On Some Inequalities with Higher Fractional Orders

The novelty herein pertains to a class of fractional differential equations involving the Hadamard fractional derivative of higher order. Our investigation encompasses the fractional integral operator of a logarithmic function. The mathematical tools utilized in this study are derived from an important function, wherein its behavior in terms of maximum value facilitates the establishment of bounds necessary for proving the existence of solutions, specifically through Green’s function. Based on this, we endeavor to estimate the bounds of Green’s function as well as analyze its properties within the considered interval. This approach enables us to establish the Hartman–Wintner- and Lyapunov-type inequalities for a class of fractional Hadamard problems. Furthermore, we introduce a novel technique to determine the maximum value of Green’s function. Finally, we illustrate these findings through two applications.