Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 5 2 2024 03 01 Analysis of a certain Pseudo $q$-calculus and its applications in integral inequalities 205 232 202703 10.30504/jims.2024.455086.1177 EN H. Fallah Andevari Department of‎ ‎Mathematics, Babol Noshirvani University of Technology‎, ‎Shariati Ave., Babol‎, ‎Iran. A. Babakhani Department of‎ ‎Mathematics, Babol Noshirvani University of Technology‎, ‎Shariati Ave., Babol‎, ‎Iran. 0000-0002-5342-1322 D. S. Oliveira Institute of Science and Technology, Federal University of São Paulo, Shariati Ave., São José dos Campos–SP, Brazil. Journal Article 2024 04 30 We define three new operators which‎, ‎in exceptional cases‎, ‎are reduced to\linebreak $q$-integral/derivative $q_a$-integral/derivative and ${}^bq$-integral/derivative operators‎. ‎Fundamental properties emerge among these specific $q$-operators‎, ‎like fractional calculus‎. By utilizing these three newly introduced operators‎, ‎we can prove various inequalities‎, ‎the most notable of which are the Chebyshev and Hermite-Hadamard types‎. ‎Consequently‎, ‎these new operators provide a generalized approach to many problems in classical inequalities using classical fractional calculus‎. https://jims.ims.ir/article_202703_9bce2de6c7bdf1f46b753d74007f2229.pdf