Quantum Maclaurin-type inequalities for $q$-differentiable convex functions and their applications

This study presents novel quantum analogs of Maclaurin-type inequalities for functions possessing convex $q$-derivatives. We present and validate the principal results by obtaining a new quantum integral identity. Furthermore, we utilize these inequalities to establish error bounds for the Maclaurin quadrature formula and for specific means, including arithmetic and $p$-logarithmic means, therefore illustrating their importance in mathematical analysis. The conclusions gained not only generalize current inequalities but also establish a foundation for further investigation in integral inequalities and their quantum counterparts.