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On Two Different Concepts of Subquadraticity

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Inequalities and Applications 2010

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 161))

Abstract

In the recent years, subquadratic functions have been investigated by several authors. However, two different concepts of subquadraticity have been considered. Based on a simple modification of the geometric notion of concave functions a function f:[0,∞[ →ℝ is called subquadratic if, for each x≥0, there exists a constant c x∈ℝ such that the inequality

$$ f(y)-f(x)\leq c_x(y-x)+f\big(|y-x|\big) $$

is valid for all nonnegative y.

Related to the concept of quadratic functions, a function f:ℝ→ℝ is said to be subquadratic if it fulfils the inequality

$$ f(x+y)+f(x-y)\leq 2f(x)+2f(y) $$

for all x,y∈ℝ. In the present paper, the connections between these two concepts are described and a third inequality related to these concepts is studied.

The research of the first and second authors has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK-81402 and by TÁMOP 4.2.1./B-09/1/KONV-2010-0007/IK/IT project. The project is implemented through the New Hungary Development Plan co-financed by the European Social Fund and the European Regional Development Fund.

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References

  1. Abramovich, S.: Superquadracity of functions and rearrangements of sets. JIPAM. J. Inequal. Pure Appl. Math. 8, Article 46 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Abramovich, S.: 1. Remark. In: Bandle, C., Gilányi, A., Losonczi, L., Páles, Zs., Plum, M. (eds.) Problems and Remarks, Inequalities and Applications, p. xli. Birkhäuser Verlag, Boston, Berlin (2009)

    Google Scholar 

  3. Abramovich, S.: On superquadracity. J. Math. Inequal. 3, 329–339 (2009)

    Article  MathSciNet  Google Scholar 

  4. Abramovich, S., Banić, S., Matic, M.: Superquadratic functions in several variables. J. Math. Anal. Appl. 327, 1444–1460 (2007)

    Article  MathSciNet  Google Scholar 

  5. Abramovich, S., Barić, J., Pečarić, J.: Fejér and Hermite-Hadamard type inequalities for superquadratic functions. J. Math. Anal. Appl. 344, 1048–1056 (2008)

    Article  MathSciNet  Google Scholar 

  6. Abramovich, S., Dragomir, S.S.: Normalized Jensen functional, superquadracity and related topics. In: Bandle, C., Gilányi, A., Losonczi, L., Páles, Zs., Plum, M. (eds.) Inequalities and Applications, pp. 217–228. Birkhäuser Verlag, Basel, Boston, Berlin (2009)

    Chapter  Google Scholar 

  7. Abramovich, S., Ivelić, S., Pečarić, J.: Improvement of Jensen–Steffensen’s inequality for superquadratic functions. Banach J. Math. Anal. 4, 159–169 (2010)

    Article  MathSciNet  Google Scholar 

  8. Abramovich, S., Jameson, G., Sinnamon, G.: Refining of Jensen’s inequality. Bull. Math. Soc. Sci. Math. Roum. 47(95), 3–14 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Abramovich, S., Jameson, G., Sinnamon, G.: Inequalities for averages of convex and superquadratic functions. JIPAM. J. Inequal. Pure Appl. Math. 5, Article 91 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Abramovich, S., Klaričić Bakula, M., Banić, S.: A variant of Jensen-Steffensen’s inequality for convex and superquadratic functions. JIPAM. J. Inequal. Pure Appl. Math. 7, Article 70 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Abramovich, S., Persson, L.-E., Pečarić, J., Varošanec, S.: General inequalities via isotonic subadditive functionals. Math. Inequal. Appl. 10, 15–28 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Banić, S.: Superquadratic functions. PhD thesis, Zagreb (2007)

    Google Scholar 

  13. Banić, S., Pečarić, J., Varošanec, S.: Superquadratic functions and refinements of some classical inequalities. J. Korean Math. Soc. 45, 513–525 (2008)

    Article  MathSciNet  Google Scholar 

  14. Gilányi, A.: 5. Remark (Remark on subquadratic functions, related to Shoshana Abramovich’s talk and remark). In: Bandle, C., Gilányi, A., Losonczi, L., Páles, Zs., Plum, M. (eds.) Problems and Remarks, Inequalities and Applications, pp. xliii–xlv. Birkhäuser Verlag, Basel, Boston, Berlin (2009)

    Google Scholar 

  15. Gilányi, A., Troczka-Pawelec, K.: Regularity of weakly subquadratic functions. J. Math. Anal. Appl. 382, 814–821 (2011)

    Article  MathSciNet  Google Scholar 

  16. Kominek, Z., Troczka, K.: Some remarks on subquadratic functions. Demonstr. Math. 39, 751–758 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Państwowe Wydawnictwo Naukowe – Uniwersytet Śląski, Warszawa–Kraków–Katowice (1985)

    MATH  Google Scholar 

  18. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities, 2 edn. Birkhäuser Verlag, Basel (2009)

    Book  Google Scholar 

  19. Oguntuase, J.A., Persson, L.-E.: Refinement of Hardy’s inequalities via superquadratic and subquadratic functions. J. Math. Anal. Appl. 339, 1305–1312 (2008)

    Article  MathSciNet  Google Scholar 

  20. Rosenbaum, R.A.: Subadditive functions. Duke Math. J. 17, 227–242 (1950)

    Article  MathSciNet  Google Scholar 

  21. Smajdor, W.: Subadditive and Subquadratic Set-Valued Functions, vol. 889. Scientific Publications of the University of Silesia, Katowice (1987)

    MATH  Google Scholar 

  22. Troczka-Pawelec, K.: Some inequalities connected with a quadratic functional equation. Pr. Nauk. Akad. Jana Długosza Czest. Mat. 13, 73–79 (2008)

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Informatics, University of Debrecen, Pf. 12, 4010, Debrecen, Hungary

    Attila Gilányi

  2. Institute of Mathematics, University of Debrecen, Pf. 12, 4010, Debrecen, Hungary

    Csaba Gábor Kézi

  3. Institute of Mathematics and Computer Science, Jan Długosz University of Częstochowa, al. Armii Krajowej 13/15, 42-200, Częstochowa, Poland

    Katarzyna Troczka-Pawelec

Authors
  1. Attila Gilányi
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  2. Csaba Gábor Kézi
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  3. Katarzyna Troczka-Pawelec
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Corresponding author

Correspondence to Csaba Gábor Kézi .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Basel, Rheinsprung 21, Basel, 4051, Switzerland

    Catherine Bandle

  2. Faculty of Informatics, University of Debrecen, Pf. 12, Debrecen, 4010, Hungary

    Attila Gilányi

  3. Department of Economic Analysis, and Information Technology for Business, University of Debrecen, Kassai út 26, Debrecen, 4028, Hungary

    László Losonczi

  4. , Institute of Analysis, KIT, Kaiserstr. 12, Karlsruhe, 76128, Germany

    Michael Plum

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Dedicated to the memory of Wolfgang Walter

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Gilányi, A., Kézi, C.G., Troczka-Pawelec, K. (2012). On Two Different Concepts of Subquadraticity. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_16

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