Some estimates on the AM-GM inequality and its applications

Subject Areas : Functional analysis

F. P. Mohebbi ¹ (Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.)
M. Hassani ² (Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran)
M. Erfanian Omidvar ³ (Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.)
H. Moradi ⁴ (Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran)

Received: 2024-03-01 Accepted : 2024-05-12 Published : 2024-05-15

Keywords: Young inequality, arithmetic-geometric mean inequality, logarithmic mean,

Abstract :

‎The present paper seeks to establish an approximation of the arithmetic-geometric mean inequality (AM-GM) using a logarithmically concave function‎. ‎We utilized the specific properties of this class of functions to derive modified versions of the AM-GM inequality as a specific example‎. ‎These findings present a fresh perspective on the subject‎.

References:

[1] S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), 471-478.

[2] Y. Feng, Refinements of the Heinz inequalities, J. Inequal. Appl. (2012), 2012:18.

[3] S. Furuichi, H. R. Moradi, On further refinements for Young inequalities, Open Math. 16 (1) (2018), 1478-1482.

[4] S. Furuichi, H. R. Moradi, Some refinements of classical inequalities, Rocky Mountain J. Math. 48 (7) (2018), 2289-2309.

[5] S. Furuichi, H. R. Moradi, M. Sababheh, New inequalities for interpolational operator means, J. Math. Inequal. 15 (1) (2021), 107-116.

[6] F. Kittaneh, Y. Manasrah, Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl. 361 (2010), 262-269.

[7] F. Kittaneh, Y. Manasrah, Reverses Young and Heinz inequalities for matrices, Linear Multilinear Algebra. 59 (2011), 1031-1037.

[8] H. R. Moradi, M. E. Omidvar, Complementary inequalities to improved AM-GM inequality, Acta. Math. Sin. 33 (2017), 1609-1616.

[9] H. R. Moradi, M. Sababheh, S. Furuichi, On the operator Hermite-Hadamard inequality, Complex Anal. Oper. Theory. (2021), 15:122.

[10] R. Pal, M. Singh, M. S. Moslehian, J. S. Aujla, A new class of operator monotone functions via operator means, Linear Multilinear Algebra. 64 (12) (2016), 2463-2473.

[11] M. Sababheh, S. Furuichi, Z. Heydarbeygi, H. R. Moradi, On the arithmetic-geometric mean inequality, J. Math. Inequal. 15 (3) (2021), 1255-1266.

[12] M. Sababheh, H. R. Moradi, I. H. Gümü¸ s, Further inequalities for convex functions, Bull. Malays. Math. Sci. Soc. (2023), 46:42.

[13] M. Sababheh, S. Sheybani, H. R. Moradi, Matrix Fejér and Levin-Steˇ ckin inequalities, Kragujev. J. Math. 49 (5) (2025), 807-821.

[14] K. Yanagi, K. Kuriyama, S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Algebra Appl. 394 (2005), 109-118.