Abstract
A new class of mappings that includes the class of Lipschitzian mappings is introduced. For this kind of mappings, new integral inequalities of Hadamard’s type are obtained. Our results are extensions of many previous contributions related to integral inequalities for Lipschitzian mappings.
Highlights
A function ω : I ⊂ R → R is said to be L-Lipschitzian, where L > 0, if jωðσÞ − ωðκÞj ≤ Ljσ − κj, ð1Þ for all σ, κ ∈ I
For other works related to integral inequalities for Lipschitzian functions, see, for example, [5,6,7,8,9,10,11] and the references therein
Motivated by the above mentioned results, in this paper, we obtain some Hadamard-type integral inequalities for a new class of functions which includes the class of Lipschitzian functions
Summary
A function ω : I ⊂ R → R is said to be L-Lipschitzian, where L > 0, if jωðσÞ − ωðκÞj ≤ Ljσ − κj, ð1Þ for all σ, κ ∈ I. In [1, 2], some integral inequalities of Hadamard’s type involving L-Lipschitzian functions were derived. Notice that if ðP, Q, RÞ = ðα, ðα + βÞ/2, βÞ and ðξ, ξ2, ξ3Þ = ð1/4, 1/2, 1/4Þ, (7) reduces to the Bullen-type inequality ωðβÞ!. If ðP, Q, RÞ = ðα, ðα + βÞ/2, βÞ and ðξ, ξ2, ξ3Þ = ð1/6, 2/3, 1/6Þ, (7) reduces to the Bullen-type inequality (see [4]). For other works related to integral inequalities for Lipschitzian functions, see, for example, [5,6,7,8,9,10,11] and the references therein. Motivated by the above mentioned results, in this paper, we obtain some Hadamard-type integral inequalities for a new class of functions which includes the class of Lipschitzian functions
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