| Summary: | In this article, we establish some new generalized inequalities of the Hilbert-type on time scales’ delta calculus, which can be considered similar to formulas for inequalities of Hilbert type. The major innovation point is to establish some dynamic inequalities of the Hilbert-type on time scales’ delta calculus for delta differentiable functions of one variable and two variables. In this paper, we use the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>s</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="4pt"></mspace><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>s</mi><mi>j</mi></msub><mo>,</mo><msub><mi>z</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>w</mi><mi>j</mi></msub><mo>,</mo><msub><mi>n</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula>. These inequalities will be proved by applying Hölder’s inequality, the chain rule on time scales, and the mean inequality. As special cases of our results (when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>), we obtain the discrete and continuous inequalities. Also, we can obtain other inequalities in different time scales, like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><msup><mi>q</mi><mover><mi mathvariant="double-struck">Z</mi><mo>−</mo></mover></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>.
|
|---|