A new version of Boole's formula type inequalities in multiplicative calculus with application to quadrature formula

This paper presents a rigorous proof of novel multiplicative integral identity and utilize it to establish new Boole's type inequalities for multiplicatively convex functions. These newly established inequalities can be helpful in finding the bounds for Boole's formula within the framework of multiplicative calculus. Moreover, Boole's type inequalities provide best optimal approximations for polynomials of degree six. Finding an error term using the first derivative is an excellent achievement in inequality theory because the class of first-time differentiable functions is more extensive than that of bounded functions with six derivatives. Numerical examples and graphical analysis are conducted to validate the effectiveness of the newly derived findings. Furthermore, the derived results are applied to the quadrature formula and special means of real numbers, demonstrating their practical utility within the context of multiplicative calculus. This research highlights their potential impact on computational mathematics and related fields. The establishment of Boole's type inequalities for multiplicatively convex functions extends our understanding of inequalities in multiplicative calculus, opening avenues for future research and applications.