This paper presents new generalizations of the Hermite-Hadamard inequality for convex functions via (p,q)-quantum integrals. First, based on the definitions of (p,q)-derivatives and integrals over finite intervals, we establish a unified (p,q)-Hermite-Hadamard inequality framework, combining midpoint-type and trapezoidal-type inequalities into a single form. Furthermore, by introducing a parameter λ, we propose a generalized (p,q)-integral inequality, whose special cases reduce to classical quantum Hermite-Hadamard inequalities and existing results in the literature. Furthermore, using hybrid integral techniques, we construct refined inequalities that incorporate (p,q)-integral
terms, and by adjusting λ, we demonstrate their improvements and extensions to known inequalities. Specific examples are provided to validate the applicability of the results. The findings indicate that the proposed (p,q)-integral approach offers more flexible mathematical tools for the estimation of numerical integration error, convex optimization problems, and analysis of system performance in control theory, thus enriching the research results of quantum calculus in the field of inequalities.

This paper presents new generalizations of the Hermite-Hadamard inequality for convex functions via (p,q)-quantum integrals. First, based on the definitions of (p,q)-derivatives and integrals over finite intervals, we establish a unified (p,q)-Hermite-Hadamard inequality framework, combining midpoint-type and trapezoidal-type inequalities into a single form. Furthermore, by introducing a parameter λ, we propose a generalized (p,q)-integral inequality, whose special cases reduce to classical quantum Hermite-Hadamard inequalities and existing results in the literature. Furthermore, using hybrid integral techniques, we construct refined inequalities that incorporate (p,q)-integral
terms, and by adjusting λ, we demonstrate their improvements and extensions to known inequalities. Specific examples are provided to validate the applicability of the results. The findings indicate that the proposed (p,q)-integral approach offers more flexible mathematical tools for the estimation of numerical integration error, convex optimization problems, and analysis of system performance in control theory, thus enriching the research results of quantum calculus in the field of inequalities.