Some New Methods for Generating Convex Functions

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


We present some new methods for constructing convex functions. One of the methods is based on the composition of a convex function of several variables which is separately monotone with convex and concave functions. Using several well-known results on the composition of convex and quasi-convex functions we build new convex, quasi-convex, concave, and quasi-concave functions. The third section is dedicated to the study of convexity property of symmetric Archimedean functions. In the fourth section the asymmetric Archimedean function is considered. A classical example of such a function is the Bellman function. The fifth section is dedicated to the study of convexity/concavity of symmetric polynomials. In the sixth section a new proof of Chandler–Davis theorem is given. Starting from symmetric convex functions defined on finite dimensional spaces we build several convex functions of hermitian matrices. The seventh section is dedicated to a generalization of Muirhead’s theorem and to some applications of it. The last section is dedicated to the construction of convex functions based on Taylor remainder series.

2010 AMS Subject Classification

26A51; 26B25; 26D05; 26D07; 26D10; 41A36; 41A58 


  1. 1.
    M. Aguiar, C. Andre, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I.M. Isaacs, A. Jedwab, et al., Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. Adv. Math. 229(4), 2310–2337 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    E.E. Allen, The descent monomials and a basis for the diagonally symmetric polynomials. J. Algebra Combin. 3, 5–16 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    D. Andrica, M.O. Drimbe, On some inequalities involving isotonic functionals. Math. Anal. Numer. Theorie Approx. 17(1), 1–7 (1988)MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Becheanu, International Mathematical Olympiads 1959–2000, Problems, Solutions, Results (Academic Distribution Center, Freeland, 2001)Google Scholar
  5. 5.
    R. Bellman, On an inequality concerning an indefinite form. Am. Math. Mon. 63, 108–109 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    H. Bergstrom, A triangle inequality for matrices, in Den Ilte Skandinauiske Matematikerkongress (1949), pp. 264–267Google Scholar
  7. 7.
    C. Bertone, The Euler characteristic as a polynomial in the Chern classes. Int. J. Algebra 2, 757–769 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    E.C. Boadi, Symmetric Polynomials, Combinatorics and Mathematical, Master Thesis, University of Ottawa, Canada, 2016.
  9. 9.
    J.M. Borwein, A.S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples (Springer, Berlin, 2010)Google Scholar
  10. 10.
    J.M. Borwein, J.D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples (Cambridge University Press, Cambridge, 2010)zbMATHCrossRefGoogle Scholar
  11. 11.
    W.Y.C. Chen, C. Krattenthaler, A.L.B. Yang, The flagged Cauchy determinant. Graphs Combin. 21, 51–62 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    K.M. Chong, Spectral order preserving matrices and Muirhead’s theorem. Trans. Am. Math. Soc. 200, 437–444 (1974)MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. Curnier, Q.C. He, P. Zysset, Conewise linear elastic materials. J. Elasticity 37, 1–38 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Z. Cvetkovski, Inequalities.Theorems, Techniques and Selected Problems (Springer, Berlin, 2012)zbMATHCrossRefGoogle Scholar
  15. 15.
    C. Davis, All convex invariant functions of Hermitian matrices. Arch. Math. 8(4), 276–278 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    D.E. Daykin, Generalisation of the Muirhead-Rado inequality. Proc. Am. Math. Soc. 30(1), 84–86 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    D. Djukić, V. Janković, I. Matić, N. Petrović, The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959–2004 (Springer, Berlin, 2006)zbMATHGoogle Scholar
  18. 18.
    M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-I. Appl. Math. Comput. 145, 23–32 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-II. Appl. Math. Comput. 146, 759–769 (2003)zbMATHGoogle Scholar
  20. 20.
    M.E.A. El-Mikkawy, Explicit inverse of a generalized Vandermonde matrix. Appl. Math. Comput. 146, 643–651 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    M.E.A. El-Mikkawy, T. Sogabe, Notes on particular symmetric polynomials with applications. Appl. Math. Comput. 215, 3311–3317 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    R. Fletcher, A new variational result for quasi-Newton formulae. SIAM J. Optim. 1, 18–21 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    D. Gao, P. Neff, I. Roventa, C. Thiel, On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor. J. Elast. 127, 303–308 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    D.J.H. Garling, Inequalities: A Journey into Linear Analysis (Cambridge University Press, Cambridge, 2007)zbMATHCrossRefGoogle Scholar
