Hilfer fractional derivatives and their role in Wirtinger-type inequalities with applications

In this paper, we explore novel Wirtinger-type inequalities for the Hilfer fractional derivatives. By formulating suitable integral identities and employing Hölder’s inequality, we propose a class of generalized Hilfer fractional Wirtinger-type inequalities for the Lₛ space with $s>1$ . Several special cases for the Riemann-Liouville and Caputo fractional integral inequalities are also presented. The validity of the findings is illustrated through examples and graphical representations. Moreover, applications are given in the context of inequalities involving the arithmetic mean and geometric mean.