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Annales Mathematicae Silesianae

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Inequalities Of Lipschitz Type For Power Series In Banach Algebras

Sever S. Dragomir
  • Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
  • School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa, URL: http://rgmia.org/dragomir
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Published Online: 2015-09-30 | DOI: https://doi.org/10.1515/amsil-2015-0006


Let f(z)=n=0αnzn be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that f(y)f(x)yx01fa((1t)x+ty)dt where fa(z)=n=0|αn|zn . Inequalities for the commutator such as f(x)f(y)f(y)f(x)2fa(M)fa(M)yx, if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

(2010) Mathematics Subject Classification:: 47A63; 47A99

Key words and phrases:: Banach algebras; Power series; Lipschitz type inequalities; Hermite-Hadamard type inequalities


  • [1] Azpeitia A.G., Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), no. 1, 7–12.Google Scholar

  • [2] Bhatia R., Matrix analysis, Springer-Verlag, New York, 1997.Google Scholar

  • [3] Cheung W.-S., Dragomir S.S., Vector norm inequalities for power series of operators in Hilbert spaces, Tbilisi Math. J. 7 (2014), no. 2, 21–34.Google Scholar

  • [4] Dragomir S.S., Cho Y.J., Kim S.S., Inequalities of Hadamard’s type for Lipschitzian mappings and their applications, J. Math. Anal. Appl. 245 (2000), no. 2, 489–501.Google Scholar

  • [5] Dragomir S.S., A mapping in connection to Hadamard’s inequalities, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 128 (1991), 17–20.Google Scholar

  • [6] Dragomir S.S., Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167 (1992), 49–56.Google Scholar

  • [7] Dragomir S.S., On Hadamard’s inequalities for convex functions, Math. Balkanica 6 (1992), 215–222.Google Scholar

  • [8] Dragomir S.S., An inequality improving the second Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), no. 3, Art. 35.Google Scholar

  • [9] Dragomir S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), 471–476.CrossrefGoogle Scholar

  • [10] Dragomir S.S., Gomm I., Bounds for two mappings associated to the Hermite–Hadamard inequality, Aust. J. Math. Anal. Appl. 8 (2011), Art. 5, 9 pp.Google Scholar

  • [11] Dragomir S.S., Gomm I., Some new bounds for two mappings related to the Hermite–Hadamard inequality for convex functions, Numer. Algebra Cont Optim. 2 (2012), no. 2, 271–278.Google Scholar

  • [12] Dragomir S.S., Milośević D.S., Sándor J., On some refinements of Hadamard’s inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math. 4 (1993), 21–24.Google Scholar

  • [13] Dragomir S.S., Pearce C.E.M., Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, 2000. Available at http://rgmia.org/monographs/hermite_hadamard.html

  • [14] Guessab A., Schmeisser G., Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory 115 (2002), no. 2, 260–288.Google Scholar

  • [15] Kilianty E., Dragomir S.S., Hermite–Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl. 13 (2010), no. 1, 1–32.Google Scholar

  • [16] Matić M., Pečarić J., Note on inequalities of Hadamard’s type for Lipschitzian mappings, Tamkang J. Math. 32 (2001), no. 2, 127–130.Google Scholar

  • [17] Merkle M., Remarks on Ostrowski’s and Hadamard’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 113–117.Google Scholar

  • [18] Mikusiński J., The Bochner integral, Birkhäuser Verlag, Basel, 1978.Google Scholar

  • [19] Pearce C.E.M., Rubinov A.M., P-functions, quasi-convex functions, and Hadamard type inequalities, J. Math. Anal. Appl. 240 (1999), no. 1, 92–104.Google Scholar

  • [20] Pečarić J., Vukelić A., Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions, in: Functional equations, inequalities and applications, Kluwer Acad. Publ., Dordrecht, 2003, pp. 105–137.Google Scholar

  • [21] Toader G., Superadditivity and Hermite–Hadamard’s inequalities, Studia Univ. Babeş-Bolyai Math. 39 (1994), no. 2, 27–32.Google Scholar

  • [22] Yang G.-S., Hong M.-C., A note on Hadamard’s inequality, Tamkang J. Math. 28 (1997), no. 1, 33–37.Google Scholar

  • [23] Yang G.-S., Tseng K.-L., On certain integral inequalities related to Hermite–Hadamard inequalities, J. Math. Anal. Appl. 239 (1999), no. 1, 180–187.Google Scholar

About the article

Received: 2014-10-21

Revised: 2015-03-04

Published Online: 2015-09-30

Published in Print: 2015-09-01

Citation Information: Annales Mathematicae Silesianae, Volume 29, Issue 1, Pages 61–83, ISSN (Online) 0860-2107, DOI: https://doi.org/10.1515/amsil-2015-0006.

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© Sever S. Dragomir. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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