Integral inequalities of Hermite-Hadamard type for multiplicatively harmonic 𝒔-convex functions

Our study consists of five sections. The introduction section provides information on inequality theory and the historical process and literature studies of convex functions. The general information section includes some definitions, theorems, inequalities, and means necessary for the following sections, and also includes the expressions and proofs of the Hermite-Hadamard inequality, which forms the basis of the study, for certain types of convex functions. The section that inspired our study is the materials and methods section. In this section, the concepts of multiplicative calculus and multiplicative integral are introduced, and Hermite-Hadamard type integral inequalities are given for multiplicative convex functions. In the fourth section, first, the Hermite-Hadamard inequality is given for multiplicatively harmonic 𝑠-convex functions, and Hermite-Hadamard type inequalities are obtained for the product and quotient of two multiplicatively harmonic 𝑠-convex functions. Then, new integral-based inequalities are obtained for the product and quotient of harmonic convex and multiplicatively harmonic 𝑠-convex functions. In addition, certain upper limits are obtained for the class of multiplicatively harmonic 𝑠-convex functions. The study is supported by examples, and the obtained values and figures of these examples are given. Finally, in the fifth section, the evaluation of the study is made, and the results are presented.