Some inequalities based on a general quantum difference operator

In this paper, some integral inequalities based on the general quantum difference operator D β are deduced. Here, D β is defined by D β f ( t ) = ( f ( β ( t ) ) − f ( t ) ) / ( β ( t ) − t ) , where β is a strictly increasing continuous function, defined on an interval I ⊆ R , that has one fixed po...

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Bibliographic Details
Published in:Journal of inequalities and applications 2015-02, Vol.2015 (1), p.1-12, Article 38
Main Authors: Hamza, Alaa E, Shehata, Enas M
Format: Article
Language:English
Subjects:
Analysis
Applications of Mathematics
Inequalities
Integrals
Intervals
Mathematics
Mathematics and Statistics
Operators
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Summary:In this paper, some integral inequalities based on the general quantum difference operator D β are deduced. Here, D β is defined by D β f ( t ) = ( f ( β ( t ) ) − f ( t ) ) / ( β ( t ) − t ) , where β is a strictly increasing continuous function, defined on an interval I ⊆ R , that has one fixed point s 0 ∈ I . The β -Hölder and β -Minkowski inequalities are proved. Also, the β -Gronwall, β -Bernoulli, and some related inequalities are shown. Finally, the β -Lyapunov inequality is established.
ISSN:1029-242X
1025-5834
1029-242X
DOI:10.1186/s13660-015-0566-y