(k,s)-fractional integral operators in multiplicative calculus

Abstract

The research here endeavors to delve into the trapezoid-type inequalities pertaining to multiplicative $(k,s)$-fractional integrals. To this end, we introduce a class of operators, called the multiplicative $(k,s)$-fractional integrals, and subsequently give an analysis of these newly minted operators, examining their characteristics including boundedness, continuity, commutative properties, semigroup property, and several others. Moreover, leveraging the fractional integral identity, we derive three trapezoid-type inequalities of multiplicative type, where the function $U^{\ast }$ possesses multiplicative convexity or ${\left(\ln U^{\ast }\right)}^q$ maintains convexity for $q>1$.