The book "Exploring Mathematical Analysis, Approximation Theory, and Optimization" discusses various mathematical inequalities and principles that are crucial in optimization and its applications. Key topics include:
Jensen-Steffensen Inequality: This inequality for convex functions is extended and refined, providing insights into the behavior of convex functions and their applications in optimization.
Ostrowski and Trapezoid Type Inequalities: These inequalities generalize the classical results for Riemann integrals to Riemann-Liouville fractional integrals, useful in approximation theory and optimization problems involving fractional calculus.
Hardy's Inequality: This inequality is improved, and its relation to Legendre polynomials is discussed, which is significant in numerical analysis and optimization.
Strong Maximum Principle for General Nonlinear Operators: This principle provides conditions under which a function attains its maximum value, essential for proving existence and uniqueness of solutions in optimization problems.
Ergodic Theory: The application of ergodic theory to alternating series expansions for real numbers is explored, offering insights into the behavior of dynamical systems and their applications in optimization.
These inequalities and principles are fundamental in optimization as they help in deriving bounds, proving existence and uniqueness of solutions, and designing efficient algorithms for solving optimization problems across various fields, including engineering, economics, and computer science.