This article estimates the deviation of a function taking values in a real Banach space from its Bochner integral mean by establishing new Ostrowski-type inequalities for three times differentiable functions. A new integral identity involving a parameter and the third derivative is presented, proved by means of the Bochner integration by parts formula. Based on this identity, three Ostrowski-type inequalities are derived under convexity assumptions imposed on the norm of the third derivative, by employing Hölder's inequality and the power mean inequality. Classical results from the literature are recovered as particular cases of the inequalities obtained.
Citation: Ohud Bulayhan Almutairi. New inequalities for differentiable functions with values in a Banach space[J]. AIMS Mathematics, 2026, 11(8): 24179-24193. doi: 10.3934/math.2026976
This article estimates the deviation of a function taking values in a real Banach space from its Bochner integral mean by establishing new Ostrowski-type inequalities for three times differentiable functions. A new integral identity involving a parameter and the third derivative is presented, proved by means of the Bochner integration by parts formula. Based on this identity, three Ostrowski-type inequalities are derived under convexity assumptions imposed on the norm of the third derivative, by employing Hölder's inequality and the power mean inequality. Classical results from the literature are recovered as particular cases of the inequalities obtained.
| [1] |
A. Ostrowski, Über die Absolutabweichung einer differenzierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1937), 226–227. https://doi.org/10.1007/BF01214290 doi: 10.1007/BF01214290
|
| [2] |
E. Set, A. Ekinci, On some generalized integral inequalities for functions whose second derivatives absolute value are convex, Numer. Methods Partial Differ. Equ., 40 (2024), e22758. https://doi.org/10.1002/num.22758 doi: 10.1002/num.22758
|
| [3] |
N. S. Barnett, C. Buşe, P. Cerone, S. S. Dragomir, Ostrowski's inequality for vector-valued functions and applications, Comput. Math. Appl., 44 (2002), 559–572. https://doi.org/10.1016/S0898-1221(02)00171-2 doi: 10.1016/S0898-1221(02)00171-2
|
| [4] |
S. S. Dragomir, A refinement of the Ostrowski inequality for absolutely continuous functions and applications, Acta Math. Vietnam., 27 (2002), 203–217. https://doi.org/10.14510/LM.V22I0.885.G757 doi: 10.14510/LM.V22I0.885.G757
|
| [5] | M. Z. Sarıkaya, E. Set, M. E. Özdemir, On some integral inequalities for twice differentiable mappings, Stud. Univ. Babeş-Bolyai Math., 59 (2014), 11–24. |
| [6] | S. S. Dragomir, Ostrowski type inequalities for functions with values in Banach spaces, RGMIA Res. Rep. Coll., 24 (2021), 21. |
| [7] |
G. A. Anastassiou, Ostrowski and Landau inequalities for Banach space valued functions, Math. Comput. Model., 55 (2012), 312–329. https://doi.org/10.1016/j.mcm.2011.08.003 doi: 10.1016/j.mcm.2011.08.003
|
| [8] |
M. A. Latif, O. B. Almutairi, Ostrowski-type inequalities for functions of two variables in Banach spaces, Mathematics, 12 (2024), 2748. https://doi.org/10.3390/math12172748 doi: 10.3390/math12172748
|
| [9] | H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2011. https://doi.org/10.1007/978-0-387-70914-7 |
| [10] | P. Cerone, S. S. Dragomir, Mathematical inequalities, New York: Chapman and Hall/CRC, 2010. https://doi.org/10.1201/b10483 |
| [11] | S. S. Dragomir, T. M. Rassias, Ostrowski type inequalities and applications in numerical integration, Springer Dordrecht, 2002. https://doi.org/10.1007/978-94-017-2519-4 |
| [12] |
S. S. Dragomir, S. Wang, A new inequality of Ostrowski's type in $L_1$-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. Math., 28 (1997), 239–244. https://doi.org/10.5556/j.tkjm.28.1997.4320 doi: 10.5556/j.tkjm.28.1997.4320
|
| [13] |
M. E. Özdemir, H. Kavurmacı, E. Set, Ostrowski's type inequalities for $(\alpha, m)$-convex functions, Kyungpook Math. J., 50 (2010), 371–378. https://doi.org/10.5666/KMJ.2010.50.3.371 doi: 10.5666/KMJ.2010.50.3.371
|
| [14] |
E. Set, New inequalities of Ostrowski type for mappings whose derivatives are $s$-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147–1154. https://doi.org/10.1016/j.camwa.2011.12.023 doi: 10.1016/j.camwa.2011.12.023
|
| [15] |
G. Rahman, M. Vivas-Cortez, Ç. Yildiz, M. Samraiz, S. Mubeen, M. F. Yassen, On the generalization of Ostrowski-type integral inequalities via fractional integral operators with application to error bounds, Fractal Fract., 7 (2023), 683. https://doi.org/10.3390/fractalfract7090683 doi: 10.3390/fractalfract7090683
|
| [16] |
X. Zhan, A. Mateen, M. Toseef, M. A. Ali, Some Simpson- and Ostrowski-type integral inequalities for generalized convex functions in multiplicative calculus with their computational analysis, Mathematics, 12 (2024), 1721. https://doi.org/10.3390/math12111721 doi: 10.3390/math12111721
|
| [17] |
H. M. Rezk, A. I. Saied, M. Ali, G. AlNemer, M. Zakarya, Inequalities of Ostrowski type for functions whose derivative module is relatively convex on time scales, Axioms, 13 (2024), 235. https://doi.org/10.3390/axioms13040235 doi: 10.3390/axioms13040235
|
| [18] | S. Kovač, J. Pečarić, M. Ribičić Penava, Ostrowski type inequalities for $3$-convex functions, Dolomites Res. Notes Approx., 18 (2025), 41–48. |
| [19] | J. Diestel, J. J. Uhl, Jr., Vector measures: Mathematical surveys and monographs, Volume 15, American Mathematical Society, 1977. http://doi.org/10.1090/surv/015 |
Ohud Bulayhan Almutairi. New inequalities for differentiable functions with values in a Banach space[J]. AIMS Mathematics, 2026, 11(8): 24179-24193. doi: 10.3934/math.2026976
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