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Volume 7, Issue 4
Entanglement of Several Classical and Dynamic Estimates with Unified Approach on Time Scales

Faryal Chaudhry & Muhammad Jibril Shahab Sahir

DOI: 10.12150/jnma.2025.1142

J. Nonl. Mod. Anal., 7 (2025), pp. 1142-1152.

Published online: 2025-07

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In this research article, we present several generalizations of Qi’s inequality on time scales. We establish dynamic versions of Callebaut’s inequality and Cauchy-Schwarz’s inequality on time scales. To establish our results, we apply the diamond-alpha integral and the time scale $∆$ or $∇$-Riemann-Liouville type fractional integrals. Our findings unify and extend discrete, continuous and quantum analogues.

  • Keywords

Time scales, fractional Riemann-Liouville integrals, Qi’s, Callebaut’s and Cauchy-Schwarz’s inequalities.

  • AMS Subject Headings

26D15, 26E70, 34N05, 26A33

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JNMA-7-1142, author = {Chaudhry , Faryal and Sahir , Muhammad Jibril Shahab}, title = {Entanglement of Several Classical and Dynamic Estimates with Unified Approach on Time Scales}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2025}, volume = {7}, number = {4}, pages = {1142--1152}, abstract = {

In this research article, we present several generalizations of Qi’s inequality on time scales. We establish dynamic versions of Callebaut’s inequality and Cauchy-Schwarz’s inequality on time scales. To establish our results, we apply the diamond-alpha integral and the time scale $∆$ or $∇$-Riemann-Liouville type fractional integrals. Our findings unify and extend discrete, continuous and quantum analogues.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2025.1142}, url = {http://global-sci.org/intro/article_detail/jnma/24229.html} }
TY - JOUR T1 - Entanglement of Several Classical and Dynamic Estimates with Unified Approach on Time Scales AU - Chaudhry , Faryal AU - Sahir , Muhammad Jibril Shahab JO - Journal of Nonlinear Modeling and Analysis VL - 4 SP - 1142 EP - 1152 PY - 2025 DA - 2025/07 SN - 7 DO - http://doi.org/10.12150/jnma.2025.1142 UR - https://global-sci.org/intro/article_detail/jnma/24229.html KW - Time scales, fractional Riemann-Liouville integrals, Qi’s, Callebaut’s and Cauchy-Schwarz’s inequalities. AB -

In this research article, we present several generalizations of Qi’s inequality on time scales. We establish dynamic versions of Callebaut’s inequality and Cauchy-Schwarz’s inequality on time scales. To establish our results, we apply the diamond-alpha integral and the time scale $∆$ or $∇$-Riemann-Liouville type fractional integrals. Our findings unify and extend discrete, continuous and quantum analogues.

Chaudhry , Faryal and Sahir , Muhammad Jibril Shahab. (2025). Entanglement of Several Classical and Dynamic Estimates with Unified Approach on Time Scales. Journal of Nonlinear Modeling and Analysis. 7 (4). 1142-1152. doi:10.12150/jnma.2025.1142
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