Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
E.E. Allen, The descent monomials and a basis for the diagonally symmetric polynomials. J. Algebra Combin. 3, 5–16 (1994)
D. Andrica, M.O. Drimbe, On some inequalities involving isotonic functionals. Math. Anal. Numer. Theorie Approx. 17(1), 1–7 (1988)
M. Becheanu, International Mathematical Olympiads 1959–2000, Problems, Solutions, Results (Academic Distribution Center, Freeland, 2001)
R. Bellman, On an inequality concerning an indefinite form. Am. Math. Mon. 63, 108–109 (1957)
H. Bergstrom, A triangle inequality for matrices, in Den Ilte Skandinauiske Matematikerkongress (1949), pp. 264–267
C. Bertone, The Euler characteristic as a polynomial in the Chern classes. Int. J. Algebra 2, 757–769 (2008)
E.C. Boadi, Symmetric Polynomials, Combinatorics and Mathematical, Master Thesis, University of Ottawa, Canada, 2016. www.Physicsmysite.science.uottawa.ca/hsalmasi/report/thesis-evans.pdf
J.M. Borwein, A.S. Lewis, Convex Analysis and Nonlinear Optimization: Theory and Examples (Springer, Berlin, 2010)
J.M. Borwein, J.D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples (Cambridge University Press, Cambridge, 2010)
W.Y.C. Chen, C. Krattenthaler, A.L.B. Yang, The flagged Cauchy determinant. Graphs Combin. 21, 51–62 (2005)
K.M. Chong, Spectral order preserving matrices and Muirhead’s theorem. Trans. Am. Math. Soc. 200, 437–444 (1974)
A. Curnier, Q.C. He, P. Zysset, Conewise linear elastic materials. J. Elasticity 37, 1–38 (1995)
Z. Cvetkovski, Inequalities.Theorems, Techniques and Selected Problems (Springer, Berlin, 2012)
C. Davis, All convex invariant functions of Hermitian matrices. Arch. Math. 8(4), 276–278 (1957)
D.E. Daykin, Generalisation of the Muirhead-Rado inequality. Proc. Am. Math. Soc. 30(1), 84–86 (1971)
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)
M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-I. Appl. Math. Comput. 145, 23–32 (2003)
M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-II. Appl. Math. Comput. 146, 759–769 (2003)
M.E.A. El-Mikkawy, Explicit inverse of a generalized Vandermonde matrix. Appl. Math. Comput. 146, 643–651 (2003)
M.E.A. El-Mikkawy, T. Sogabe, Notes on particular symmetric polynomials with applications. Appl. Math. Comput. 215, 3311–3317 (2010)
R. Fletcher, A new variational result for quasi-Newton formulae. SIAM J. Optim. 1, 18–21 (1991)
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)
D.J.H. Garling, Inequalities: A Journey into Linear Analysis (Cambridge University Press, Cambridge, 2007)
I.M. Gessel, Symmetric functions and P-recursiveness. J. Combin.Theory Ser. A 53, 257–285 (1990)
V. Gorin, G. Panova, Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43(6), 3052–3132 (2015)
G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, 2nd edn. (Cambridge University Press, Cambridge, 1952)
S. Helgason, Differential Geometry and Symmetric Spaces, vol. 341 (American Mathematical Society, Providence, 2001)
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)
R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1994)
D.L. Hydorn, R.J. Muirhead, Polynomial estimation of eigenvalues. Commun. Stat. Theory Meth. 28 , 581–596 (1999)
R. Jozsa, G. Mitchison, Symmetric polynomials in information theory: entropy and subentropy. J. Math. Phys. 56(6), 062201 (2015)
E.C. Kemble, The Fundamental Principles of Quantum Mechanics (Dover, New York, 1958)
B. Kimelfeld, A generalization of Muirhead’s theorem. Linear Algebra Appl. 216, 205–209 (1995)
E.P. Klement, R. Mesiar, E. Pap, Generated triangular norms. Kybernetika, 36(3), 363–377 (2000)
X. Lachaume, On the concavity of a sum of elementary symmetric polynomials. ArXiv e-prints, arXiv:1712.10327 (2017)
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)
A.S. Lewis, Convex analysis on the Hermitian matrices. SIAM J. Optim. 6, 164–177 (1996)
A.S. Lewis, Derivatives of spectral functions. Math. Oper. Res. 6, 576–588 (1996)
A.S. Lewis, The mathematics of eigenvalue optimization. Math. Programm. 97(1–2), 155–176 (2003)
A.S. Lewis, M.L. Overton, Eigenvalue optimization. Acta Numer. 5 , 149–190 (1996)
A.S. Lewis, H. Sendov, Twice differentiable spectral functions. SIAM J. Matrix Anal. Appl. 23(2), 368–386 (2001)
L. Losonczi, Z. Pales, Inequalities for indefinite forms. J. Math. Anal. Appl. 205, 148–156 (1997)
S.V. Lyudkovskii, Compact relationships between invariants of classical Lie groups and elementary symmetric polynomials. Theory Math. Phys. 89, 1281–1286 (1991)
