Caputo-Hadamard fractional Wirtinger-type inequalities via Taylor expansion with applications to classical means

In this paper, we explored Caputo-Hadamard fractional Wirtinger-type inequalities using Taylor's formula. The main findings were derived by utilizing Hölder's inequality to derive results for Caputo-Hadamard fractional derivatives in terms of $L_q$ norms for $q>1$. Through graphical interpretation, we confirmed the validity of the results. A flowchart summarizing the logical progression from lemma to theorem was added for clarity. Furthermore, inequalities were also derived for Hadamard fractional derivatives. Finally, we discussed the applications of Wirtinger-type inequalitie which incorporates arithmetic mean and geometric mean-type inequalities.