Some additive reverses of Callebaut and Hölder inequalities for isotonic functionals

Year 2023, Volume: 6 Issue: 4, 249 - 259, 15.12.2023

Sever Dragomır

https://doi.org/10.33205/cma.1362691

Cited By: 1

Abstract

In this paper, we obtain some reverses of Callebaut and Hölder inequalities for isotonic functionals via a reverse of Young’s inequality we have established recently. Applications for integrals and n-tuples of real numbers are provided as well.

Keywords

Discrete inequalities , Isotonic Functionals , Hölder's inequality , Schwarz’s inequality , Callebaut’s inequality , Integral inequalities

References

• D. Andrica, C. Badea: Grüss’ inequality for positive linear functionals, Periodica Math. Hung., 19 (1998), 155–167.
• P. R. Beesack, J. E. Peˇcari´c: On Jessen’s inequality for convex functions, J. Math. Anal. Appl., 110 (1985), 536–552.
• D. K. Callebaut: Generalization of Cauchy-Schwarz inequality, J. Math. Anal. Appl., 12 (1965), 491–494.
• S. S. Dragomir: A refinement of Hadamard’s inequality for isotonic linear functionals, Tamkang J. Math.(Taiwan), 24 (1992), 101–106.
• S. S. Dragomir: On a reverse of Jessen’s inequality for isotonic linear functionals, J. Ineq. Pure & Appl. Math., 2 (3) (2001), Article 36.
• S. S. Dragomir: On the Jessen’s inequality for isotonic linear functionals, Nonlinear Anal. Forum, 7 (2) (2002), 139–151.
• S. S. Dragomir: On the Lupa¸s-Beesack-Peˇcari´c inequality for isotonic linear functionals, Nonlinear Funct. Anal. & Appl., 7 (2) (2002), 285–298.
• S.S. Dragomir: Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc., 74 (3) (2006), 417–478.
• S. S. Dragomir: Additive refinements and reverses of Young’s operator inequality with applications, Preprint RGMIA Res. Rep. Coll., 18 (2015), Art. A 165.
• S. S. Dragomir: Inequalities for Synchronous Functions and Applications, Constr. Math. Anal., 2 (3) (2019), 109–123.
• S. S. Dragomir, Ostrowski’s Type Inequalities for the Complex Integral on Paths, Constr. Math. Anal., 3 (4) (2020), 125–138.
• S. S. Dragomir, N. M. Ionescu: On some inequalities for convex-dominated functions, L’Anal. Num. Théor. L’Approx., 19 (1) (1990), 21–27.
• S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University (2000).
• S. S. Dragomir, C. E. M. Pearce and J. E. Peˇcari´c: On Jessen’s and related inequalities for isotonic sublinear functionals, Acta. Sci. Math.(Szeged), 61 (1995), 373–382.
• S. Furuichi: Refined Young inequalities with Specht’s ratio, J. Egyptian Math. Soc., 20 (2012), 46–49.
• F. Kittaneh, Y. Manasrah: Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl., 361 (2010), 262–269.
• F. Kittaneh, Y. Manasrah: Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra, 59 (2011), 1031–1037.
• W. Liao, J. Wu and J. Zhao: New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math., 19 (2) (2015), 467–479.
• A. Lupa¸s: A generalisation of Hadamard’s inequalities for convex functions, Univ. Beograd. Elek. Fak., 544/576 (1976), 115–121.
• J .E. Peˇcari´c: On Jessen’s inequality for convex functions (III), J. Math. Anal. Appl., 156 (1991), 231–239.
• J. E. Peˇcari´c, P. R. Beesack: On Jessen’s inequality for convex functions (II), J. Math. Anal. Appl., 118 (1986), 125–144.
• J. E. Peˇcari´c, S. S. Dragomir: A generalisation of Hadamard’s inequality for isotonic linear functionals, Radovi Mat.(Sarjevo), 7 (1991), 103–107.
• J. E. Peˇcari´c, I. Ra¸sa: On Jessen’s inequality, Acta. Sci. Math.(Szeged), 56 (1992), 305–309.
• W. Specht: Zer Theorie der elementaren Mittel, Math. Z., 74 (1960), 91–98.
• G. Toader, S. S. Dragomir: Refinement of Jessen’s inequality, Demonstr. Math., 28 (1995), 329–334.
• M. Tominaga: Specht’s ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583–588.
• G. Zuo, G. Shi and M. Fujii: Refined Young inequality with Kantorovich constant, J. Math. Inequal., 5 (2011), 551-556.