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Some inequalities for interval-valued functions on time scales

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Abstract

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

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Acknowledgements

This study was funded by Fundamental Research Funds for the Central Universities (Grant Nos. 2017B19714, 2017B07414). Torres was supported by FCT and CIDMA, Project UID/MAT/04106/2013. The authors are very grateful to two anonymous referees, for several valuable and helpful comments, suggestions and questions, which helped them to improve the paper into present form.

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Authors and Affiliations

  1. College of Science, Hohai University, Nanjing, 210098, Jiangsu, People’s Republic of China

    Dafang Zhao, Guoju Ye & Wei Liu

  2. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, Hubei, People’s Republic of China

    Dafang Zhao

  3. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193, Aveiro, Portugal

    Delfim F. M. Torres

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  1. Dafang Zhao
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  2. Guoju Ye
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  3. Wei Liu
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  4. Delfim F. M. Torres
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Correspondence to Dafang Zhao.

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Zhao, D., Ye, G., Liu, W. et al. Some inequalities for interval-valued functions on time scales. Soft Comput 23, 6005–6015 (2019). https://doi.org/10.1007/s00500-018-3538-6

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