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Some inequalities for interval-valued functions on time scales

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Dafang Zhao, Guoju Ye, Wei Liu, Delfim F. M. Torres

Classical Analysis and ODEs · Submitted 11 Sep 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral.

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arXiv ID1809.03709 (version 1)
Pith IDYBNE6NUG, Pith's permanent record of this paper and its review
CitationDafang Zhao, Delfim F. M. Torres, Guoju Ye, Wei Liu. "Some inequalities for interval-valued functions on time scales." arXiv:1809.03709 (2018). https://pith.science/paper/YBNE6NUG
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We introduce the interval Darboux delta integral (shortly, the $ID$ $\Delta$-integral) and the interval Riemann delta integral (shortly, the $IR$ $\Delta$-integral) for interval-valued functions on time scales. Fundamental properties of $ID$ and $IR$ $\Delta$-integrals and examples are given. Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral. Also, some examples are given to illustrate our theorems.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce the interval Darboux delta integral (shortly, the $ID$ $\Delta$-integral) and the interval Riemann delta integral (shortly, the $IR$ $\Delta$-integral) for interval-valued functions on time scales. contribution 1.00 0.95 0.08 asserted 1.00
s1 Fundamental properties of $ID$ and $IR$ $\Delta$-integrals and examples are given. contribution 0.90 0.94 0.06 asserted 0.99
s2 Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral. result 0.80 0.96 0.77 asserted 1.00
s3 Also, some examples are given to illustrate our theorems. contribution 0.67 0.92 0.03 asserted 1.00

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Original abstract

We introduce the interval Darboux delta integral (shortly, the $ID$ $\Delta$-integral) and the interval Riemann delta integral (shortly, the $IR$ $\Delta$-integral) for interval-valued functions on time scales. Fundamental properties of $ID$ and $IR$ $\Delta$-integrals and examples are given. Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral. Also, some examples are given to illustrate our theorems.

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  • More recently, some of classical inequalities have been extended to set-valued functions by Nikodem et al. [ref] , Štrboja et al. [ref] , and Zhang et al. [ref] , especially to interval-valued functions by Chalco-Cano et al. [ref] , Costa et al. [ref] , Flores-Franulič et al. [ref] , Román-Flores et al. [ref] , and Zhao et al. [ref] .

    Discrete Opial type inequalities for interval-valued functions · Introduction · 0.83

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