Related Experiment Video
Updated: Apr 24, 2026
13:44
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
Published on: August 30, 2013
42.6K
Parameterized hilbert-type integral inequalities in the whole plane
Qiliang Huang1, Shanhe Wu2, Bicheng Yang1
1Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, China.
Thescientificworldjournal
|September 13, 2014
Summary
Researchers derived new Hilbert-type integral inequalities using real analysis. These inequalities feature nonhomogeneous kernels and multiparameters, with best possible constant factors.
Area of Science:
- Real Analysis
- Mathematical Inequalities
Background:
- Hilbert-type integral inequalities are fundamental in analysis.
- Existing inequalities often have limitations regarding kernel types and parameters.
Purpose of the Study:
- To establish novel Hilbert-type integral inequalities in the whole plane.
- To analyze inequalities with nonhomogeneous kernels and multiple parameters.
- To determine the optimality of constant factors.
Main Methods:
- Application of real analysis techniques.
- Estimation of weight functions.
- Derivation of integral inequalities.
Main Results:
- New Hilbert-type integral inequalities in the whole plane were established.
- Constant factors involving hypergeometric and beta functions were proven to be the best possible.
- Equivalent forms, reverses, and special cases with homogeneous kernels were considered.
Conclusions:
- The study provides significant advancements in the theory of integral inequalities.
- The derived inequalities offer powerful tools for various mathematical applications.
- The best possible nature of the constant factors enhances the utility of these inequalities.
Related Concept Videos
01:28Improper Integrals: Discontinuous Integrands
163
Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
163
01:29Improper Integrals: Infinite Intervals
264
An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
264
01:29Application of Nonlinear Inequalities
357
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...
357
01:28Graphical Representation of Inequalities
444
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...
444
01:27Area Between Curves: Problem Solving
198
A region can be enclosed by three curves: a square root function, a reflected cube root function, and a linear function. The linear function intersects each of the other two curves, and these intersection points determine where the boundary of the enclosed region changes. Because different curves serve as the upper and lower boundaries in different parts of the graph, the area cannot be found using a single setup over the entire interval.To compute the area, the region is first divided into two...
198
01:23Absolute Value Inequalities
436
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∣ ≤...
436