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Generalization of Some Bounds containing Entropies on Time Scales

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Abstract

In this paper, weighted Jensen’s inequality for diamond integrals is utilized to get some new inequalities containing entropies. Shanon entropy, triangular discrimination, Jeffreys distance, Bhattacharyya coefficient, Hellinger discrimination and Rényi entropy are introduced and their bounds are derived using diamond–integral formalism. We discussed the classical, discrete and q-analogue of obtained results by restricting the time scale to \({\mathbb {R}}\), \(h{\mathbb {Z}}\) for \(h>0\) and \(q^{{\mathbb {N}}_{0}}\) for \(q>1.\) For different divergence measures, the entropic bounds are also deduced. Furthermore, resultant inequalities are also discussed considering the Zipf law and the Zipf–Mandelbrot law to establish some new inequalities for isolated points. The new established results are the generalizations of results proved in Ansari et al. (J Inequal Appl 2021:1–21, 2021), Horváth et al. (Bull Malays Math Sci Soc 42:933-946, 2019), Matić, Pearce and Pečarić (Shannon’s and related inequalities in information theory. Survey on classical inequalities, Springer, Dordrech, 2000).

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Acknowledgements

The authors wish to thanks the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Sargodha, Sargodha, Pakistan

    Muhammad Bilal, Khuram Ali Khan & Ammara Nosheen

  2. Government Associate College Miani, Sargodha, Pakistan

    Muhammad Bilal

  3. Croatian Academy of Science and Arts HR, Zagreb, Croatia

    Josip Pečarić

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Contributions

MB initiated the work and made calculations. KAK supervised and validated the draft. AN deduced the existing results and finalized the draft. JP dealt with the formal analysis and investigation. All the authors read and approved the final manuscript.

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Correspondence to Muhammad Bilal.

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Muhammad Bilal, Khuram Ali Khan, Ammara Nosheen, and Josip Pečarić have contributed equally to this work.

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Bilal, M., Khan, K.A., Nosheen, A. et al. Generalization of Some Bounds containing Entropies on Time Scales. Qual. Theory Dyn. Syst. 22, 71 (2023). https://doi.org/10.1007/s12346-023-00768-1

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