Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination

• Main Authors: Waggas Galib Atshan, Ibtihal Abdul Ridha Rahman, Alina Alb Lupaş
• Format: Article
• Language: English
• Published: MDPI AG 2021-09-01
• Series: Symmetry
• Subjects: subordination; bi-univalent function; analytic function; quasi-subordination; hurwitz–lerch zeta function
• Online Access: https://www.mdpi.com/2073-8994/13/9/1653

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.