Steffensen type inequalities involving convex functions

Pečarić, Josip; Kalamir, Ksenija Smoljak

doi:10.7153/mia-18-26

Cited by 6 publications

(16 citation statements)

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“…Similar result was proved in [8] so we omit the proof. If the assumptions of Theorem 3.1 hold for all n ∈ N, the following corollary is valid.…”

Section: Lemma 31 a Function ψ : I → R Is Convex If And Only If Thesupporting

confidence: 49%

“…We continue with the result needed for the application to Stolarsky type means. Again, similar result was obtained in [8] so we give it without the proof. 1) and (2.2).…”

Section: Lemma 31 a Function ψ : I → R Is Convex If And Only If Thementioning

confidence: 55%

“…In the following theorem we recall the connection between the class of functions M c 1 [a, b] and the class of convex functions proved in [8]. In the following theorems we obtain new Steffensen type inequalities for class of functions that are convex at point c. …”

Section: Resultsmentioning

confidence: 99%

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New Steffensen Type Inequalities Involving Convex Functions

Pečarić

Kalamir

2014

Results. Math.

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We prove weaker conditions for Steffensen type inequalities obtained by Masjed-Jamei, Qi and Srivastava. Moreover, we extend these inequalities to the class of convex functions. Further, we give an application of new inequalities to obtain Stolarsky type means. Mathematics SubjectClassification. Primary 26D15; Secondary 26A51. The well-known Steffensen inequality reads, [10]: Theorem 0.1. Suppose that f is nonincreasing and g is integrable on [a, b] with 0 ≤ g ≤ 1 and λ = b a g(t)dt. Then we haveThe inequalities are reversed for f nondecreasing.In [5] Masjed-Jamei, Qi and Srivastava obtained the following Steffensen type inequalities:Theorem 0.2. If f and g are integrable functions such that f is nonincreasing andon (a, b), where q = 0 and

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“…Similar result was proved in [8] so we omit the proof. If the assumptions of Theorem 3.1 hold for all n ∈ N, the following corollary is valid.…”

Section: Lemma 31 a Function ψ : I → R Is Convex If And Only If Thesupporting

confidence: 49%

“…We continue with the result needed for the application to Stolarsky type means. Again, similar result was obtained in [8] so we give it without the proof. 1) and (2.2).…”

Section: Lemma 31 a Function ψ : I → R Is Convex If And Only If Thementioning

confidence: 55%

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New Steffensen Type Inequalities Involving Convex Functions

Pečarić

Kalamir

2014

Results. Math.

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“…Let us begin by introducing a class of functions that extends the class of convex functions. As noted in [7] we can describe the property from Definition 6 as "convexity at point . " In [7] Pečarić and Smoljak also proved that there is a connection between the class of functions M 1 [ , ] and the class of convex functions.…”

Section: Resultsmentioning

confidence: 99%

Generalized Steffensen Type Inequalities Involving Convex Functions

Pečarić

Kalamir

2014

Journal of Function Spaces

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In this paper generalized Steffensen type inequalities related to the class of functions that are “convex at pointc” are derived and as a consequence inequalities involving the class of convex functions are obtained. Moreover, linear functionals from the difference of the right- and left-hand side of the obtained generalized inequalities are constructed and new families of exponentially convex functions related to constructed functionals are derived.

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“…Futher, some newer works on this subject include extension of Steffensen's inequality and its generalizations to the class of convex functions (see [7,8,9]). We recall the following generalization obtained by Pečarić in [5].…”

Section: Introductionmentioning

confidence: 99%