Refinements and Extensions of Classical Integral Inequalities: Hölder, Hardy, Minkowski, Clarkson, and Schweitzer

Integral inequalities are a fundamental part of modern mathematical analysis and the theory of function spaces. In this paper, we present several refinements and extensions to classical integral inequalities, with a particular focus on those of Hölder, Hardy, Minkowski, Clarkson, and Schweitzer. First, we apply Hölder's inequality to find new refined bounds. Then, we establish Hölder-type inequalities using extended Young's inequalities. Consequently, we derive Hardy-type derivative inequalities with an optimal weight factor. After that, we introduce the Minkowski-Clarkson relation and variation for two functions. Lastly, we formulate a weighted generalisation of Schweitzer's inequality incorporating parametric functions. Concrete examples involving the beta and gamma functions demonstrate the sharpness and applicability of the proposed bounds, showing measurable improvements upon their classical counterparts.