Linear Differential Equations on Some Classes of Weighted Function Spaces
Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type...
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MDPI AG
2021-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/7/1113 |
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| author | Ahmed El-Sayed Ahmed Amnah E. Shammaky |
| author_facet | Ahmed El-Sayed Ahmed Amnah E. Shammaky |
| author_sort | Ahmed El-Sayed Ahmed |
| collection | DOAJ |
| description | Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>h</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>+</mo><msub><mi>K</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><msup><mi>h</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><msup><mi>h</mi><mo>′</mo></msup><mo>+</mo><msub><mi>K</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, will be discussed in this current research, whereas the considered coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>K</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are holomorphic in the disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Γ</mo><mi>R</mi></msub><mo>=</mo><mrow><mo>{</mo><mi>υ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>|</mo><mi>υ</mi><mo>|</mo><mo><</mo><mi>R</mi><mo>}</mo></mrow><mo>,</mo><mspace width="0.277778em"></mspace><mn>0</mn><mo><</mo><mi>R</mi><mo>≤</mo><mo>∞</mo><mo>.</mo></mrow></semantics></math></inline-formula> In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature. |
| first_indexed | 2024-03-10T10:09:21Z |
| format | Article |
| id | doaj.art-069328c019554b7aa7713ee3a78ab5f7 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T10:09:21Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-069328c019554b7aa7713ee3a78ab5f72023-11-22T01:15:22ZengMDPI AGSymmetry2073-89942021-06-01137111310.3390/sym13071113Linear Differential Equations on Some Classes of Weighted Function SpacesAhmed El-Sayed Ahmed0Amnah E. Shammaky1Mathematics Department, Faculty of Science, Taif University, Taif 21944, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi ArabiaSome weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>h</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>+</mo><msub><mi>K</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><msup><mi>h</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><msup><mi>h</mi><mo>′</mo></msup><mo>+</mo><msub><mi>K</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, will be discussed in this current research, whereas the considered coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>K</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>υ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are holomorphic in the disc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Γ</mo><mi>R</mi></msub><mo>=</mo><mrow><mo>{</mo><mi>υ</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>|</mo><mi>υ</mi><mo>|</mo><mo><</mo><mi>R</mi><mo>}</mo></mrow><mo>,</mo><mspace width="0.277778em"></mspace><mn>0</mn><mo><</mo><mi>R</mi><mo>≤</mo><mo>∞</mo><mo>.</mo></mrow></semantics></math></inline-formula> In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.https://www.mdpi.com/2073-8994/13/7/1113complex linear differential equationanalytic functionsgrowth order |
| spellingShingle | Ahmed El-Sayed Ahmed Amnah E. Shammaky Linear Differential Equations on Some Classes of Weighted Function Spaces Symmetry complex linear differential equation analytic functions growth order |
| title | Linear Differential Equations on Some Classes of Weighted Function Spaces |
| title_full | Linear Differential Equations on Some Classes of Weighted Function Spaces |
| title_fullStr | Linear Differential Equations on Some Classes of Weighted Function Spaces |
| title_full_unstemmed | Linear Differential Equations on Some Classes of Weighted Function Spaces |
| title_short | Linear Differential Equations on Some Classes of Weighted Function Spaces |
| title_sort | linear differential equations on some classes of weighted function spaces |
| topic | complex linear differential equation analytic functions growth order |
| url | https://www.mdpi.com/2073-8994/13/7/1113 |
| work_keys_str_mv | AT ahmedelsayedahmed lineardifferentialequationsonsomeclassesofweightedfunctionspaces AT amnaheshammaky lineardifferentialequationsonsomeclassesofweightedfunctionspaces |