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Triple Diamond-Alpha integral and Hölder-type inequalities
1College of Science and Technology, North China Electric Power University, Baoding, P.R. China.
Journal of Inequalities and Applications
|May 19, 2018
Summary
This study defines the triple Diamond-Alpha integral and extends Hölder and Minkowski inequalities for functions on time scales. These new generalizations advance integral inequality theory on dynamic systems.
Area of Science:
- Mathematical Analysis
- Real Analysis
- Dynamic Equations on Time Scales
Background:
- Integral inequalities are fundamental in mathematical analysis.
- Time scales calculus provides a unified framework for discrete and continuous analysis.
- Generalizing existing inequalities to new integral forms is crucial for broader applications.
Purpose of the Study:
- To introduce the novel concept of the triple Diamond-Alpha integral for functions of three variables.
- To establish and generalize Hölder and reverse Hölder inequalities for this integral on time scales.
- To derive a new generalization of the Minkowski inequality using the established results.
Main Methods:
- Definition of the triple Diamond-Alpha integral.
- Application of time scales calculus principles.
- Deductive mathematical reasoning to establish inequality properties.
Main Results:
- The definition of the triple Diamond-Alpha integral is formally presented.
- New formulations of Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales are derived.
- A generalized Minkowski inequality for the triple Diamond-Alpha integral on time scales is obtained.
Conclusions:
- The paper successfully introduces and analyzes the triple Diamond-Alpha integral.
- The derived inequalities offer significant extensions to existing mathematical tools on time scales.
- These findings contribute to the advancement of integral inequality theory and its applications.
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