolimits} \left\{ {{e^{i\lambda }}{{zf'\left( z \right)} \over {g\alpha \left( z \right)}}} \right\}</tex-math>
</alternatives>
</inline-formula> for <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2021-0019_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mo>|</mo> <mi>λ</mi> <mo>|</mo></mrow><mo>&lt;</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow></math>
<tex-math>\left| \lambda \right| &lt; {\pi \over 2}</tex-math>
</alternatives>
</inline-formula>, cos(<italic>λ</italic>) <italic>> δ</italic>, 0 ≤ <italic>δ &lt;</italic> 1, 0 ≤ <italic>α</italic> ≤ 1 and <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gm-2021-0019_eq_004.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>g</mi></mrow><mi>α</mi></msub><mo>=</mo><mfrac><mi>z</mi><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>α</mi><mi>z</mi></mrow><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math>
<tex-math>{g_\alpha } = {z \over {{{\left( {1 - \alpha z} \right)}^2}}}</tex-math>
</alternatives>
</inline-formula>. The aim of the present paper is to find the upper bound of the Fekete-Szego functional |<italic>a</italic>3 − <italic>µa</italic>22| for the class <italic>C</italic>g<italic>α</italic>(<italic>λ, δ</italic>). The results obtained in this paper is significant in the sense that it can be used in future research in this field, particularly in solving coefficient inequalities such as the Hankel determinant problems and also the Fekete-Szego problems for other subclasses of univalent functions.
</abstract>

This paper deals with the class S containing functions which are analytic and univalent in the open unit disc U = {z ∈ ℂ : |z| < 1}. Functions f in S are normalized by f(0) = 0 and f′(0) = 1 and has the Taylor series expansion of the form 


f(z)=z+∑n=2∞anzn
f\left( z \right) = z + \sum\limits_{n = 2}^\infty {{a_n}{z^n}}

. In this paper we investigate on the subclass of S of close-to-convex functions denoted as Cgα(λ, δ) where function f ∈ Cgα(λ, δ) satisfies 


Re{ eiλzf′(z)gα(z) }
{\mathop{\rm Re}\nolimits} \left\{ {{e^{i\lambda }}{{zf'\left( z \right)} \over {g\alpha \left( z \right)}}} \right\}

 for 


| λ |<π2
\left| \lambda \right| < {\pi \over 2}

, cos(λ) > δ, 0 ≤ δ < 1, 0 ≤ α ≤ 1 and 


gα=z(1−αz)2
{g_\alpha } = {z \over {{{\left( {1 - \alpha z} \right)}^2}}}

. The aim of the present paper is to find the upper bound of the Fekete-Szego functional |a3 − µa22| for the class Cgα(λ, δ). The results obtained in this paper is significant in the sense that it can be used in future research in this field, particularly in solving coefficient inequalities such as the Hankel determinant problems and also the Fekete-Szego problems for other subclasses of univalent functions.

The Fekete-Szego Theorem for Close-to-convex Functions Associated with The Koebe Type Function

Faculty of Computer and Mathematical Sciences, Department of Mathematics

General Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

This paper deals with the class S containing functions which are analytic and univalent in the open unit disc U = {z ∈ ℂ : |z| δ, 0 ≤ δ < 1, 0 ≤ α ≤ 1 and 


gα=z(1−αz)2
{g_\alpha } = {z \over {{{\left( {1 - \alpha z} \right)}^2}}}

. The aim of the present paper is to find the upper bound of the Fekete-Szego functional |a3 − µa22| for the class Cgα(λ, δ). The results obtained in this paper is significant in the sense that it can be used in future research in this field, particularly in solving coefficient inequalities such as the Hankel determinant problems and also the Fekete-Szego problems for other subclasses of univalent functions.

This paper deals with the class S containing functions which are analytic and univalent in the open unit disc U = {z ∈ ℂ : |z| < 1}. Functions f in S are...

The Fekete-Szego Theorem for Close-to-convex Functions Associated with The Koebe Type Function

Data publikacji: 30 mar 2022

Zakres stron: 127 - 136

Otrzymano: 30 paź 2021

Przyjęty: 25 lis 2021

DOI: https://doi.org/10.2478/gm-2021-0019

Słowa kluczowe
univalent functions, Fekete-szego problems, close-to-convex function, Koebe function

© 2021 Sidik Rathi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The Fekete-Szego Theorem for Close-to-convex Functions Associated with The Koebe Type Function

Data publikacji: 30 mar 2022

Zakres stron: 127 - 136

Otrzymano: 30 paź 2021

Przyjęty: 25 lis 2021

DOI: https://doi.org/10.2478/gm-2021-0019

Słowa kluczoweunivalent functions, Fekete-szego problems, close-to-convex function, Koebe function

© 2021 Sidik Rathi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Słowa kluczowe
univalent functions, Fekete-szego problems, close-to-convex function, Koebe function