# Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

**Theorem**

**9.**

## 2. Preliminaries

**Lemma**

**1.**

**Definition**

**1.**

**Lemma**

**2.**

## 3. Main Results

**Lemma**

**3.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

## 4. Some Examples and Consequences

**Corollary**

**2.**

**Corollary**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Dragomir, S.S.; Pearce, C.E.M. Selected topics on Hermite-Hadamard inequalities and applications. In RGMIA Monographs; Victoria University: Melbourne, VIC, Australia, 2000. [Google Scholar]
- Mitrinović, D.S. Analytic Inequalities; Springer: Berlin/Heidelberg, Germany, 1970. [Google Scholar]
- Pečarić, J.; Proschan, F.; Tong, Y. Convex Functions, Partial Orderings, and Statistical Applications; Academic Press, Inc.: Cambridge, MA, USA, 1992. [Google Scholar]
- Sarikaya, M.Z.; Set, E.; Özdemir, M.E. On new inequalities of Simpson’s type for convex functions. Res. Group Math. Inequalities Appl. Res. Rep. Coll.
**2010**, 13, 2. [Google Scholar] [CrossRef] - Kirmaci, U.S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput.
**2004**, 147, 137–146. [Google Scholar] [CrossRef] - Hezenci, F.; Budak, H.; Kara, H. New version of fractional Simpson type inequalities for twice differentiable functions. Adv. Differ. Equ.
**2021**, 2021, 460. [Google Scholar] [CrossRef] - Ozdemir, M.E.; Ardic, A.A. Some companions of Ostrowski type inequality for functions whose second derivatives are convex and concave with applications. Arab J. Math. Sci.
**2015**, 21, 53–66. [Google Scholar] [CrossRef] - Afzal, W.; Shabbir, K.; Treanţă, S.; Nonlaopon, K. Jensen and Hermite-Hadamard type inclusions for harmonical h-Godunova-Levin functions. AIMS Math.
**2023**, 8, 3303–3321. [Google Scholar] [CrossRef] - Afzal, W.; Shabbir, K.; Botmart, T. Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued (h
_{1}, h_{2})-Godunova-Levin functions. AIMS Math.**2022**, 7, 19372–19387. [Google Scholar] [CrossRef] - Afzal, W.; Nazeer, W.; Botmart, T.; Treanţă, S. Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation. AIMS Math.
**2023**, 8, 1696–1712. [Google Scholar] [CrossRef] - Butt, S.I.; Tariq, M.; Aslam, A.; Ahmad, H.; Nofal, T.A. Hermite–Hadamard type inequalities via generalized harmonic exponential convexity and applications. J. Funct. Spaces
**2021**, 2021, 5533491. [Google Scholar] [CrossRef] - Chandola, A.; Agarwal, R.; Pandey, M.R. Some New Hermite–Hadamard, Hermite–Hadamard Fejer and Weighted Hardy Type Inequalities Involving (k-p) Riemann– Liouville Fractional Integral Operator. Appl. Math. Inf. Sci.
**2022**, 16, 287–297. [Google Scholar] - Chen, H.; Katugampola, U.N. Hermite–Hadamard and Hermite–Hadamard–Fejr type inequalities for generalized fractional integrals. J. Math. Anal. Appl.
**2017**, 446, 1274–1291. [Google Scholar] [CrossRef] - Stojiljković, V.; Ramaswamy, R.; Abdelnaby, O.A.A.; Radenović, S. Some Refinements of the Tensorial Inequalities in Hilbert Spaces. Symmetry
**2023**, 15, 925. [Google Scholar] [CrossRef] - Dragomir, S.S. Refinements and Reverses of Tensorial Hermite–Hadamard Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces; ResearchGate: Berlin, Germany, 2022; ResearchGate Preprint. [Google Scholar]
- Dragomir, S.S. Inequalities for normal operators in Hilbert spaces. Appl. Anal. Discret. Math.
**2007**, 1, 92–110. [Google Scholar] [CrossRef] - Dragomir, S.S. The Hermite-Hadamard type Inequalities for Operator Convex Functions. Appl. Math. Comput.
**2011**, 218, 766–772. [Google Scholar] - Dragomir, S.S. An Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces; Researchgate: Berlin, Germany, 2022. [Google Scholar]
- Stojiljkovic, V. Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces. Electron. J. Math. Anal. Appl.
**2023**, 11, 1–15. [Google Scholar] [CrossRef] - Stojiljkovic, V. Twice Differentiable Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces. Eur. J. Pure Appl. Math.
**2023**, 16, 1421–1433. [Google Scholar] [CrossRef] - Stojiljkovic, V.; Dragomir, S. Differentiable Ostrowski type tensorial norm inequality for continuous functions of selfadjoint operators in Hilbert spaces. Gulf J. Math.
**2023**, 15, 40–55. [Google Scholar] [CrossRef] - Araki, H.; Hansen, F. Jensenís operator inequality for functions of several variables. Proc. Am. Math. Soc.
**2000**, 128, 20. [Google Scholar] [CrossRef] - Koranyi, A. On some classes of analytic functions of several variables. Trans. Am. Math. Soc.
**1961**, 101, 520–554. [Google Scholar] [CrossRef] - Guo, H. What Are Tensors Exactly? World Scientific: Singapore, 2021. [Google Scholar] [CrossRef]
- Dragomir, S.S. Tensorial Norm Inequalities for Taylor’s Expansions of Functions of Selfadjoint Operators in Hilbert Spaces; ResearchGate: Berlin, Germany, 2022. [Google Scholar]
- Sarikaya, M.Z.; Set, E.; Yaldiz, H.; Başak, N. Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model.
**2013**, 57, 2403–2407. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Stojiljković, V.; Mirkov, N.; Radenović, S.
Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces. *Symmetry* **2024**, *16*, 121.
https://doi.org/10.3390/sym16010121

**AMA Style**

Stojiljković V, Mirkov N, Radenović S.
Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces. *Symmetry*. 2024; 16(1):121.
https://doi.org/10.3390/sym16010121

**Chicago/Turabian Style**

Stojiljković, Vuk, Nikola Mirkov, and Stojan Radenović.
2024. "Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces" *Symmetry* 16, no. 1: 121.
https://doi.org/10.3390/sym16010121