Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination
In this paper, we introduce new subclasses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">R</mi><mrow><mi mathvariant="sans-ser...
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2021-09-01
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doaj-272917ac8b1444ab8406de770b6f32052021-09-26T01:31:15ZengMDPI AGSymmetry2073-89942021-09-01131653165310.3390/sym13091653Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-SubordinationWaggas Galib Atshan0Ibtihal Abdul Ridha Rahman1Alina Alb Lupaş2Department of Mathematics, University of Al-Qadisiyah, Diwaniyah 58001, IraqDepartment of Mathematics, University of Al-Qadisiyah, Diwaniyah 58001, IraqDepartment of Mathematics and Computer Science, University of Oradea, 1 Universitatii Street, 410087 Oradea, RomaniaIn this paper, we introduce new subclasses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">R</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">K</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula> of bi-univalent functions in the open unit disk <i>U</i> by using quasi-subordination conditions and determine estimates of the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>a</mi><mn>2</mn></msub></mfenced></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>a</mi><mn>3</mn></msub></mfenced></semantics></math></inline-formula> for functions of these subclasses. We discuss the improved results for the associated classes involving many of the new and well-known consequences. We notice that there is symmetry in the results obtained for the new subclasses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">R</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">K</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula>, as there is a symmetry for the estimations of the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>3</mn></msub></semantics></math></inline-formula> for all the subclasses defind in our this paper.https://www.mdpi.com/2073-8994/13/9/1653subordinationbi-univalent functionanalytic functionquasi-subordinationhurwitz–lerch zeta function |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Waggas Galib Atshan Ibtihal Abdul Ridha Rahman Alina Alb Lupaş |
spellingShingle |
Waggas Galib Atshan Ibtihal Abdul Ridha Rahman Alina Alb Lupaş Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination Symmetry subordination bi-univalent function analytic function quasi-subordination hurwitz–lerch zeta function |
author_facet |
Waggas Galib Atshan Ibtihal Abdul Ridha Rahman Alina Alb Lupaş |
author_sort |
Waggas Galib Atshan |
title |
Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination |
title_short |
Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination |
title_full |
Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination |
title_fullStr |
Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination |
title_full_unstemmed |
Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination |
title_sort |
some results of new subclasses for bi-univalent functions using quasi-subordination |
publisher |
MDPI AG |
series |
Symmetry |
issn |
2073-8994 |
publishDate |
2021-09-01 |
description |
In this paper, we introduce new subclasses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">R</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">K</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula> of bi-univalent functions in the open unit disk <i>U</i> by using quasi-subordination conditions and determine estimates of the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>a</mi><mn>2</mn></msub></mfenced></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="|" close="|"><msub><mi>a</mi><mn>3</mn></msub></mfenced></semantics></math></inline-formula> for functions of these subclasses. We discuss the improved results for the associated classes involving many of the new and well-known consequences. We notice that there is symmetry in the results obtained for the new subclasses <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">R</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>τ</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="fraktur">K</mi><mrow><mi mathvariant="sans-serif">Σ</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>α</mi></mrow></msubsup><mfenced separators="" open="(" close=")"><mi>λ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi mathvariant="sans-serif">Φ</mi></mfenced></mrow></semantics></math></inline-formula>, as there is a symmetry for the estimations of the coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>3</mn></msub></semantics></math></inline-formula> for all the subclasses defind in our this paper. |
topic |
subordination bi-univalent function analytic function quasi-subordination hurwitz–lerch zeta function |
url |
https://www.mdpi.com/2073-8994/13/9/1653 |
work_keys_str_mv |
AT waggasgalibatshan someresultsofnewsubclassesforbiunivalentfunctionsusingquasisubordination AT ibtihalabdulridharahman someresultsofnewsubclassesforbiunivalentfunctionsusingquasisubordination AT alinaalblupas someresultsofnewsubclassesforbiunivalentfunctionsusingquasisubordination |
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