Quadratic refinements of Young type inequalities

Yonghui Ren; Pengtong Li; Guoqing Hong

doi:10.1515/ms-2017-0416

Published by De Gruyter September 27, 2020

Quadratic refinements of Young type inequalities

Yonghui Ren , Pengtong Li and Guoqing Hong

From the journal Mathematica Slovaca

https://doi.org/10.1515/ms-2017-0416

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Abstract

In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely:

(va+(1−v)b)2−v∑j=1N2j(b−ab2j−12j)2≤(avb1−v)2+v2(a−b)2

for v ∉ [0, 12N+1], N ∈ ℕ, a, b > 0; and

(va+(1−v)b)2−(1−v)∑j=1N2j(a−a2j−1b2j)2≤(avb1−v)2+(1−v)2(a−b)2

for v ∉ [1 − 12N+1, 1], N ∈ ℕ, a, b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.

Keywords: Young’s inequality; operator inequality; unitarily invariant norms

MSC 2010: Primary 15A45; Secondary 47A30

This research is supported by the National Natural Science Foundation of China (11671201).

(Communicated by Emanuel Chetcuti)

References

[1] Bakherad, M.—Krnic, M.—Moslehian, M.: Reverse Young-type inequalities for matrices and operators, Rocky Mountain J. Math. 46 (2016), 1089–1105.10.1216/RMJ-2016-46-4-1089Search in Google Scholar

[2] Bakherad, M.—Moslehian, M.: Reverses and variations of Heinz inequality, Linear Multilinear Algebra 63 (2014), 1972–1980.10.1080/03081087.2014.880433Search in Google Scholar

[3] Furuichi, S.: Refined Young inequalities with Specht’s ratio, J. Egypt. Math. Soc. 20 (2012), 46–49.10.1016/j.joems.2011.12.010Search in Google Scholar

[4] Kórus, P.: A refinement of Young’s inequality, Acta Math. Hungar. 153 (2017), 430–435.10.1007/s10474-017-0735-1Search in Google Scholar

[5] Liao, W.—Wu, J.—Zhao, J.: New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math. 19 (2015), 467–479.10.11650/tjm.19.2015.4548Search in Google Scholar

[6] Pečarić, J.—Furuta, T.—Hot, J.—Seo, Y.: Mond-Pečarić Method in Operator Inequalities, Element, Zagreb, 2005.Search in Google Scholar

[7] Sababheh, M.—Choi, D.: A complete refinement of Young’s inequality, J. Math. Anal. Appl. 440 (2016), 379–393.10.1016/j.jmaa.2016.03.049Search in Google Scholar

[8] Sababheh, M.—Moslehian, M.: Advanced refinements of Young and Heinz inequalities, J. Number Theory 172 (2016), 178–199.10.1016/j.jnt.2016.08.009Search in Google Scholar

[9] Tominaga, M.: Specht’s ratio in the Young inequality, Sci. Math. Japon. 55 (2002), 583–588.Search in Google Scholar

[10] Yang, C.—Gao, Y.—Lu, F.: Some refinements of Young type inequality for positive linear map, Math. Slovaca 69 (2019), 919–930.10.1515/ms-2017-0277Search in Google Scholar

[11] Yang, C.—Ren, Y.: A reverse of Young inequalities, J. Math. Inequal. 13 (2019), 53–63.10.7153/jmi-2019-13-04Search in Google Scholar

[12] Yang, C.—Ren, Y.: Some results of Heron mean and Young inequalities, J. Inequal. Appl. 172 (2018), 1–9.Search in Google Scholar

[13] Zhao, X.—Li, L.—Zuo, H.: Further improved Young inequalities for operators and matrices, J. Math. Inequal. 11 (2017), 1023–1029.10.7153/jmi-2017-11-78Search in Google Scholar

[14] Zhao, J.—Wu, J.: Operator inequalities involving improved Young and its reverse inequalities, J. Math. Anal. Appl. 421 (2015), 1779–1789.10.1016/j.jmaa.2014.08.032Search in Google Scholar

[15] Zou, L.: Improved logarithmic-geometric mean inequality and its application, Bull. Iran. Math. Soc. 43 (2017), 2323–2326.Search in Google Scholar

[16] Zou, L.—Jiang, Y.: Improved arithmetic-geometric mean inequality and its application, J. Math. Inequal. 9 (2015), 107–111.10.7153/jmi-09-10Search in Google Scholar

[17] Zuo, H.—Shi, G.—Fujii, M.: Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5 (2011), 551–556.10.7153/jmi-05-47Search in Google Scholar

Received: 2019-09-26

Accepted: 2020-01-17

Published Online: 2020-09-27

Published in Print: 2020-10-27

Quadratic refinements of Young type inequalities

Abstract

References

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