  25. 25.
    I.M. Gessel, Symmetric functions and P-recursiveness. J. Combin.Theory Ser. A 53, 257–285 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    V. Gorin, G. Panova, Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43(6), 3052–3132 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, 2nd edn. (Cambridge University Press, Cambridge, 1952)zbMATHGoogle Scholar
  28. 28.
    S. Helgason, Differential Geometry and Symmetric Spaces, vol. 341 (American Mathematical Society, Providence, 2001)zbMATHGoogle Scholar
  29. 29.
    T. Hoang, A. Seeger, On conjugate functions, subgradients, and directional derivatives of a class of optimality criteria in experimental design. Statistics 22, 349–368 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1994)zbMATHGoogle Scholar
  31. 31.
    D.L. Hydorn, R.J. Muirhead, Polynomial estimation of eigenvalues. Commun. Stat. Theory Meth. 28 , 581–596 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    R. Jozsa, G. Mitchison, Symmetric polynomials in information theory: entropy and subentropy. J. Math. Phys. 56(6), 062201 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    E.C. Kemble, The Fundamental Principles of Quantum Mechanics (Dover, New York, 1958)zbMATHGoogle Scholar
  34. 34.
    B. Kimelfeld, A generalization of Muirhead’s theorem. Linear Algebra Appl. 216, 205–209 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    E.P. Klement, R. Mesiar, E. Pap, Generated triangular norms. Kybernetika, 36(3), 363–377 (2000)MathSciNetzbMATHGoogle Scholar
  36. 36.
    X. Lachaume, On the concavity of a sum of elementary symmetric polynomials. ArXiv e-prints, arXiv:1712.10327 (2017)Google Scholar
  37. 37.
    S. Lehmich, P. Neff, J. Lankeit, On the convexity of the function C f(det C) on positive-definite matrices. Math. Mech. Solids 19(4), 369–375 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    A.S. Lewis, Convex analysis on the Hermitian matrices. SIAM J. Optim. 6, 164–177 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A.S. Lewis, Derivatives of spectral functions. Math. Oper. Res. 6, 576–588 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    A.S. Lewis, The mathematics of eigenvalue optimization. Math. Programm. 97(1–2), 155–176 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    A.S. Lewis, M.L. Overton, Eigenvalue optimization. Acta Numer. 5 , 149–190 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    A.S. Lewis, H. Sendov, Twice differentiable spectral functions. SIAM J. Matrix Anal. Appl. 23(2), 368–386 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    L. Losonczi, Z. Pales, Inequalities for indefinite forms. J. Math. Anal. Appl. 205, 148–156 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    S.V. Lyudkovskii, Compact relationships between invariants of classical Lie groups and elementary symmetric polynomials. Theory Math. Phys. 89, 1281–1286 (1991)MathSciNetCrossRefGoogle Scholar
  45. 45.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. (The Clarendon Press, Oxford University Press, New York, 1995)Google Scholar
  46. 46.
    P. Major, The limit behavior of elementary symmetric polynomials of I.I.D. random variables when their order tends to infinity. Ann. Probab. 27, 1980–2010 (1999)MathSciNetzbMATHGoogle Scholar
  47. 47.
    R.B. Manfrino, J.A.G. Ortega, R.V. Delgado, Inequalities: A Mathematical Olympiad Approach (Springer, Berlin, 2010)zbMATHGoogle Scholar
  48. 48.
    M. Marcus, L. Lopes, Symmetric functions and Hermitian matrices. Can. J. Math. 9, 305–312 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    A.W. Marshal, I. Olkin, B.C. Arnold, Inequalities : Theory of Majorization and Its Applications 2nd edn. (Springer, Berlin, 2011)CrossRefGoogle Scholar
  50. 50.
    A.W. Marshall, F. Proschan, An inequality for convex functions involving majorization. J. Math. Anal. Appl. 12, 87–90 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    C.D. Meyer, Matrix analysis and Applied Linear Algebra (SIAM, Philadelphia, 2000)CrossRefGoogle Scholar
  52. 52.
    D.S. Mitrinović, J. Pečarić, Unified treatment of some inequalities for mixed means. sterreich. Akad. Wiss. Math. Nat. Kl. Sitzungsber. II, 197(8–10), 391–397 (1988)Google Scholar
  53. 53.
    D.S. Mitrinović, J. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis (Kluwer, Dordrecht, 1993)zbMATHCrossRefGoogle Scholar
  54. 54.
