On proportional Caputo-hybrid fractional maclaurin-type inequalities for convex stochastic processes

Raouf Fakhfakh; Rabab Alzahrani; Fatimah Alshahrani; Hend Aljahani; Foued Mtiri; Abdellatif Ben Makhlouf; Raouf Fakhfakh; Rabab Alzahrani; Fatimah Alshahrani; Hend Aljahani; Foued Mtiri; Abdellatif Ben Makhlouf

doi:10.3934/math.20261215

AIMS Mathematics

2026, Volume 11, Issue 9: 30658-30688. doi: 10.3934/math.20261215

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Research article

On proportional Caputo-hybrid fractional maclaurin-type inequalities for convex stochastic processes

1.
Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia
2.
Department of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia; Email: r.alzahrani@psau.edu.sa
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia
5.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
6.
Mathematics Education Section, Faculty of Education and arts, Sohar University, P.O. Box 44, Postal Code 311, Sohar, Sultanate of Oman
7.
Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia

Received: 12 May 2026 Revised: 30 August 2026 Accepted: 14 September 2026 Published: 20 September 2026
MSC : 26D10, 26D15, 60E15

In this paper, we investigated Maclaurin-type inequalities for convex stochastic processes by means of proportional Caputo-hybrid fractional operators. To this end, we introduced the left-sided and right-sided stochastic mean-square proportional Caputo-hybrid operators and established a new integral identity involving these operators. This identity served as the main analytical tool for deriving our estimates. Based on this representation formula, we obtained new fractional Caputo-hybrid Maclaurin-type inequalities for convex stochastic processes under appropriate differentiability and integrability assumptions. The obtained results may be regarded as stochastic fractional extensions of the classical Maclaurin-type inequalities. In this way, the present work enriched the theory of stochastic fractional calculus and contributed to the ongoing development of generalized integral inequalities for stochastic processes.
- stochastic proportional Caputo-hybrid operators,
- Maclaurin's inequality,
- convex stochastic processes,
- Hölder's inequality,
- power mean inequality
Citation: Raouf Fakhfakh, Rabab Alzahrani, Fatimah Alshahrani, Hend Aljahani, Foued Mtiri, Abdellatif Ben Makhlouf. On proportional Caputo-hybrid fractional maclaurin-type inequalities for convex stochastic processes[J]. AIMS Mathematics, 2026, 11(9): 30658-30688. doi: 10.3934/math.20261215

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Abstract

In this paper, we investigated Maclaurin-type inequalities for convex stochastic processes by means of proportional Caputo-hybrid fractional operators. To this end, we introduced the left-sided and right-sided stochastic mean-square proportional Caputo-hybrid operators and established a new integral identity involving these operators. This identity served as the main analytical tool for deriving our estimates. Based on this representation formula, we obtained new fractional Caputo-hybrid Maclaurin-type inequalities for convex stochastic processes under appropriate differentiability and integrability assumptions. The obtained results may be regarded as stochastic fractional extensions of the classical Maclaurin-type inequalities. In this way, the present work enriched the theory of stochastic fractional calculus and contributed to the ongoing development of generalized integral inequalities for stochastic processes.

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