Abstract
In this paper, we establish a reversed Hölder inequality via an \(\alpha,\beta\)-symmetric integral, which is defined as a linear combination of the α-forward and the β-backward integrals, and then we give some generalizations of the \(\alpha,\beta\)-symmetric integral Hölder inequality which is due to Brito da Cruz et al.; some related inequalities are also given.
Highlights
Let ak ≥, bk ≥ (k =, . . . , n), p >, /p + /q =
In Section, we recall some basic definitions and properties of α, β-symmetric integral, which can be found in [, ]; in Section, we establish a α, β-symmetric integral reverse Hölder inequality and give some generalizations of the α, β-symmetric integral Hölder inequality, we apply the obtained results to establish the reverse Minkowski inequality, Dresher inequality, and its reverse form involving α, β-symmetric integral, some extensions of the Minkowski and Dresher inequalities are given; in Section, we establish some further generalizations and refinements of the α, β-symmetric integral Hölder inequality; in Section, we present a subdividing of the α, β-symmetric integral Hölder inequality
Β-symmetric integral which is defined as a linear combination of the α-forward and the β-backward integrals
Summary
The α-forward and β-backward differences are defined as follows (see [ ]):. Let g : I → R be a nonnegative α-forward integrable function on [a, b]. (see [ ]) Assume that f : I → R and let |f | be α-forward integrable on [a, b]. Β-symmetric integral which is defined as a linear combination of the α-forward and the β-backward integrals. If f is α-forward and β-backward integrable on [a, b], α, β ≥ with α + β > , we define the α, βsymmetric integral of f from a to b by b α b β b a f (t) dα,β t = α + β f (t).
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