Generalized Hermite–Hadamard–Mercer-type inequalities and related estimates for multiplicative convex functions with applications

Abstract

UDC 517.51, 517.16

The theory of multiplicative calculus has recently gained loads of attention. In this framework, the operations of addition and subtraction are systematically replaced with multiplication and division, respectively. The interest in this system is to obtain a multiplicative analog of the results obtained in the classical sense. Our aim is to contribute to the body of knowledge in this direction. Specifically, we establish some novel generalizations of the Hermite–Hadamard–Mercer-type inequalities involving new multiplicative tempered fractional integrals. A new lemma is also established. By using this lemma, Hölder's inequality, and the power-mean inequality, we obtain more inequalities of the Trapezoid–Mercer type. Our results generalize many other results available from the literature. We present some numerical examples, utilizing the Wolfram Mathematica software for computations and graphing, to prove the validity of our results by comparing the outcomes for different values of the operating parameter $\lambda\in[0,1]$ with 0.1 as the step size. Finally, we obtain additional estimates by applying our results to special means, such as arithmetic, harmonic, logarithmic, and $p$-logarithmic means.