Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc
Motivated by q-calculus, we define a new family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Σ</mo></semantics></math></inline-formula>, which is the family of bi-univalent anal...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-04-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/14/4/758 |
| _version_ | 1797443472261644288 |
|---|---|
| author | Alaa H. El-Qadeem Mohamed A. Mamon Ibrahim S. Elshazly |
| author_facet | Alaa H. El-Qadeem Mohamed A. Mamon Ibrahim S. Elshazly |
| author_sort | Alaa H. El-Qadeem |
| collection | DOAJ |
| description | Motivated by q-calculus, we define a new family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Σ</mo></semantics></math></inline-formula>, which is the family of bi-univalent analytic functions in the open unit disc <i>U</i> that is related to the Einstein function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We establish estimates for the first two Taylor–Maclaurin coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, and the Fekete–Szegö inequality <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><mo>−</mo><mi>μ</mi><msubsup><mi>a</mi><mn>2</mn><mn>2</mn></msubsup></mfenced></semantics></math></inline-formula> for the functions that belong to these families. |
| first_indexed | 2024-03-09T12:56:32Z |
| format | Article |
| id | doaj.art-1294ad77646f42ea86191b2a49bb9f4a |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T12:56:32Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1294ad77646f42ea86191b2a49bb9f4a2023-11-30T21:59:27ZengMDPI AGSymmetry2073-89942022-04-0114475810.3390/sym14040758Application of Einstein Function on Bi-Univalent Functions Defined on the Unit DiscAlaa H. El-Qadeem0Mohamed A. Mamon1Ibrahim S. Elshazly2Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, EgyptDepartment of Mathematics, Faculty of Science, Tanta University, Tanta 31527, EgyptDepartment of Basic Sciences, Common First Year, King Saud University, Alriyad 11451, Saudi ArabiaMotivated by q-calculus, we define a new family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Σ</mo></semantics></math></inline-formula>, which is the family of bi-univalent analytic functions in the open unit disc <i>U</i> that is related to the Einstein function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We establish estimates for the first two Taylor–Maclaurin coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><msub><mi>a</mi><mn>3</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, and the Fekete–Szegö inequality <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><mo>−</mo><mi>μ</mi><msubsup><mi>a</mi><mn>2</mn><mn>2</mn></msubsup></mfenced></semantics></math></inline-formula> for the functions that belong to these families.https://www.mdpi.com/2073-8994/14/4/758analytic functionEinstein functionbernoulli numbersbi-univalent functionquantum calculussubordination |
| spellingShingle | Alaa H. El-Qadeem Mohamed A. Mamon Ibrahim S. Elshazly Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc Symmetry analytic function Einstein function bernoulli numbers bi-univalent function quantum calculus subordination |
| title | Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc |
| title_full | Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc |
| title_fullStr | Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc |
| title_full_unstemmed | Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc |
| title_short | Application of Einstein Function on Bi-Univalent Functions Defined on the Unit Disc |
| title_sort | application of einstein function on bi univalent functions defined on the unit disc |
| topic | analytic function Einstein function bernoulli numbers bi-univalent function quantum calculus subordination |
| url | https://www.mdpi.com/2073-8994/14/4/758 |
| work_keys_str_mv | AT alaahelqadeem applicationofeinsteinfunctiononbiunivalentfunctionsdefinedontheunitdisc AT mohamedamamon applicationofeinsteinfunctiononbiunivalentfunctionsdefinedontheunitdisc AT ibrahimselshazly applicationofeinsteinfunctiononbiunivalentfunctionsdefinedontheunitdisc |