Some New Bullen-Type Inequalities Obtained via Fractional Integral Operators

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Holder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well.

Automated summary

In this paper, a new auxiliary identity of the Bullen type is established for twice-differentiable functions using fractional integral operators. Based on this new identity, generalized Bullen-type inequalities are derived by exploiting convexity properties. Concrete examples are provided to illustrate the results, and their correctness is confirmed through graphical analysis. Estimations of bounds are analyzed, showing that improved Holder and power mean inequalities yield better upper-bound results compared to classical inequalities. Furthermore, applications to quadrature rules, modified Bessel functions, and digamma functions are discussed.