Notes on judgment criteria of convex functions of several variables

Yıl 2021, , 235 - 243, 31.12.2021

Huannan Shi , Pei Wang Jian Zhang Wei-shih Du

https://doi.org/10.53006/rna.986088

Cited By: 1

Öz

By transferring the judgment of convex functions of several variables into the judgment of convex functions
of one variable, the authors discuss the convexity of some convex functions of several variables.

Anahtar Kelimeler

convex functions of several variables, one variable convex function, judgment method

Kaynakça

• [1] X.-D. Chen, Remarks on convex functions, Journal of Western Chongqing University Natural Science Edition 2 (2003), no. 2, 37-40; available online at http://dx.chinadoi.cn/10.3969/j.issn.1673-8012.2003.04.012. (Chinese)
• [2] S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite?Hadamard Inequalities and Applications, Amended version, RGMIA Monographs, Victoria University, 2002; available online at http://rgmia.org/monographs/hermite_hadamard. html.
• [3] N. Elezovi¢ and J. Pecaric, A note on Schur-convex functions, Rocky Mountain J. Math. 30 (2000), no. 3, 853?856; available online at https://doi.org/10.1216/rmjm/1021477248.
• [4] A.W. Marshall, I. Olkin, and B.C. Arnold, Inequalities: Theory of Majorization and its Applications, 2nd Ed., Springer Ver- lag, New York-Dordrecht-Heidelberg-London, 2011; available online at http://dx.doi.org/10.1007/978-0-387-68276-1.
• [5] F. Qi, Inequalities for an integral, Math. Gaz. 80 (1996), no. 488, 376-377; available online at https://doi.org/10.2307/ 3619581.
• [6] F. Qi, Pólya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications, Math. Inequal. Appl. 18 (2015), no. 1, 1-38; available online at https://doi.org/10.7153/mia-18-01.
• [7] F. Qi, J. Sándor, S.S. Dragomir, and A. Sofo, Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math. 9 (2005), no. 3, 411-420; available online at https://doi.org/10.11650/twjm/1500407849.
• [8] H.-N. Shi, S.-H. Wu, and F. Qi, An alternative note on the Schur-convexity of the extended mean values, Math. Inequal. Appl. 9 (2006), no. 2, 219-224; available online at http://dx.doi.org/10.7153/mia-09-22.
• [9] Y. Shuang and F. Qi, Integral inequalities of Hermite-Hadamard type for extended s-convex functions and applications, Mathematics 6 (2018), no. 11, Article 223, 12 pages; available online at https://doi.org/10.3390/math6110223.
• [10] B.-Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, 1990. (Chinese)
• [11] S.-G. Wang, M.-X. Wu, and Z.-Z. Jia, Inequalities in Matrix Theory, Second Ed., Science Press, Beijing, 2006.
• [12] J.-J. Wu and Y.-G. Zhu, Several methods for determining the convexity of a function, Journal of Communication University of China Science and Technology, 27 (2002), no. 6, 79-83. (Chinese)
• [13] D.E. Wulbert, Favard's inequality on average values of convex functions, Math. Comput. Modelling 37 (2003), no. 12-13, 1383?1391; available online at https://doi.org/10.1016/S0895-7177(03)90048-3.
• [14] X.M. Zhang and Y.M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math. 40 (2010), no. 3, 1061-1068; available online at https://doi.org/10.1216/RMJ-2010-40-3-1061.
• [15] N. G. Zheng, X.M. Zhang, and Y.M. Chu, Convexity and geometrical convexity of exponential and logarithmic means in N variables, Acta Math. Sci. Ser. A (Chin. Ed.) 28 (2008), no. 6, 1173?1180. (Chinese)