Abstract

Let fc(r)=∑n=0∞ecnrnf_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n be an analytic function; c=(cn)∈l∞{\bf c}=(c_n)\in l_\infty. We assume that rr is some logarithmically convex and lower semicontinuous functional on a locally convex topological space LL. In this paper we derive a formula on the Legendre-Fenchel transform of a functional λ^(c,φ)=ln⁡fc(eλ(φ)) \widehat{\lambda}({\bf c},\varphi)= \ln f_{\bf c}(e^{\lambda(\varphi)})\, where λ(φ)=ln⁡r(φ)\lambda(\varphi)=\ln r(\varphi) (φ∈L\varphi\in L). In this manner we generalize to the infinite case Theorem 3.1 of the paper of U. Ostaszewska and K. Zajkowski ["Legendre-Fenchel transform of the spectral exponent of polynomials of weighted composition operators", Positivity, DOI 10.1007/s11117-009-0023-6].

Author information

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

Krzysztof Zajkowski

Institute of Mathematics, University of Bialystok, Akademicka 2, 15-267 Bialystok, Poland

kryza@math.uwb.edu.pl

Keywords

  • Legendre-Fenchel transform
  • logarithmic convexity
  • log-exponential function
  • entropy function
  • spectral radius
  • weighted composition operators

Mathematics Subject Classification

44A15, 47A10, 47B37

References

  1. A. Antonevich, V. Bakhtin, A. Lebedev: Thermodynamics and spectral radius, Nonlinear Phenom. Complex Syst. 4(4) (2001) 318–321.Google Scholar
  2. A. Antonevich, V. Bakhtin, A. Lebedev: On t-entropy and variational principle for the spectral radii of transfer and weighted shift operators, Ergodic Theory Dyn. Syst. 31(4) (2011) 995–1042.https://doi.org/10.1017/S0143385710000210Google Scholar
  3. C. Combari, M. Laghdir, L. Thibault: Sous-différentiels de fonctions convexes composées, Ann. Sci. Math. Qué. 18 (1994) 119–148.Google Scholar
  4. C. Combari, M. Laghdir, L. Thibault: A note on subdifferentials of convex composite functionals, Arch. Math. 67 (1996) 239–252.https://doi.org/10.1007/BF01195240Google Scholar
  5. J.-B. Hiriart-Urruty: A note on the Legendre-Fenchel transform of convex composite functions, in: Nonsmooth Mechanics and Analysis, P. Alart et al. (ed.), Adv. Mech. Math. 12, Springer, New York (2006) 35–46.https://doi.org/10.1007/0-387-29195-4_4Google Scholar
  6. J.-E. Martínez-Legaz, I. Singer: Some conjugation formulas and subdifferential formulas of convex analysis revisited, J. Math. Anal. Appl. 313 (2006) 717–729.https://doi.org/10.1016/j.jmaa.2005.03.068Google Scholar
  7. U. Ostaszewska, K. Zajkowski: Legendre-Fenchel transform of the spectral exponent of polynomials of weighted composition operators, Positivity 14(3) (2010) 373–381.https://doi.org/10.1007/s11117-009-0023-6Google Scholar
  8. U. Ostaszewska, K. Zajkowski: Variational principle for the spectral exponent of polynomials of weighted composition operators, J. Math. Anal. Appl. 361 (2010) 246–251.https://doi.org/10.1016/j.jmaa.2009.08.057Google Scholar
  9. U. Ostaszewska, K. Zajkowski: Legendre-Fenchel transform of the spectral exponent of analytic functions of weighted composition operators, J. Convex Analysis 18(2) (2011) 367–377.https://doi.org/10.68381/jca18020Google Scholar
  10. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Grundlehren der mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  11. H. H. Schaefer: Banach Lattices and Positive Operators, Springer, Berlin (1974).https://doi.org/10.1007/978-3-642-65970-6Google Scholar

Suggested citation

K. Zajkowski. “Convex Conjugates of Analytic Functions of Logarithmically Convex Functionals.” Journal of Convex Analysis 20 (2013), No. 1, 243–252. https://doi.org/10.68381/jca20015

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.