On The ( <i>k</i> , <i>s</i> )‐Hilfer Fractional Derivatives via Wirtinger‐Type Inequalities and Their Analytical Applications

ABSTRACT This article introduces a novel definition of ‐Hilfer fractional derivative. Based on this derivative, some fractional Wirtinger type inequalities are established for the spaces, where by using Hölder's inequality. Various related special cases are also presented. To validate our main results, examples with graphical representations are provided. Applications of ‐Hilfer fractional Wirtinger‐type inequalities are demonstrated in terms of arithmetic mean and geometric mean‐type inequality.