A Summarize of Research on Schur Convexity Related to Hadamard Integral Inequality
DOI:
https://doi.org/10.63286/jima.034Keywords:
Integral mean of convex functions, Schur-convexity, majorization, summarizeAbstract
In 2000, Croatian mathematicians Elezovic and Pecaric investigated the Schur-convexity of the integral mean of a convex function with respect to the upper and lower limits of integration, obtaining an important and pioneering result. Building on this work, numerous scholars at home and abroad have since carried out a series of generalizations and extensions. This paper presents a survey of these developments, with the aim of promoting deeper research on integral inequalities via majorization theory.
References
[1] A. M. Marshall and I. Olkin, Inequalities: theory of majorization and its application, Academies Press, New York, 1979.
[2] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: theory of majorization and its application (Second Edition), Springer, New York–Dordrecht–Heidelberg–London, 2011.
[3] B. Y. Wang, Foundations of Majorization Inequalities (Kong zhi bu deng shi ji chu), Beijing Normal University Press, Beijing, China, 1990.
[4] H. N. Shi, Schur-Convex Functions and Inequalities, Harbin Institute of Technology Press Ltd, Harbin, Heilongjiang and Walter de Gruyter GmbH, Berlin/Boston, 2019.
DOI: 10.1515/9783110607840
[5] X. M. Zhang and Y. M. Chu, New discussion to analytic inequalities, Harbin Institute of Technology Press, Harbin, 2009.
[6] Y. M. Chu, G. D. Wang, and X. H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Mathematische Nachrichten, 284(5–6), 2011, 653–663.
DOI: 10.1002/mana.200810197
[7] Z.-H. Yang, Schur power convexity of Gini means, Bulletin of the Korean Mathematical Society, 50(2), 2013, 485–498.
DOI: 10.4134/BKMS.2013.50.2.485
[8] N. G. Zheng, Two methods of proving Hadamard inequality for convex functions, Journal of Huzhou Normal University, 27(2), 2005, 15–17.
[9] N. Elezovic and J. Pečarić, A note on Schur-convex functions, Rocky Mountain Journal of Mathematics, 30(3), 2000, 853–856.
DOI: 10.1216/rmjm/1021477248
[10] F. Qi, Schur-convexity of the extended mean values, RGMIA Research Report Collection, 4(4), 2001, 1–6.
[11] X. M. Zhang and Y. M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain Journal of Mathematics, 40(3), 2010, 1061–1068.
DOI: 10.1216/RMJ-2010-40-3-1061
[12] D. E. Wulbert, Favard's inequality on average values of convex functions, Mathematical and Computer Modelling, 37, 2003, 1383–1391.
DOI: 10.1016/S0895-7177(03)90048-3
[13] B. Y. Long, Y. P. Jiang, and Y. M. Chu, Schur convexity properties of the weighted arithmetic integral mean and Chebyshev functional, Journal of Numerical Analysis and Approximation Theory, 42(1), 2013, 72–81.
DOI: 10.33993/jnaat421-983
[14] J. Sun, Z. L. Sun, B. Y. Xi, and F. Qi, Schur-geometric and Schur-harmonic convexity of an integral mean for convex functions, Turkish Journal of Analysis and Number Theory, 3(3), 2015, 87–89.
DOI: 10.12691/tjant-3-3-4
[15] F. Qi, J. Sándor, S. S. Dragomir, and A. Sofo, Notes on the Schur-convexity of the extended mean values, Taiwanese Journal of Mathematics, 9(3), 2005, 411–420.
DOI: 10.11650/twjm/1500407849
[16] V. Culjak, I. Franjic, R. Ghulam, and J. Pečarić, Schur-convexity of averages of convex functions, Journal of Inequalities and Applications, 2011, Art. 581918, pp. 1–25.
DOI: 10.1155/2011/581918
[17] Y. M. Chu, G. D. Wang, and X. H. Zhang, Schur convexity and Hadamard's inequality, Mathematical Inequalities & Applications, 13(4), 2010, 725–731.
[18] S. H. Wang and S. P. Bai, Schur-convexity for a class of symmetric functions, Turkish Journal of Analysis and Number Theory, 4(1), 2016, 16–19.
DOI: 10.12691/tjant-4-1-3
[19] V. Culjak, I. Franjic, R. Ghulam, and J. Pečarić, Schur-convexity of averages of convex functions, Journal of Inequalities and Applications, 2011, Art. 581918, pp. 1–25.
DOI: 10.1155/2011/581918
[20] I. Franjic and J. Pečarić, Schur-convexity and the Simpson formula, Applied Mathematics Letters, 24, 2011, 1565–1568.
DOI: 0.1016/j.aml.2011.03.047
[21] H. N. Shi, Schur-Convex Functions relate to Hadamard-type inequalities, Journal of Mathematical Inequalities, 1(1), 2007, 127–136.
[22] H. N. Shi, D. M. Li, and C. H. Gu, The Schur-convexity of the mean of a convex function, Applied Mathematics Letters, 22, 2009, 932–937.
DOI: 10.1016/j.aml.2008.04.017
[23] L. He, Two new mappings associated with inequalities of Hadamard-type for convex functions, Journal of Inequalities in Pure and Applied Mathematics, 10(3), 2009, 1–11.
[24] H. N. Shi, Two Schur-convex functions relate to Hadamard-type integral inequalities, Publicationes Mathematicae Debrecen, 78(2), 2011, 393–403.
DOI: 10.5486/PMD.2011.4777
[25] H. N. Shi, D. S. H. Wang, and C. H. R. Fu, Schur-convexity of the mean of convex functions for two variables, Axioms, 11, 2022, 681.
DOI: 10.3390/axioms11120681
[26] M. R. Chowdappa, K. N. Udaya Kumara, S. R. Perla, S. Padmanabhan, and K. M. Nagaraju, Schur harmonic convexity of the mean of convex functions for two variables, 3rd IEEE International Conference on Knowledge Engineering and Communication Systems (ICKECS 2025), 28–29 April 2025, Chickballapur, Karnataka.
[27] H. N. Shi and J. Zhang, Schur-power convexity of integral mean for convex functions on the co-ordinates, Open Mathematics, 21, 2023, Art. 20230157, pp. 1–9.
DOI: 10.1515/math-2023-0157
[28] N. Safaei and A. Barani, Schur-convexity related to co-ordinated convex functions in plane, Journal of Mathematical Inequalities, 15(2), 2021, 467–476.
DOI: 10.7153/jmi-2021-15-35
[29] N. Safaei and A. Barani, Schur-harmonic convexity related to co-ordinated harmonically convex functions in plane, Journal of Inequalities and Applications, 2019, Art. 297, pp. 1–13.
DOI: 10.1186/s13660-019-2249-6
[30] N. Safaei and A. Barani, Schur-convexity of integral arithmetic means of co-ordinated convex functions in $R^3$, Mathematical Analysis & Convex Optimization, 1(1), 2020, 15–24.
DOI: 10.29252/maco.1.1.3
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