I.G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. (The Clarendon Press, Oxford University Press, New York, 1995)
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)
R.B. Manfrino, J.A.G. Ortega, R.V. Delgado, Inequalities: A Mathematical Olympiad Approach (Springer, Berlin, 2010)
M. Marcus, L. Lopes, Symmetric functions and Hermitian matrices. Can. J. Math. 9, 305–312 (1957)
A.W. Marshal, I. Olkin, B.C. Arnold, Inequalities : Theory of Majorization and Its Applications 2nd edn. (Springer, Berlin, 2011)
A.W. Marshall, F. Proschan, An inequality for convex functions involving majorization. J. Math. Anal. Appl. 12, 87–90 (1965)
C.D. Meyer, Matrix analysis and Applied Linear Algebra (SIAM, Philadelphia, 2000)
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)
D.S. Mitrinović, J. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis (Kluwer, Dordrecht, 1993)
V.V. Monov, A family of symmetric polynomials of the eigenvalues of a matrix. Linear Algebra Appl. 429, 2199–2208 (2008)
W.W. Muir, Inequalities concerning the inverses of positive definite matrices. Proc. Edinb. Math. Soc 19,109–113 (1974)
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)
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)
C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach (Springer, Berlin, 2018)
A. Pazman, Foundations of optimum experimental design, in Mathematics and its Applications. East European Series (D. Reidel, Boston, 1986)
J.E. Pečarić, Remark on an inequality of S. Gabler. J. Math. Anal. Appl. 184(1), 19–21 (1994)
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)
J.E. Pečarić, V. Volenec, Interpolation of the Jensen inequality with some applications. Sitzungsber. Oesterr. Akad. Wiss. Abt. II 197, 463–467 (1988)
F. Proschan, J. Sethuraman, Two generalizations of Muirhead’s theorem. Bull. Calcutta Math. Soc. 69, 341–344 (1977)
T. Puong, Diamonds in Mathematical Inequalities (Hanoi Publishing House, 2007)
A.W. Roberts, D.E. Varberg, Convex Functions (Academic Press, London, 1973)
J.V. Ryff, On Muirhead’s theorem. Pacific J. Math. 21(3), 567–576 (1967)
L.I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955)
H.J. Schmidt, J. Schnack, Partition functions and symmetric polynomials. Am. J. Phys. 70, 53–57 (2002)
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. 147–153
L.J. Schulman, Muirhead-Rado inequality for compact groups. Positivity 13, 559–574 (2009)
I. Schur, Uber eine Klasse von Mittelbildungen mit Anwendungdie Determinanten. Theorie Sitzungsber. Berlin. Math. Gesellschaft 22, 9–29 (1923)
A. Seeger, Convex analysis of spectrally defined matrix functions. SIAM J. Optim. 7, 679–696 (1997)
M. Silhavy, The convexity of C →h(C). Tech. Mech. 35(1), 60–61 (2015)
T. Sogabe, M.E.A. El-Mikkawy, On a problem related to the Vandermonde determinant. Disc. Appl. Math. 157, 2997–2999 (2009)
S.J. Spector, A note on the convexity of C →h(C). J. Elast. 118(2), 251–256 (2015)
S. Sra, New concavity and convexity results for symmetric polynomials and their ratios. Linear Multilinear Algebra 1–9 (2018). http://dx.doi.org/10.1080/03081087.2018.1527891
R. Stanley, Some combinatorial properties of Jack symmetric functions. Adv. Math. 77, 76–115 (1989)
L. Tibiletti, Quasi-concavity property of multivariate distribution functions. Ratio Math. 9, 27–36 (1995)
J. Tkadlec, Triangular norms with continuous diagonals. Tatra Mt. Math. Publ. 16, 187–195 (1999)
M. Torki, First- and second-order epi-differentiability in eigenvalue optimization. J. Math. Anal. Appl. 234, 391–416 (1999)
N.K. Tsing, M.K.H. Fan, E.I. Verriest, On analyticity of functions involving eigenvalues. Linear Algebra Appl. 207, 159–180 (1994)
L. Vandenberghe, S. Boyd, S.P. Wu, Determinant maximization with linear matrix inequality constraints. SIAM J. Matrix Anal. Appl. 19(2), 499–533 (1998)
H. Wolkowicz, Measures for symmetric rank-one updates. Math. Oper. Res. 19, 815–830 (1994)
B.J. Venkatachala, Inequalities. An Approach Through Problems (Springer, Berlin, 2018)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Andrica, D., Rădulescu, S., Rădulescu, M. (2019). Some New Methods for Generating Convex Functions. In: Andrica, D., Rassias, T. (eds) Differential and Integral Inequalities. Springer Optimization and Its Applications, vol 151. Springer, Cham. https://doi.org/10.1007/978-3-030-27407-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-27407-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-27406-1
Online ISBN: 978-3-030-27407-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)