    V.V. Monov, A family of symmetric polynomials of the eigenvalues of a matrix. Linear Algebra Appl. 429, 2199–2208 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    W.W. Muir, Inequalities concerning the inverses of positive definite matrices. Proc. Edinb. Math. Soc 19,109–113 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    R.F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinb. Math. Soc. 21, 144–157 (1903)zbMATHCrossRefGoogle Scholar
  57. 57.
    Y.E. Nesterov, A.S. Nemirovskii, Optimization over positive semidefinite matrices: Mathematical Background and User’s Manual (USSR Academy Science Center Economics and Mathematical Institute, Moscow, 1990)Google Scholar
  58. 58.
    C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach (Springer, Berlin, 2018)zbMATHCrossRefGoogle Scholar
  59. 59.
    A. Pazman, Foundations of optimum experimental design, in Mathematics and its Applications. East European Series (D. Reidel, Boston, 1986)Google Scholar
  60. 60.
    J.E. Pečarić, Remark on an inequality of S. Gabler. J. Math. Anal. Appl. 184(1), 19–21 (1994)zbMATHCrossRefGoogle Scholar
  61. 61.
    J.E. Pečarić, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings, and Statistical Applications. Mathematics in Science and Engineering, vol. 187 (Academic Press, London, 1992)Google Scholar
  62. 62.
    J.E. Pečarić, V. Volenec, Interpolation of the Jensen inequality with some applications. Sitzungsber. Oesterr. Akad. Wiss. Abt. II 197, 463–467 (1988)MathSciNetzbMATHGoogle Scholar
  63. 63.
    F. Proschan, J. Sethuraman, Two generalizations of Muirhead’s theorem. Bull. Calcutta Math. Soc. 69, 341–344 (1977)MathSciNetzbMATHGoogle Scholar
  64. 64.
    T. Puong, Diamonds in Mathematical Inequalities (Hanoi Publishing House, 2007)Google Scholar
  65. 65.
    A.W. Roberts, D.E. Varberg, Convex Functions (Academic Press, London, 1973)zbMATHGoogle Scholar
  66. 66.
    J.V. Ryff, On Muirhead’s theorem. Pacific J. Math. 21(3), 567–576 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    L.I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955)zbMATHGoogle Scholar
  68. 68.
    H.J. Schmidt, J. Schnack, Partition functions and symmetric polynomials. Am. J. Phys. 70, 53–57 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    H.J. Schmidt, J. Schnack, Symmetric polynomials in physics, in ed. by J.-P. Gazeau et al., GROUP 24, Physical and Mathematical Aspects of Symmetries (Institute of Physics Publishing, Bristol and Philadelphia, 2002), pp. 147–153Google Scholar
  70. 70.
    L.J. Schulman, Muirhead-Rado inequality for compact groups. Positivity 13, 559–574 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    I. Schur, Uber eine Klasse von Mittelbildungen mit Anwendungdie Determinanten. Theorie Sitzungsber. Berlin. Math. Gesellschaft 22, 9–29 (1923)Google Scholar
  72. 72.
    A. Seeger, Convex analysis of spectrally defined matrix functions. SIAM J. Optim. 7, 679–696 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    M. Silhavy, The convexity of C →h(C). Tech. Mech. 35(1), 60–61 (2015)Google Scholar
  74. 74.
    T. Sogabe, M.E.A. El-Mikkawy, On a problem related to the Vandermonde determinant. Disc. Appl. Math. 157, 2997–2999 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    S.J. Spector, A note on the convexity of C →h(C). J. Elast. 118(2), 251–256 (2015)zbMATHCrossRefGoogle Scholar
  76. 76.
    S. Sra, New concavity and convexity results for symmetric polynomials and their ratios. Linear Multilinear Algebra 1–9 (2018).
  77. 77.
    R. Stanley, Some combinatorial properties of Jack symmetric functions. Adv. Math. 77, 76–115 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    L. Tibiletti, Quasi-concavity property of multivariate distribution functions. Ratio Math. 9, 27–36 (1995)Google Scholar
  79. 79.
    J. Tkadlec, Triangular norms with continuous diagonals. Tatra Mt. Math. Publ. 16, 187–195 (1999)MathSciNetzbMATHGoogle Scholar
  80. 80.
    M. Torki, First- and second-order epi-differentiability in eigenvalue optimization. J. Math. Anal. Appl. 234, 391–416 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    N.K. Tsing, M.K.H. Fan, E.I. Verriest, On analyticity of functions involving eigenvalues. Linear Algebra Appl. 207, 159–180 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    L. Vandenberghe, S. Boyd, S.P. Wu, Determinant maximization with linear matrix inequality constraints. SIAM J. Matrix Anal. Appl. 19(2), 499–533 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    H. Wolkowicz, Measures for symmetric rank-one updates. Math. Oper. Res. 19, 815–830 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    B.J. Venkatachala, Inequalities. An Approach Through Problems (Springer, Berlin, 2018)Google 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

Personalised recommendations