Skip to main content
SpringerLink
Log in

Applications of Quasiconvexity

  • Chapter
  • First Online:
Contributions in Mathematics and Engineering

Abstract

This survey deals with functions called γ-quasiconvex functions and their relations to convexity and superquadracity. For γ-quasiconvex functions and for superquadratic functions, we get analogs of inequalities satisfied by convex functions and we get refinements for those convex functions which are also γ-quasiconvex as well as superquadratic. We show in which cases the refinements by γ-quasiconvex functions are better than those obtained by superquadratic functions and convex functions. The power functions defined on x ≥ 0 where the power is greater or equal to two are examples of convex, quasiconvex, and superquadratic functions.

In Honor of Constantin Carathéodory

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abramovich, S., Persson, L.E.: Some new scales of refined Hardy type inequalities via functions related to superquadracity. Math. Inequal. Appl. 16, 679–695 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Abramovich, S., Persson, L.E.: Some new refined Hardy type inequalities with breaking points p = 2 or p = 3. In: Proceedings of the IWOTA 2011, Operator Theory: Advances and Applications, vol. 236, pp. 1–10. Birkhäuser/Springer, Basel/Berlin (2014)

    Google Scholar 

  3. Abramovich, S., Persson, L.E.: Inequalities for averages of quasiconvex and superquadratic functions. Math. Inequal. Appl. 19 (2), 535–550 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Abramovich, S., Jameson, G., Sinnamon, G.: Inequalities for averages of convex and superquadratic functions. J. Inequal. Pure Appl. Math. 5 (4), Article 91 (2004)

    Google Scholar 

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

    Google Scholar 

  6. Abramovich, S., Bakula M.K., Matić M., Pečarić, J.: A variant of Jensen-Steffensen’s inequality and quazi arithmetic means. J. Math. Anal. Appl. 307, 370–386 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Abramovich, S., Krulic, K., Person, L.-E., Pečarić, J.: Some new refined Hardy type inequalities with general kernels via superquadratic and subquadratic functions. Aequationes Math. 79 (1), 157–172 (2010) (On line March 2010)

    Google Scholar 

  8. Abramovich, S., Persson, L.E., Samko, S.: Some new scales of refined Hardy type inequalities. Math. Inequal. Appl. 17, 1105–1114 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Abramovich, S., Persson, L.E., Samko, S.: On γ-quasiconvexity, superquadracity and two-sided reversed Jensen type inequalities. Math. Inequal. Appl. 18 (2), 615–628 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Anwar, M., Jakšetić, J., Pečarić, J., Rehman, A.: Exponential convexity, positive semi-definite matrices and fundamental inequalities. J. Math. Inequal. 4 (2), 171–189 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Banić, S., Varošanec, S.: Functional inequalities for superquadratic functions. Int. J. Pure Appl. Math. 43 (4), 5037–549 (2008)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Bibi, R., Bohner, M., Pečarić, J.: Cauchy type means and exponential and logarithmic convexity for superquadratic functions on time scale. Ann. Funct. Anal. 6 (1), 59–83 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gilányi, A., Kézi, C.G., Troczka-Pawelec, K.: On two different concept of superquadracity. Inequalities and Applications, pp. 209–215. Birkhauser, Basel (2010)

    Google Scholar 

  16. Hardy, G.H.: Notes on a theorem of Hilbert. Math. Z. 6, 314–317 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mitroi, F.-C.: On the Jensen-Steffensen inequality and superquadracity. Anale Universitatii, Oradea, Fasc. Mathematica, Tom XVIII, pp. 269–275 (2011)

    Google Scholar 

  18. 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  MATH  Google Scholar 

  19. Oguntuase, J.A., Persson, L.-E.: Hardy type inequalities via convexity-the journey so far. Aust. J. Math. Anal. Appl. 7 (2), Art. 18, 19 (2010)

    Google Scholar 

  20. Oguntuase, J.A., Persson, L.E.: Time scale Hardy type inequalities via superquadracity. Ann. Funct. Anal. 5 (2), 61–73 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Oguntuase, J.A., Popoola, B.A.: Refinement of Hardy’s inequalities involving many functions via superquadratic functions. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (2), 271–283 (2011)

    Google Scholar 

  22. Oguntuase, J.A., Persson, L.E., Essel, E.K., Popoola, B.A.: Refined multidimensional Hardy-type inequalities via superquadracity. Banach J. Math. Anal. 2 (2), 129–139 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Pečarić, J., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic, Boston (1992)

    MATH  Google Scholar 

  24. Persson, L.E., Samko, N.: What should have happened if Hardy had discovered this? J. Inequal. Appl. 2012:29, 11 pp (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, 31905, Israel

    S. Abramovich

Authors
  1. S. Abramovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Abramovich .

Editor information

Editors and Affiliations

  1. Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Mathematics, National Technical Unversity of Athens Department of Mathematics, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Abramovich, S. (2016). Applications of Quasiconvexity. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31317-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation