Analysis of Newton-type inequalities for differentiable hyperbolic $ p $-convex functions via RL-integrals

Saima Riaz; Khuram Ali Khan; Tamador Alihia; Ghulam Abbas; Tahreem Akram; Saima Riaz; Khuram Ali Khan; Tamador Alihia; Ghulam Abbas; Tahreem Akram

doi:10.3934/math.2026648

AIMS Mathematics

2026, Volume 11, Issue 6: 15745-15764. doi: 10.3934/math.2026648

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Research article

Analysis of Newton-type inequalities for differentiable hyperbolic $ p $-convex functions via RL-integrals

1.
Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan
2.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
3.
Department of Mathematics, Govt. Graduate College Bhalwal, 40410 Bhalwal, Pakistan

Received: 30 March 2026 Revised: 18 May 2026 Accepted: 25 May 2026 Published: 04 June 2026
MSC : 26A33, 26D10, 26D15, 41A55

In this paper, new generalizations of Newton-type inequalities for the class of hyperbolic $ p $-convex functions by utilizing Riemann-Liouville fractional integrals are established. By means of an auxiliary identity connected with the Riemann-Liouville fractional integrals, new estimates are obtained for functions that are hyperbolic $ p $-convex. The established inequalities are further improved by the effective use of Hölder's inequality and the power-mean inequality. Examples along with graphs are provided to demonstrate the validity of the newly established inequalities and comparisons with existing results. The results of this paper will open up new avenues of research and may be generalized to other types of fractional operators and generalized convex functions.
- convex functions,
- Riemann-Liouville fractional integrals,
- Simpson inequality,
- mathematical operators,
- hyperbolic $ p $-convex function
Citation: Saima Riaz, Khuram Ali Khan, Tamador Alihia, Ghulam Abbas, Tahreem Akram. Analysis of Newton-type inequalities for differentiable hyperbolic $ p $-convex functions via RL-integrals[J]. AIMS Mathematics, 2026, 11(6): 15745-15764. doi: 10.3934/math.2026648

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Abstract

In this paper, new generalizations of Newton-type inequalities for the class of hyperbolic $ p $-convex functions by utilizing Riemann-Liouville fractional integrals are established. By means of an auxiliary identity connected with the Riemann-Liouville fractional integrals, new estimates are obtained for functions that are hyperbolic $ p $-convex. The established inequalities are further improved by the effective use of Hölder's inequality and the power-mean inequality. Examples along with graphs are provided to demonstrate the validity of the newly established inequalities and comparisons with existing results. The results of this paper will open up new avenues of research and may be generalized to other types of fractional operators and generalized convex functions.

References

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