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Convexity Revisited: Methods, Results, and Applications

  • Dorin AndricaEmail author
  • Sorin Rădulescu
  • Marius Rădulescu
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 151)

Abstract

We present some new aspects involving strong convexity, the pointwise and uniform convergence on compact sets of sequences of convex functions, circular symmetric inequalities and bistochastic matrices with examples and applications, the convexity properties of the multivariate monomial, and Schur convexity.

2010 AMS Subject Classification

41A36; 26D20 

References

  1. 1.
    D. Andrica, The Refinements of Some Geometric Inequalities (Romanian) (Lucrarile seminarului “Theodor Angheluta”, Cluj-Napoca, 1983), pp. 1–2Google Scholar
  2. 2.
    D. Andrica, On a maximum problem, in Proceedings of the Colloquium on Approximation and Optimization (Cluj-Napoca, October 25–27, 1984) (Univ. of Cluj-Napoca, Cluj-Napoca, 1985), pp. 173–177Google Scholar
  3. 3.
    D. Andrica, The Refinement of Some Geometric Inequalities, Seminar on Geometry, Cluj-Napoca, Preprint Nr. 10 (1986), pp. 71–76MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Andrica, An abstract result in approximation theory, in Approximation and Optimization: Proceedings of the International Conference on Approximation and Optimization – ICAOR, Cluj-Napoca, vol. II, July 29–Aug 1, 1996, pp. 9–12Google Scholar
  5. 5.
    D. Andrica, Note on an abstract approximation theorem, in Approximation Theory and Applications, ed. by Th. M. Rassias (Hadronic Press, Palm Harbor, 1999), pp. 1–10zbMATHGoogle Scholar
  6. 6.
    D. Andrica, C. Badea, Jensen’s and Jessen’s Inequality, Convexity-preserving and Approximating Polynomial Operators and Korovkin’s Theorem, “Babes-Bolyai” University, Cluj-Napoca, Seminar on Mathematical Analysis, Preprint Nr. 4 (1986), pp. 7–16zbMATHGoogle Scholar
  7. 7.
    D. Andrica, M.O. Drimbe, On some inequalities involving isotonic functionals. Mathematica – Rev. Anal. Numér. Théor. Approx. 17(1), 1–7 (1988)MathSciNetzbMATHGoogle Scholar
  8. 8.
    D. Andrica, L. Mare, An inequality concerning symmetric functions and some applications, in Recent Progress in Inequalities. Mathematics and Its Applications, vol. 430 (Kluwer Academic, Dordrecht, 1998), pp. 425–431CrossRefGoogle Scholar
  9. 9.
    D. Andrica, D.-Şt. Marinescu, New interpolation inequalities to Euler’s R ≥ 2r. Forum. Geom. 17, 149–156 (2017)Google Scholar
  10. 10.
    D. Andrica, C. Mustăţa, An abstract Korovkin type theorem and applications. Studia Univ. Babeş-Bolyai Math. 34, 44–51 (1989)MathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Andrica, I. Rasa, The Jensen inequality: refinements and applications. Mathematica- L’Analyse Numerique et la Theorie de l’Approximation 14(2), 105–108 (1985)MathSciNetzbMATHGoogle Scholar
  12. 12.
    D. Andrica, I. Rasa, Gh. Toader, On some inequalities involving convex sequences. Mathematica-Revue L’analyse numerique et la theorie de l’approximation 13(1), 5–7 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    P.R. Beesak, J. Pečarić, On Jessen’s inequality for convex functions. J. Math. Anal. Appl. 110, 536–552 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    R. Bojanic, J. Roulier, Approximation of convex functions by convex splines and convexity-preserving continuous linear operators. L’Analyse Numerique et la Theorie de l’Approximation 3, 143–150 (1974)MathSciNetzbMATHGoogle Scholar
  15. 15.
    S. Bontas, Estimation for lower bounds for a symmetric functions, Aalele St. Univ. “Al.I.Cuza” Iasi, Tomul XLVII, s.I a, Matematica (2001), f.1, 41–50Google Scholar
  16. 16.
