On multiplicative conformable fractional integrals: theory and applications

In this paper, we first introduce the multiplicative conformable left and right fractional integrals, followed by the derivation of key properties, such as integrability, boundedness, continuity, and the semi-group property, for the newly defined multiplicative conformable fractional integrals. Then, we establish the Hermite–Hadamard inequalities in three distinct senses for multiplicative conformable fractional integrals. Moreover, we present several corresponding midpoint and trapezoidal inequalities for the obtained Hermite-Hadamard inequalities including multiplicative conformable fractional integrals. By special cases, we present the relations between newly obtained inequalities for multiplicative conformable fractional integrals and existing results for multiplicative Riemann–Liouville fractional integrals and multiplicative integrals. Furthermore, we give some new Hermite–Hadamard type, trapezoid type and midpoint type inequalities or multiplicative Riemann–Liouville fractional integrals. Finally, we give several examples and 3D graphs to illustrate the main results.