Comprehensive study of Weddle’s formula-type inequalities for convex functions within the framework of quantum calculus and their computational analysis

Abstract

Abstract This paper presents novel extensions of Weddle’s formula-type inequalities for convex functions within the framework of quantum calculus along with their computational analysis. Weddle’s formula, traditionally used in numerical integration, plays a significant role in approximating integrals and improving the accuracy of numerical methods. By extending this formula to handle convex functions, we provide new inequalities that offer more precise bounds for integrals in classical and quantum calculus contexts. Due to their wide application in optimization, economics, and various branches of mathematics, convex functions are particularly suited for this type of analysis. First, we establish an integral identity for q-differentiable convex functions. Then, with the help of this newly established identity, we prove Weddle’s formula-type integral inequalities specifically designed for q-differentiable convex functions. Moreover, we present applications of the newly established inequalities to numerical quadrature formulas and special means of real numbers, demonstrating their potential impact on computational mathematics and related fields. Numerical examples, computational analysis, and graphical representations further confirm the validity and effectiveness of these inequalities in the context of q-calculus.

Keywords

• Weddle’s formula
• Convex functions
• Quantum calculus
• Numerical integration
• Error bounds