Dinh, Trung-Hoa; Dinh, Thanh-Duc; Vo, Bich-Khue T. A new type of operator convexity. (English) Zbl 1491.47013 Acta Math. Vietnam. 43, No. 4, 595-605 (2018). Summary: Let \(r, s\) be positive numbers. We define a new class of operator \((r, s)\)-convex functions by the following inequality \[ f \left( \left[\lambda A^{r} + (1-\lambda)B^{r}\right]^{1/r}\right) \leq [\lambda f(A)^{s} +(1-\lambda)f(B)^{s}]^{1/s}, \] where \(A, B\) are positive definite matrices and for any \(\lambda \in [0,1]\). We prove the Jensen, Hansen-Pedersen, and Rado type inequalities for such functions. Some equivalent conditions for a function \(f\) to become operator \((r, s)\)-convex are established. Cited in 3 Documents MSC: 47A63 Linear operator inequalities Keywords:operator \((r,s)\)-convex functions; operator Jensen-type inequality; operator Hansen-Pedersen-type inequality; operator Rado-type inequality PDFBibTeX XMLCite \textit{T.-H. Dinh} et al., Acta Math. Vietnam. 43, No. 4, 595--605 (2018; Zbl 1491.47013) Full Text: DOI References: [1] Ando, T.; Hiai, F., Operator log-convex functions and operator means, Math. Ann., 350, 611-630, (2011) · Zbl 1221.47028 · doi:10.1007/s00208-010-0577-4 [2] Audenaert, KMR; Hiai, F., On matrix inequalities between the power means: counterexamples, Linear Algebra Appl., 439, 1590-1604, (2013) · Zbl 1283.15061 · doi:10.1016/j.laa.2013.04.012 [3] Dinh, T-H; Vo, B-KT, Some inequalities for operator \((p, h)\)(p,h)-convex functions, Linear Multilinear Algebra, 66, 580-592, (2018) · doi:10.1080/03081087.2017.1307914 [4] Gill, PM; Pearce, CEM; Pečarić, J., Hadamard’s inequality for \(r\)-convex functions, J. Math. Anal. Appl., 215, 461-470, (1997) · Zbl 0891.26013 · doi:10.1006/jmaa.1997.5645 [5] Neumark, MA, On a representation of additive operator set functions, Dok. Akad. Nauk SSSR, 41, 373-375, (1943) · Zbl 0061.25410 [6] Tikhonov, OE, A note on definition of matrix convex functions, Linear Algebra Appl., 416, 773-775, (2006) · Zbl 1100.15011 · doi:10.1016/j.laa.2005.12.019 [7] Zhang, KS; Wan, JP, p-convex functions and their properties, Pure Appl. Math. (Xi’an), 23, 130-133, (2007) · Zbl 1165.26312 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.