Related Experiment Video
Updated: Jul 31, 2026
12:25
2D and 3D Matrices to Study Linear Invadosome Formation and Activity
Published on: June 2, 2017
Some complementary inequalities to Jensen's operator inequality
Jadranka Mićić1, Hamid Reza Moradi2, Shigeru Furuichi3
11Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia.
Journal of Inequalities and Applications
|February 6, 2018
Summary
Researchers present improved complementary inequalities for self-adjoint operators and positive mappings, building on Jensen's inequality. These advancements utilize an enhanced Mond-Pečarić method, leading to new quasi-arithmetic mean inequalities.
Area of Science:
- Functional Analysis
- Operator Theory
- Mathematical Inequalities
Background:
- Jensen's inequality is a fundamental concept in mathematical analysis.
- Complementary inequalities refine existing inequalities by providing tighter bounds.
- Self-adjoint operators and positive linear mappings are key objects in operator theory.
Purpose of the Study:
- To establish new complementary inequalities related to Jensen's inequality.
- To improve existing methods for deriving such inequalities.
- To apply these findings to quasi-arithmetic means.
Main Methods:
- Utilizing an improved Mond-Pečarić method.
- Applying techniques from functional analysis and operator theory.
- Leveraging properties of real-valued twice differentiable functions.
Main Results:
- New complementary inequalities for self-adjoint operators and positive linear mappings are derived.
- The presented inequalities offer improvements over existing results.
- The study provides novel inequalities involving quasi-arithmetic means.
Conclusions:
- The improved Mond-Pečarić method effectively generates refined complementary inequalities.
- The findings contribute to the theory of inequalities in operator theory.
- Applications to quasi-arithmetic means demonstrate the practical utility of the new inequalities.
Related Concept Videos
01:28Inequalities
Inequalities express mathematical relationships where two values are not equal and are compared using symbols such as <, >, ≤, or ≥. These expressions define a range of possible solutions rather than a single value. Interval notation provides a concise way to express these solution sets, especially when the variable spans a continuous range. An open interval, written as (a, b), excludes the endpoints, while a closed interval [a, b] includes them. There are also half-open intervals, such...
01:23Absolute Value Inequalities
The absolute value is a mathematical tool that represents the distance of a number from zero on the number line, regardless of its sign. In the context of inequalities, absolute value expressions help define a range of permissible values or boundaries for a variable. These inequalities are commonly used in scientific modeling and data interpretation, where variability within or beyond a certain threshold must be captured precisely.An absolute value inequality of the form ∣x∣ ≤ a, where a ≥ 0,...
01:29Application of Nonlinear Inequalities
A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality: can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values are 3...
01:28Graphical Representation of Inequalities
The graph of the equation where y equals x squared forms a curve known as a parabola. This curve acts as a boundary in the coordinate plane, dividing it into distinct regions based on the relative position of points.When the equality sign in the equation is replaced with an inequality—such as greater than, less than, greater than or equal to, or less than or equal to—the graphical representation changes from a single curve into a broader shaded area that signifies the set of all points...
01:28Lagrange Multipliers: Two Constraints
The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
01:25Introduction to Nonlinear Inequalities
Linear and nonlinear inequalities are fundamental for analyzing variable relationships and identifying ranges satisfying specific conditions. A linear inequality involves variables raised only to the first power, resulting in a straight-line graph. This line partitions the coordinate plane into two distinct regions: one that satisfies the inequality and one that does not. Each region represents a set of solutions where the linear relationship holds true under the specified constraint.Nonlinear...