| Summary: | The primary goal of this research is to prove some new Hardy-type ∇-conformable dynamic inequalities by employing product rule, integration by parts, chain rule and <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mi>γ</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula>-nabla Hölder inequality on time scales. The inequalities proved here extend and generalize existing results in the literature. Further, in the case when <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we obtain some well-known time scale inequalities due to Hardy inequalities. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional <i>h</i>-sum inequalities, new conformable fractional <i>q</i>-sum inequalities and new classical conformable fractional integral inequalities.
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