Jensen-Type Inequalities, Montgomery Identity and Higher-Order Convexity

Bošnjak, Marija; Krnić, Mario; Pečarić, Josip

doi:10.1007/s00009-022-02133-z

Cited by 7 publications

(8 citation statements)

References 9 publications

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“…The classical Jensen inequality [11][12][13][14][15][16][17][18] can be expressed as follows. Let the function f : I → R be a strictly convex function [12][13][14][15].…”

Section: Basic Concepts and Classical Resultsmentioning

confidence: 99%

“…In this paper, we will generalize the Chebyshev inequalities (11) and (13), and establish Chebyshev-Jensen-type inequalities which involving Chebyshev products.…”

Section: Basic Concepts and Classical Resultsmentioning

confidence: 99%

“…Then, by ( 5) and ( 6), we have the following Corollary 3.1. Therefore, Theorem 3.1 is a generalization of the Chebyshev inequality (11).…”

Section: Now Let's Prove Lemma 32mentioning

confidence: 96%

“…Classical Jensen type inequalities and their applications were also investigated by many authors (see, e.g. [11][12][13][14][15][16][17][18]). In [15], the authors considered that comparing two integral means for absolutely continuous functions whose absolute value of the derivative are convex and displayed its applications.…”

Section: Introductionmentioning

confidence: 99%

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Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Wen

Han

Yuan

2023

Preprint

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This paper will establish Chebyshev-Jensen-type inequalities which involving the Chebyshev products as follows:\[\langle X^{1}, X^{2}, \ldots, X^{m}\rangle_{\chi}\geq \left[ X^{1}, X^{2}, \ldots, X^{m}\right]_{\chi}\geq \chi\left( \overline{X^{1}}, \overline{X^{2}}, \ldots, \overline{X^{m}}\right)\]and\[\langle f_{1}, f_{2}, \ldots, f_{m}\rangle_{\chi}\geq \left[f_{1}, f_{2}, \ldots,f_{m}\right]_{\chi}\geq \chi\left( \overline{f_{1}}, \overline{f_{2}}, \ldots, \overline{f_{m}}\right),\]as well as display their applications in probability theory and surround system, especially, the observation angles inequalities are obtained as follows:\[1\geq \frac{1}{n}\sum_{i=1}^{n}P\left(R_{i}\right)\geq \frac{1}{n^{m}}\sum_{1\leq i_{1},\ldots,i_{m}\leq n}P(R_{i_{1},\ldots,i_{m}})\geq\left(\frac{2}{n}\right)^{m+|\alpha|}.\]The research methods of the paper are based on the mathematical induction, reorder method and the dimension reduction method. 2010 MSC: 26D15, 26E60, 51K05, 52A40

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“…The classical Jensen inequality [11][12][13][14][15][16][17][18] can be expressed as follows. Let the function f : I → R be a strictly convex function [12][13][14][15].…”

Section: Basic Concepts and Classical Resultsmentioning

confidence: 99%

“…In this paper, we will generalize the Chebyshev inequalities (11) and (13), and establish Chebyshev-Jensen-type inequalities which involving Chebyshev products.…”

Section: Basic Concepts and Classical Resultsmentioning

confidence: 99%

“…Then, by ( 5) and ( 6), we have the following Corollary 3.1. Therefore, Theorem 3.1 is a generalization of the Chebyshev inequality (11).…”

Section: Now Let's Prove Lemma 32mentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Wen

Han

Yuan

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…In [3], the authors established the following Chebyshev type inequality: The Jensen type inequalities and their applications were also investigated by many authors [14][15][16][17][18][19][20][21][22][23][24]. In [14], the authors considered that comparing two integral means for absolutely continuous functions, whose absolute value of the derivative are convex, and displayed its applications.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev–Jensen-Type Inequalities Involving χ-Products and Their Applications in Probability Theory

Liu,

Wen,

Zhao

2024

Mathematics

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By means of the functional analysis theory, reorder method, mathematical induction and the dimension reduction method, the Chebyshev-Jensen-type inequalities involving the χ-products ⟨·⟩χ and [·]χ are established, and we proved that our main results are the generalizations of the classical Chebyshev inequalities. As applications in probability theory, the discrete with continuous probability inequalities are obtained.

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More Accurate Jensen-Type Inequalities Based on the Lidstone Interpolation Formula

Bošnjak,

Krnić,

Lovričević

et al. 2023

Results Math

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Jensen-Type Inequalities, Montgomery Identity and Higher-Order Convexity

Cited by 7 publications

References 9 publications

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Chebyshev-Jensen-type inequalities and their applications in probability theory with surround system

Chebyshev–Jensen-Type Inequalities Involving χ-Products and Their Applications in Probability Theory

More Accurate Jensen-Type Inequalities Based on the Lidstone Interpolation Formula

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