This work is a natural continuation of the ongoing research started by the authors in a previous article [Describing multiplicative convex functions, J. Convex Analysis 27 (2020) 935–942] with respect to multiplicative convex functions, their continuity properties and their generalizations. Some results regarding algebraic genericity are also presented and open questions are posed.

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

M. E. Martínez-Gómez

Dep. de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain

mariae24@ucm.es

Gustavo A. Muñoz-Fernández

Dep. de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain

gustavo_fernandez@mat.ucm.es

Juan B. Seoane-Sepúlveda

Inst. de Matemática Interdisciplinar, Dep. de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain

jseoane@ucm.es
  1. A. Aizpuru, C. Pérez-Eslava, F. J. García-Pacheco, J. B. Seoane-Sepúlveda: Lineability and coneability of discontinuous functions on R, Publ. Math. Debrecen 72/1-2 (2008) 129–139.https://doi.org/10.5486/PMD.2008.3863Google Scholar
  2. G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen: Generalized convexity and inequalities, J. Math. Analysis Appl. 335(2) (2007) 1294–1308.https://doi.org/10.1016/j.jmaa.2007.02.016Google Scholar
  3. R. M. Aron, L. Bernal González, D. M. Pellegrino, J. B. Seoane Sepúlveda: Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton (2016).https://doi.org/10.1201/b19277Google Scholar
  4. R. Aron, V. I. Gurariy, J. B. Seoane: Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133/3 (2005) 795–803.https://doi.org/10.1090/S0002-9939-04-07533-1Google Scholar
  5. E. F. Bechenbach: Convex functions, Bull. Amer. Math. Soc. 54 (1948) 439–460.https://doi.org/10.1090/S0002-9904-1948-08994-7Google Scholar
  6. L. Bernal-González, H. J. Cabana-Méndez, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: On the dimension of subspaces of continuous functions attaining their maximum finitely many times, Trans. Amer. Math. Soc. 373/5 (2020) 3063–3083.https://doi.org/10.1090/tran/8054Google Scholar
  7. L. Bernal-González, D. Pellegrino, J. B. Seoane-Sepúlveda: Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51/1 (2014) 71–130.https://doi.org/10.1090/S0273-0979-2013-01421-6Google Scholar
  8. K. C. Ciesielski, J. B. Seoane-Sepúlveda: Differentiability versus continuity: restriction and extension theorems and monstrous examples, Bull. Amer. Math. Soc. (N.S.) 56/2 (2019) 211–260.https://doi.org/10.1090/bull/1635Google Scholar
  9. P. H. Enflo, V. I. Gurariy, J. B. Seoane-Sepúlveda: Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc. 366/2 (2014) 611–625.https://doi.org/10.1090/S0002-9947-2013-05747-9Google Scholar
  10. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler: Banach Space Theory: The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics, Springer, New York (2010).https://doi.org/10.1007/978-1-4419-7515-7Google Scholar
  11. P. Jiménez-Rodríguez, M. E. Martínez-Gómez, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: Describing multiplicative convex functions, J. Convex Analysis 27/3 (2020) 935–942.https://doi.org/10.68381/jca27047Google Scholar
  12. P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, E. Sáez-Maestro, J. B. Seoane-Sepúlveda: Algebraic genericity and the differentiability of the convolution, J. Approx. Theory 241 (2019) 86–106.https://doi.org/10.1016/j.jat.2019.01.002Google Scholar
  13. K. Keimel, W. Roth: Ordered Cones and Approximation, Lecture Notes in Mathematics 1517, Springer, Berlin (1992).https://doi.org/10.1007/BFb0089190Google Scholar
  14. C. T. Ng: On midconvex functions with midconcave bounds, Proc. Amer. Math. Soc. 102 (1988) 538–540.https://doi.org/10.1090/S0002-9939-1988-0928975-7Google Scholar
  15. C. P. Niculescu: Convexity according to the geometric mean, Math. Inequal. Appl. 2 (2000) 155–167.https://doi.org/10.7153/mia-03-19Google Scholar
  16. J. B. Seoane: Chaos and Lineability of Pathological Phenomena in Analysis, PhD Thesis, Kent State University, ProQuest LLC, Ann Arbor (2006).Google Scholar

P. Jiménez-Rodríguez, M. E. Martínez-Gómez, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda. “Generalizing Multiplicative Convex Functions.” Journal of Convex Analysis 28 (2021), No. 3, 939–950. https://doi.org/10.68381/jca28054