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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Andrica, The Refinements of Some Geometric Inequalities (Romanian) (Lucrarile seminarului “Theodor Angheluta”, Cluj-Napoca, 1983), pp. 1–2
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. 173–177
D. Andrica, The Refinement of Some Geometric Inequalities, Seminar on Geometry, Cluj-Napoca, Preprint Nr. 10 (1986), pp. 71–76
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. 9–12
D. Andrica, Note on an abstract approximation theorem, in Approximation Theory and Applications, ed. by Th. M. Rassias (Hadronic Press, Palm Harbor, 1999), pp. 1–10
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. 7–16
D. Andrica, M.O. Drimbe, On some inequalities involving isotonic functionals. Mathematica – Rev. Anal. Numér. Théor. Approx. 17(1), 1–7 (1988)
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. 425–431
D. Andrica, D.-Şt. Marinescu, New interpolation inequalities to Euler’s R ≥ 2r. Forum. Geom. 17, 149–156 (2017)
D. Andrica, C. Mustăţa, An abstract Korovkin type theorem and applications. Studia Univ. Babeş-Bolyai Math. 34, 44–51 (1989)
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)
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)
P.R. Beesak, J. Pečarić, On Jessen’s inequality for convex functions. J. Math. Anal. Appl. 110, 536–552 (1985)
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)
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, 41–50
B.Y. Chen, Classification of h-homogeneous production functions with constant elasticity of substitution. Tamkang J. Math. 43, 321–328 (2012)
C.W. Cobb, P.H. Douglas, A theory of production. Amer. Econom. Rev. 18, 139–165 (1928)
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. 89–119
Z. Cvetkovski, Inequalities. Theorems, Techniques and Selected Problems (Springer, Berlin, 2012)
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)
S.S. Dragomir, J. Pečarić, L.E. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21, 335–341 (1995)
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)
M. Eliasi, On extremal properties of general graph entropies. MATCH Commun. Math. Comput. Chem. 79, 645–657 (2018)
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. 138–142
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. 739–785
J.B. Hiriart-Urruty, C. Lemarechal, Fundamentals of Convex Analysis (Springer Science & Business Media, Berlin, 2012)
B. Jessen, Bemaerkinger om Konvexe Functioner og Uligheder immelem Middelvaerdier I, Mat. Tidsskrift, B(1931), 17–28.
C. Joita, P. Stanica, Inequalities related to rearrangements of powers and symmetric polynomials. J. Inequal. Pure Appl. Math. 4, Article 37, 4 (2003) (electronic)
P. Kim Hung, The stronger mixing variables method. Math. Reflect. 6, 1–8 (2006)
P. Kosmol, D. Müller-Wichards, Stability for families of nonlinear equations. Isvetia NAN Armenii. Mathematika, 41(1), 49–58 (2006)
Y.C. Li, C.C. Yeh, Some characterizations of convex functions. Comput. Math. Appl. 59, 327–337 (2010)
Al. Lupaş, Some properties of the linear positive operators (III). Rev. Anal. Numér. Théor. Approx. 3, 47–61 (1974)
I.G. Macdonald, Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. (The Clarendon Press, Oxford University Press, New York, 1995)
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 Majoration and Its Applications, 2nd edn. (Springer, Berlin, 2011)
F. Mat, On nonnegativity of symmetric polynomials. Amer. Math. Monthly 101, 661–664 (1994)
J.B. McLeod, On four inequalities in symmetric functions. Proc. Edinb. Math. Soc. 11, 211–219 (1959)
C.D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, 2000)
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)
D.S. Mitrinović, Analytic Inequalities (Springer, Berlin, 1970)
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)
D.S. Mitrinović, J. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis (Kluwer Academic, Dordrecht, 1993)
C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications: A Contemporary Approach (Springer, Berlin, 2018)
J. Pečarić, Note on multidimensional generalization of Slater’s inequality. J. Approx. Theory 44, 292–294 (1985)
J.E. Pečarić, D. Andrica, Abstract Jessen’s inequality for convex functions and applications. Mathematica 29(52), 1, 61–65 (1987)
T. Popoviciu, Sur l’approximation des fonctions convexes d’ordre supérieur. Mathematica (Cluj) 10, 49–54 (1935)
E. Popoviciu, Teoreme de medie din analiza matematica si legatura lor cu teoria interpolarii (Romanian) (Editura Dacia, Cluj-Napoca, 1972)
T. Puong, Diamonds in Mathematical Inequalities (Hanoi Pub. House, Ha Noi City, 2007)
M. Rădulescu, S. Rădulescu, P. Alexandrescu, On Schur inequality and Schur functions. Annals Univ. Craiova, Math. Comp. Sci. Ser. 32, 202–208(2005)
M. Rădulescu, S. Rădulescu, P. Alexandrescu, On the Godunova – Levin – Schur class of functions. Math. Inequal. Appl. 12(4), 853–962 (2009)
I. Rasa, On the inequalities of Popoviciu and Rado. Mathematica-L’Analyse Numerique et la Theorie de l’Approximation, Tome 11, 147–149 (1982)
A.W. Roberts, D.E. Varberg, Convex Functions (Academic Press, New York, 1973)
M.L. Slater, A companion inequality to Jensen’s inequality. J. Approx. Theory 32, 160–166 (1981)
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. 443–451
S. Varošanec, On h-convexity. Aust. J. Math. Anal. Appl. 326(1), 303–311 (2007)
B.J. Venkatachala, Inequalities. An Approach Through Problems (Springer, Berlin, 2018)
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)
W. Wegmuller, Ausgleichung Lursch Bernestein-Polynome. Mitt. Verein. Schweiz. Versich-Math. 26, 15–59 (1938)
E.M. Wright, A generalization of Schur’s inequality. Math. Gaz. 40, 217 (1956)
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). Convexity Revisited: Methods, Results, and Applications. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-27407-8_3
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)