Abstract

If Φ:[0,∞)→R\Phi:[0,\infty )\rightarrow \mathbb{R} is convex and continuous with Φ(0)=0\Phi (0)=0 and if q∈(1,∞)q\in (1,\infty ), q′:=qq−1q^{\prime }:=\frac{q}{q-1}, we first prove that the inequality Φ(∫0∞f(r)dr)≤C∫0∞f(r)Φ′(r1/q′)dr\Phi \left( \int_{0}^{\infty }f(r)dr\right) \leq C\int_{0}^{\infty }f(r)\Phi ^{\prime }(r^{1/q^{\prime }})dr for every f∈Lq(0,∞)f\in L^{q}(0,\infty), f≥0f\geq 0 with ∣∣f∣∣q≤1||f||_{q}\leq 1 holds when C=1C=1. In general, both sides may be ±∞\pm \infty. Related inequalities for f∈L1(RN)∩Lq(RN)f\in L^{1}(\mathbb{R}^{N})\cap L^{q}(\mathbb{R}^{N}), f≠0f\neq 0 are derived. This inequality is independent of Jensen's inequality and, when q=∞q=\infty, it is an elaboration on an inequality of Steffensen which was discussed elsewhere by the author. The next goal of the paper is to identify the range of the admissible constants CC and, in particular, to characterize the optimal constant when Φ≥0\Phi \geq 0 or Φ≤0\Phi \leq 0. It turns out that C=1C=1 is ``almost always'' optimal, at least in a restricted sense, but not always when q<∞q<\infty: Given qq, the admissible constants lie on an interval containing 11 whose left (right) endpoint is the supremum (infimum) of a function defined on some (left/right dependent) subset of R2\mathbb{R}^{2}. If q=2q=2, these extrema can be calculated in a number of examples. Among other things, this reveals that C=1C=1 need not be optimal when Φ≥0\Phi \geq 0 and Φ+′(0)=0\Phi _{+}^{\prime }(0)=0 or when Φ≤0\Phi \leq 0 and Φ+′(0)=−∞\Phi _{+}^{\prime} (0)=-\infty.

Author information

Contact details are reproduced from the original publication and may be historical.

Patrick J. Rabier

Dept. of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

rabier@imap.pitt.edu

Keywords

  • Convexity
  • Steffensen's inequality
  • weighted integral

Mathematics Subject Classification

26D15, 39B62

References

  1. J. Bergh: A generalization of Steffensen’s inequality, J. Math. Anal. Appl. 41 (1973) 187–191.https://doi.org/10.1016/0022-247X(73)90193-5Google Scholar
  2. P. W. Day: Rearrangement inequalities, Canad. J. Math. 24 (1972) 930–943.https://doi.org/10.4153/CJM-1972-093-xGoogle Scholar
  3. K. Deimling: Nonlinear Functional Analysis, Springer, Berlin (1985).https://doi.org/10.1007/978-3-662-00547-7Google Scholar
  4. G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities, Cambridge University Press, Cambridge (1934).Google Scholar
  5. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms. Part 1: Fundamentals, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7_1Google Scholar
  6. M. A. Krasnosell’skiĭ, Ya. B. Ruticckiĭ: Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen (1961).Google Scholar
  7. W. A. J. Luxemburg: Rearrangement invariant Banach function spaces, Proc. Symp. Analysis, Queen’s Univ. 10 (1967) 83–144.Google Scholar
  8. P. J. Rabier: Steffensen’s inequality and L1 − L∞ estimates of weighted integrals, Proc. Amer. Math. Soc., to appear.Google Scholar
  9. J. F. Steffensen: On certain inequalities between mean values, and their application to actuarial problems, Skand. Aktuarietidskr. 1 (1918) 82–97.https://doi.org/10.1080/03461238.1918.10405302Google Scholar
  10. W. P. Ziemer: Weakly Differentiable Functions, Springer, Berlin (1989).https://doi.org/10.1007/978-1-4612-1015-3Google Scholar

Suggested citation

P. J. Rabier. “Generalized Steffensen Inequalities and Their Optimal Constants.” Journal of Convex Analysis 19 (2012), No. 2, 301–321. https://doi.org/10.68381/jca19018

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.