On Generalized Wirtinger Inequalities for (k,ψ)-Caputo Fractional Derivatives and Applications

The primary aim of this study is to establish new Wirtinger-type inequalities involving fractional derivatives, which are essential tools in analysis and applied mathematics. We derive generalized Wirtinger-type inequalities incorporating the (k,ψ)-Caputo fractional derivatives using Taylor’s expansion. The inequalities are derived in Lp spaces (p>1) through Hölder’s inequality. A detailed analytical discussion is provided to further examine the derived inequalities. The theoretical findings are validated through numerical examples and graphical representations. Furthermore, the novelty and applicability of the proposed technique are demonstrated through the applications of the resulting inequalities to derive new results related to the arithmetic–geometric mean inequality.