Some Parameterized Quantum Midpoint and Quantum Trapezoid Type Inequalities for Convex Functions with Applications

Area of Science:

• Quantum Information Theory
• Mathematical Analysis
• Numerical Analysis

Background:

• Quantum information theory integrates computer science, information theory, and cryptography.
• Convex functions are closely linked to inequalities and entropy functions.

Purpose of the Study:

• To establish new quantum midpoint, trapezoidal, and Simpson's type inequalities.
• To demonstrate the generalization of existing inequalities in quantum calculus.
• To explore applications of these new inequalities in quadrature formulas.

Main Methods:

• Utilizing a novel parameterized q-integral equality.
• Applying methods from quantum information theory and convex analysis.
• Proving new quantum integral inequalities for differentiable convex functions.

Main Results:

• The paper successfully proves novel quantum midpoint, trapezoidal, and Simpson's type inequalities.
• These newly derived inequalities are shown to be generalizations of prior results.
• New applications for quadrature formulas are presented based on the established inequalities.

Conclusions:

• The study expands the scope of quantum integral inequalities.
• The findings contribute to the interdisciplinary field of quantum information theory.
• The established inequalities provide a foundation for advanced applications in numerical analysis.