Linear Differential Equations on Some Classes of Weighted Function Spaces

• Main Authors: Ahmed El-Sayed Ahmed, Amnah E. Shammaky
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Symmetry
• Subjects: complex linear differential equation; analytic functions; growth order
• Online Access: https://www.mdpi.com/2073-8994/13/7/1113

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.