    B.Y. Chen, Classification of h-homogeneous production functions with constant elasticity of substitution. Tamkang J. Math. 43, 321–328 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    C.W. Cobb, P.H. Douglas, A theory of production. Amer. Econom. Rev. 18, 139–165 (1928)Google Scholar
  18. 18.
    J.P. Crouzeix, Criteria for generalized convexity and generalized monotonicity in the differentiable case, Chapter 2 in Handbook of Generalized Convexity and Generalized Monotonicity, ed. by N. Hadjisavvas, S. Komlósi, S. Schaible (Springer, New York, 2005), pp. 89–119Google Scholar
  19. 19.
    Z. Cvetkovski, Inequalities. Theorems, Techniques and Selected Problems (Springer, Berlin, 2012)zbMATHCrossRefGoogle Scholar
  20. 20.
    D. Djukić, V. Janković, I. Matić, N. Petrović, The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959–2004 (Springer Science Business Media, Berlin, 2006)zbMATHGoogle Scholar
  21. 21.
    S.S. Dragomir, J. Pečarić, L.E. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21, 335–341 (1995)MathSciNetzbMATHGoogle Scholar
  22. 22.
    S.S. Dragomir, J. Ŝunde, J. Asenstorfer, On an inequality by Andrica and Raşa and its application for the Shannon and Rényi’s entropy. J. Ksiam 6(2), 31–42 (2002)Google Scholar
  23. 23.
    M. Eliasi, On extremal properties of general graph entropies. MATCH Commun. Math. Comput. Chem. 79, 645–657 (2018)MathSciNetGoogle Scholar
  24. 24.
    E.K. Godunova, V.I. Levin, Neravenstva dlja funkcii širokogo klassa, soderžaščego vypuklye, monotonnye i nekotorye drugie vidy funkcii, Vyčislitel. Mat. i. Mat. Fiz. Mežvuzov. Sb. Nauč. Trudov (MGPI, Moskva, 1985) pp. 138–142Google Scholar
  25. 25.
    H.H. Gonska, J. Meier, A bibliography on approximation of functions by Berstein type operators (1955–1982), in Approximation Theory IV (Proc. Int. Symp. College Station, 1983), ed. by C.E. Chui, L.L. Schumaker, J.D. Ward (Academic Press, New York, 1983), pp. 739–785Google Scholar
  26. 26.
    J.B. Hiriart-Urruty, C. Lemarechal, Fundamentals of Convex Analysis (Springer Science & Business Media, Berlin, 2012)zbMATHGoogle Scholar
  27. 27.
    B. Jessen, Bemaerkinger om Konvexe Functioner og Uligheder immelem Middelvaerdier I, Mat. Tidsskrift, B(1931), 17–28.Google Scholar
  28. 28.
    C. Joita, P. Stanica, Inequalities related to rearrangements of powers and symmetric polynomials. J. Inequal. Pure Appl. Math. 4, Article 37, 4 (2003) (electronic)Google Scholar
  29. 29.
    P. Kim Hung, The stronger mixing variables method. Math. Reflect. 6, 1–8 (2006)Google Scholar
  30. 30.
    P. Kosmol, D. Müller-Wichards, Stability for families of nonlinear equations. Isvetia NAN Armenii. Mathematika, 41(1), 49–58 (2006)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Y.C. Li, C.C. Yeh, Some characterizations of convex functions. Comput. Math. Appl. 59, 327–337 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Al. Lupaş, Some properties of the linear positive operators (III). Rev. Anal. Numér. Théor. Approx. 3, 47–61 (1974)Google Scholar
  33. 33.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. (The Clarendon Press, Oxford University Press, New York, 1995)Google Scholar
  34. 34.
    M. Marcus, L. Lopes, Symmetric functions and Hermitian matrices. Can. J. Math. 9, 305–312 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    A.W. Marshal, I. Olkin, B.C. Arnold, Inequalities: Theory of Majoration and Its Applications, 2nd edn. (Springer, Berlin, 2011)CrossRefGoogle Scholar
  36. 36.
    F. Mat, On nonnegativity of symmetric polynomials. Amer. Math. Monthly 101, 661–664 (1994)MathSciNetCrossRefGoogle Scholar
  37. 37.
    J.B. McLeod, On four inequalities in symmetric functions. Proc. Edinb. Math. Soc. 11, 211–219 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    C.D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, 2000)CrossRefGoogle Scholar
