Multiplicative fractional Milne-Mercer Type Inequalities via Multiplicative Proportional Caputo-Hybrid-Conformable Integral Operators

This paper establishes new Milne-Mercer type inequalities in the framework of multiplicative fractional calculus. A multiplicative proportional Caputo-Hybrid-Conformable fractional integral operator is introduced, and a corresponding integral identity is derived for multiplicatively differentiable functions. By using multiplicative convexity and logarithmic transformation techniques, several Milne-Mercer type inequalities are obtained. The proposed results extend classical Milne-type inequalities and include some existing multiplicative fractional inequalities as special cases. Numerical examples and three-dimensional visualizations are also provided to verify the validity of the main inequality and to illustrate the influence of the fractional parameters.