Sun, Yan; Yang, Hai-Tao; Qi, Feng Some inequalities for multiple integrals on the \(n\)-dimensional ellipsoid, spherical shell, and ball. (English) Zbl 1272.26026 Abstr. Appl. Anal. 2013, Article ID 904721, 7 p. (2013). Summary: The authors establish some new inequalities of Pólya type for multiple integrals on the \(n\)-dimensional ellipsoid, spherical shell, and ball, in terms of bounds of the higher order derivatives of the integrands. These results generalize the main result in the paper by F. Qi, Acta Math. Hung. 84, No. 1–2, 19–26 (1999; Zbl 0963.26008)]. MSC: 26D15 Inequalities for sums, series and integrals Citations:Zbl 0963.26008 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Pólya, G., Ein mittelwertsatz für funktionen mehrerer veränderlichen, Tohoku Mathematical Journal, 19, 1-3 (1921) · JFM 48.0297.02 [2] Pólya, G.; Szegö, G., Aufgaben und Lehrsätze aus der Analysis, 1 (1925), Berlin, Germany: Springer, Berlin, Germany · JFM 51.0173.01 [3] Pólya, G.; Szegö, G., Problems and Theorems in Analysis. Problems and Theorems in Analysis, Classics in Mathematics, 1 (1972), Berlin, Germany: Springer, Berlin, Germany · Zbl 0236.00003 [4] Pólya, G.; Szego, G., Problems and Theorems in Analysis, 1 (1984) · Zbl 0338.00001 [5] Iyengar, K. S. K., Note on an inequality, Math Students, 6, 75-76 (1938) · JFM 64.0209.02 [6] Agarwal, R. P.; Dragomir, S. S., An application of Hayashi’s inequality for differentiable functions, Computers & Mathematics with Applications, 32, 6, 95-99 (1996) · Zbl 0874.26017 · doi:10.1016/0898-1221(96)00146-0 [7] Cerone, P.; Dragomir, S. S., Lobatto type quadrature rules for functions with bounded derivative, Mathematical Inequalities & Applications, 3, 2, 197-209 (2000) · Zbl 0958.26011 · doi:10.7153/mia-03-23 [8] Qi, F., Inequalities for a multiple integral, Acta Mathematica Hungarica, 84, 1-2, 19-26 (1999) · Zbl 0963.26008 · doi:10.1023/A:1006642601341 [9] Qi, F., Inequalities for an integral, The Mathematical Gazette, 80, 488, 376-377 (1996) [10] Guo, B.-N.; Qi, F., Some bounds for the complete elliptic integrals of the first and second kinds, Mathematical Inequalities & Applications, 14, 2, 323-334 (2011) · Zbl 1217.26042 · doi:10.7153/mia-14-26 [11] Guo, B.-N.; Qi, F., Estimates for an integral in \(L^p\) norm of the \((n + 1)\)-th derivative of its integrand, Inequality Theory and Applications, 127-131 (2003), Hauppauge, NY, USA: Nova Science Publishers, Hauppauge, NY, USA · Zbl 1061.26018 [12] Guo, B.-N.; Qi, F., Some estimates of an integral in terms of the \(L^p\)-norm of the \((n + 1)\) st derivative of its integrand, Analysis Mathematica, 29, 1, 1-6 (2003) · Zbl 1026.26020 · doi:10.1023/A:1022894413541 [13] Huy, V. N.; Ngô, Q. A., On an Iyengar-type inequality involving quadratures in \(n\) knots, Applied Mathematics and Computation, 217, 1, 289-294 (2010) · Zbl 1202.26030 · doi:10.1016/j.amc.2010.05.060 [14] Qi, F., Further generalizations of inequalities for an integral, Univerzitet u Beogradu Publikacije Elektrotehničkog Fakulteta, Serija: Matematika, 8, 79-83 (1997) · Zbl 1018.26012 [15] Qi, F., Inequalities for a weighted multiple integral, Journal of Mathematical Analysis and Applications, 253, 2, 381-388 (2001) · Zbl 0966.26013 · doi:10.1006/jmaa.2000.7138 [16] Qi, F.; Zhang, Y.-J., Inequalities for a weighted integral, Advanced Studies in Contemporary Mathematics, 4, 2, 93-101 (2002) · Zbl 1019.26007 [17] Qi, F.; Wei, Z. L.; Yang, Q., Generalizations and refinements of Hermite-Hadamard’s inequality, The Rocky Mountain Journal of Mathematics, 35, 1, 235-251 (2005) · Zbl 1096.26014 · doi:10.1216/rmjm/1181069779 [18] Shi, Y. X.; Liu, Z., On Iyengar type integral inequalities, Journal of Anshan University of Science and Technology, 26, 1, 57-60 (2003) [19] Kuang, J. C., Chángyòng Bùdĕng Shì (Applied Inequalities) (2004), Jinan, China: Shandong Science and Technology Press Shandong Province, Jinan, China [20] Qi, F., Polya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications, RGMIA Research Report Collection (2013) [21] Akdemir, A. O.; Özdemir, M. E.; Varošanec, S., On some inequalities for \(h\)-concave functions, Mathematical and Computer Modelling, 55, 3-4, 746-753 (2012) · Zbl 1255.26007 · doi:10.1016/j.mcm.2011.08.051 [22] Niculescu, C. P.; Persson, L. E., Convex Functions and Their Applications. Convex Functions and Their Applications, CMS Books in Mathematics (2006), New York, NU, USA: Springer, New York, NU, USA · Zbl 1100.26002 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.