  39. 39.
    G.V. Milovanovic, A.S. Cvetkovic, Some inequalities for symmetric functions and an application to orthogonal polynomials. J. Math. Anal. Appl. 311, 191–208 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    D.S. Mitrinović, Analytic Inequalities (Springer, Berlin, 1970)zbMATHCrossRefGoogle Scholar
  41. 41.
    D.S. Mitrinović, J. Pečarić, Note on a class of functions of Godunova and Levin. C. R. Math. Rep. Acad. Sci. Can. 12, 33–36 (1990)MathSciNetzbMATHGoogle Scholar
  42. 42.
    D.S. Mitrinović, J. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis (Kluwer Academic, Dordrecht, 1993)zbMATHCrossRefGoogle Scholar
  43. 43.
    C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach (Springer, Berlin, 2018)zbMATHCrossRefGoogle Scholar
  44. 44.
    J. Pečarić, Note on multidimensional generalization of Slater’s inequality. J. Approx. Theory 44, 292–294 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    J.E. Pečarić, D. Andrica, Abstract Jessen’s inequality for convex functions and applications. Mathematica 29(52), 1, 61–65 (1987)Google Scholar
  46. 46.
    T. Popoviciu, Sur l’approximation des fonctions convexes d’ordre supérieur. Mathematica (Cluj) 10, 49–54 (1935)Google Scholar
  47. 47.
    E. Popoviciu, Teoreme de medie din analiza matematica si legatura lor cu teoria interpolarii (Romanian) (Editura Dacia, Cluj-Napoca, 1972)zbMATHGoogle Scholar
  48. 48.
    T. Puong, Diamonds in Mathematical Inequalities (Hanoi Pub. House, Ha Noi City, 2007)Google Scholar
  49. 49.
    M. Rădulescu, S. Rădulescu, P. Alexandrescu, On Schur inequality and Schur functions. Annals Univ. Craiova, Math. Comp. Sci. Ser. 32, 202–208(2005)Google Scholar
  50. 50.
    M. Rădulescu, S. Rădulescu, P. Alexandrescu, On the Godunova – Levin – Schur class of functions. Math. Inequal. Appl. 12(4), 853–962 (2009)MathSciNetzbMATHGoogle Scholar
  51. 51.
    I. Rasa, On the inequalities of Popoviciu and Rado. Mathematica-L’Analyse Numerique et la Theorie de l’Approximation, Tome 11, 147–149 (1982)MathSciNetzbMATHGoogle Scholar
  52. 52.
    A.W. Roberts, D.E. Varberg, Convex Functions (Academic Press, New York, 1973)zbMATHGoogle Scholar
  53. 53.
    M.L. Slater, A companion inequality to Jensen’s inequality. J. Approx. Theory 32, 160–166 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    E.L. Stark, Berstein – Polynome, 1912–1955, in Functional Analysis and Approximation (Proc. Conf. Math. Res. Inst. Oberwolfach 1980) ed. by P.L. Buzer, B.Sz.-Nagy, E. Gorlich (Basel, Birkhauser, 1981), pp. 443–451Google Scholar
  55. 55.
    S. Varošanec, On h-convexity. Aust. J. Math. Anal. Appl. 326(1), 303–311 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    B.J. Venkatachala, Inequalities. An Approach Through Problems (Springer, Berlin, 2018)Google Scholar
  57. 57.
    V.I. Volkov, Convergence of sequences of linear positive operators in the space of continuous functions of two variables (Russian). Dokl. Akad. Nauk 115, 17–19 (1957)MathSciNetzbMATHGoogle Scholar
  58. 58.
    W. Wegmuller, Ausgleichung Lursch Bernestein-Polynome. Mitt. Verein. Schweiz. Versich-Math. 26, 15–59 (1938)zbMATHGoogle Scholar
  59. 59.
    E.M. Wright, A generalization of Schur’s inequality. Math. Gaz. 40, 217 (1956)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dorin Andrica
    • 1
    Email author
  • Sorin Rădulescu
    • 2
  • Marius Rădulescu
    • 2
  1. 1.Department of Mathematics“Babeş-Bolyai” UniversityCluj-NapocaRomania
  2. 2.Institute of Mathematical Statistics and Applied MathematicsBucharestRomania

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