New Bounds of Hadamard’s and Simpson’s Inequalities Involving Green Functions

This manuscript aims to assess some new refinements of right Hadamard’s and Simpson’s-like inequalities by bridging the concepts of Green function theory and convexity framework. It is a known fact that Green functions are convex and symmetric. By considering the identities based on Green functions for second-order differentiable functions and elementary results of inequalities, convexity and bounded variation of functions, we present various new upper estimates of trapezoidal and Simpson’s inequalities. Also, the accuracy of the results is determined by illustrative numerical examples and simulations. Lastly, we furnish some novel applications to linear combinations of means and composite error estimates.