Glasnik Matematicki, Vol. 40, No.2 (2005), 241-247.

CONVEXIFIABLE FUNCTIONS IN INTEGRAL CALCULUS

Sanjo Zlobec

McGill University, Department of Mathematics and Statistics, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6
e-mail: zlobec@math.mcgill.ca

Abstract.   A function is said to be convexifiable if it becomes convex after adding to it a strictly convex quadratic term. In this paper we extend some of the basic integral properties of convex functions to Lipschitz continuously differentiable functions on real line. In particular, we give estimates of the mean value, a "nonstandard" form of Jensen's inequality, and an explicit representation of analytic functions. It is also outlined how one can use convexification to study ordinary differential equations.

2000 Mathematics Subject Classification.   26B25, 52A40.

Key words and phrases.   Convex function, convexifiable function, integral mean value, Jensen's inequality, analytic function.

Full text (PDF) (free access)

DOI: 10.3336/gm.40.2.05

References:

  1. J. L. W. V. Jensen, Sur les fonctions convexes et les inegalites entre les valeurs moyennes, Acta Math. 30 (1906) 175-193.
    CrossRef

  2. S. Karlin and Z. Ziegler, Some applications to inequalities of the method of generalized convexity, J. d' Analyse Math. 30 (1976) 281-303.

  3. S. Kurepa, Convex functions, Glasnik Mat. Fiz. Astron. II 11 (1956) 89-93.

  4. A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, 1973.

  5. J. van Tiel, Convex Analysis, Wiley, Chichester, 1984.

  6. S. Zlobec, Estimating convexifiers in continuous optimization, Math. Comm. 8 (2003) 129-137.

  7. S. Zlobec, Jensen's inequality for non-convex functions, Math. Comm. 9 (2004) 119-124.

  8. S. Zlobec, Characterization of convexifiable functions, McGill University, June 2004, preprint.
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