Abstract

For a measure space (Ω,Σ,μ)(\Omega,\Sigma,\mu ) and a bijective increasing function φ:[0,∞)→[0,∞)\varphi:\left[ 0,\infty \right) \rightarrow \left[0,\infty \right) the LpL^{p}-like paranormed (FF-normed) function space with the paranorm of the form pφ(x)=φ−1(∫Ωφ∘∣x∣dμ)\mathbf{p}_{\varphi }(x)=\varphi ^{-1}\left( \int_{\Omega }\varphi \circ \left\vert x\right\vert d\mu \right) is considered. Main results give general conditions under which this space is uniformly convex. The Clarkson theorem on the uniform convexity of LpL^{p}-space is generalized. Under some specific assumptions imposed on φ\varphi we give not only a proof of the uniform convexity but also show the formula of a modulus of convexity. We establish the uniform convexity of all finite-dimensional paranormed spaces, generated by a strictly convex bijection φ\varphi of [0,∞)[0, \infty). However, the a contrario proof of this fact provides no information on a modulus of convexity of these spaces. In some cases it can be done, even an exact formula of a modulus can be proved. We show how to make it in the case when S=R2S={\mathbb R}^2 and φ\varphi is given by φ(t)=et−1\varphi(t)={\rm e}^t-1.

Author information

Contact details are reproduced from the original publication and may be historical.

Justyna Jarczyk

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

j.jarczyk@wmie.uz.zgora.pl

Janusz Matkowski

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

j.matkowski@wmie.uz.zgora.pl

Keywords

  • Lp-like paranorm
  • paranormed space
  • uniformly convex paranormed space
  • modulus of convexity
  • convex function
  • geometrically convex function
  • superquadratic function

Mathematics Subject Classification

46A16, 46E30; 47H09, 47H10

References

  1. A. Clarkson: Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936) 396–414.https://doi.org/10.1090/S0002-9947-1936-1501880-4Google Scholar
  2. A. Granas, J. Dugundji: Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York (2003).https://doi.org/10.1007/978-0-387-21593-8Google Scholar
  3. G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities, Cambridge University Press, Cambridge (1952).Google Scholar
  4. J. Matkowski: On a characterization of Lp -norm, Ann. Pol. Math. 50 (1989) 137–144.https://doi.org/10.4064/ap-50-2-137-144Google Scholar
  5. J. Matkowski: The converse of the Minkowski inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990) 663–675.https://doi.org/10.1090/S0002-9939-1990-1009994-0Google Scholar
  6. J. Matkowski: Lp -like paranorms, Grazer Math. Ber. 316 (1992) 103–138.Google Scholar
  7. J. Matkowski: On a generalization of Mulholland’s inequality, Abh. Math. Semin. Univ. Hambg. 63 (1993) 97–103.https://doi.org/10.1007/BF02941335Google Scholar
  8. J. Matkowski: The converse theorem for the Minkowski inequality, Indag. Math., New Ser. 15 (2004) 73–83.https://doi.org/10.1016/S0019-3577(04)90006-7Google Scholar
  9. J. Matkowski: Converse theorem for the Minkowski inequality, J. Math. Anal. Appl. 348 (2008) 277–287.https://doi.org/10.1016/j.jmaa.2008.07.044Google Scholar
  10. J. Matkowski: Pexider type generalization of the Minkowski inequality, J. Math. Anal. Appl. 393 (2012) 298–310.https://doi.org/10.1016/j.jmaa.2012.03.045Google Scholar
  11. J. Matkowski: Subhomogeneity and subadditivity of the Lp-norm like functionals, J. Math. Anal. Appl. 404 (2013) 172–184.https://doi.org/10.1016/j.jmaa.2013.03.005Google Scholar
  12. J. Matkowski: Fixed point theorem in a uniformly convex paranormed space and its application, Topology Appl. 160 (2013) 524–531.https://doi.org/10.1016/j.topol.2013.01.004Google Scholar
  13. H. P. Mulholland: On generalizations of Minkowski inequality in the form of a triangle inequality, Proc. Lond. Math. Soc., II. Ser. 51 (1950) 294–307.https://doi.org/10.1112/plms/s2-51.4.294Google Scholar
  14. J. Musielak: Introduction to Functional Analysis (in Polish), PWN, Warszawa (1989).Google Scholar
  15. L. Pasicki: A basic fixed point theorem, Bull. Pol. Acad. Sci., Math. 54 (2006) 85–88.https://doi.org/10.4064/ba54-1-8Google Scholar
  16. L. Pasicki: Bead spaces and fixed point theorems, Topology Appl. 156 (2009) 1811–1816.https://doi.org/10.1016/j.topol.2009.03.042Google Scholar
  17. L. Pasicki: Uniformly convex spaces, bead spaces, and equivalence conditions, Czech. Math. J. 61 (2011) 383–388.https://doi.org/10.1007/s10587-011-0082-2Google Scholar
  18. L. Pasicki: A comment to Matkowski’s paper, Topology Appl. 160 (2013) 951–952.https://doi.org/10.1016/j.topol.2013.02.008Google Scholar
  19. A. Wilansky: Functional Analysis, Blaisdell, New York (1964).Google Scholar
  20. W. Wnuk: Orlicz spaces cannot be normed analogously to Lp spaces, Indag. Math. 46 (1984) 357–359.https://doi.org/10.1016/1385-7258(84)90034-9Google Scholar

Suggested citation

J. Jarczyk, J. Matkowski. “Uniform Convexity of Paranormed Generalizations of L^p Spaces.” Journal of Convex Analysis 22 (2015), No. 1, 117–144. https://doi.org/10.68381/jca22007

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.