Refinements of the Jensen Inequality and Estimates of the Jensen Gap Based on Interval‐Valued Functions
ABSTRACT The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well‐known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic‐geometric, and Young's inequalit...
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| Abstract | ABSTRACT
The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well‐known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic‐geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real‐world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval‐valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH‐differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work. |
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| AbstractList | The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well‐known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic‐geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real‐world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval‐valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH‐differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work. ABSTRACT The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well‐known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic‐geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real‐world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval‐valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH‐differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work. |
| Author | Demir, İzzettin |
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| Cites_doi | 10.1016/S0096-3003(02)00657-4 10.1007/s40314-016-0396-7 10.1007/978-1-4471-0249-6 10.1016/S0893-9659(98)00086-X 10.2478/v10006-009-0033-3 10.1016/j.ins.2017.08.055 10.46793/KgJMat2104.647I 10.1007/BF03322144 10.1090/proc/14741 10.1016/j.na.2008.12.005 10.1515/dema-2005-0411 10.1007/978-3-642-69512-4 10.1186/1029-242X-2012-178 10.1016/j.aej.2024.03.093 10.1145/142920.134024 10.2991/ijcis.d.210409.001 10.1016/j.fss.2017.02.001 10.1016/j.fss.2019.06.002 10.1007/BF01214290 10.1006/jmaa.1999.6506 10.1007/s00500-014-1483-6 10.22436/jnsa.009.02.32 10.1137/1.9780898717716 10.7153/jmi-2017-11-80 |
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37 start-page: 343 issue: 4 year: 2015 ident: e_1_2_9_7_1 article-title: Inequalities of Hermite‐Hadamard Type for h‐Convex Functions on Linear Spaces publication-title: Proyecciones: Journal of Mathematics – ident: e_1_2_9_41_1 doi: 10.1137/1.9780898717716 – volume: 31 start-page: 457 year: 2012 ident: e_1_2_9_26_1 article-title: Ostrowski Type Inequalities for Interval‐Valued Functions Using Generalized Hukuhara Derivative publication-title: Computational and Applied Mathematics – ident: e_1_2_9_12_1 doi: 10.7153/jmi-2017-11-80 |
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The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key... The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in... |
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| SubjectTerms | Convexity generalized Hukuhara differentiability Inequalities Inequality interval‐valued functions Jensen inequality left and right convex interval‐valued function Ostrowski inequality |
| Title | Refinements of the Jensen Inequality and Estimates of the Jensen Gap Based on Interval‐Valued Functions |
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