Abstract
An inequality first proved by J. F. Steffensen [1] in 1918 is the subject of this Chapter. It is curious that this inequality is not included in the monograph of G. H. Hardy, J. Littlewood, and G. Pólya [2] Steffensen’s paper was not reviewed in Jahrbuch über die Fortschritte der Matematik. It is however, mentioned by G. Szegö in his review of the papers [3] and [4] by T. Hayashi.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
STEFFENSEN, J. F., On certain inequalities between mean values, and their application to actuarial problems, Skand. Aktuarietids. 1918, 8297.
HARDY, G. H., J. E. LITTLEWOOD, and G. PÔLya, “Inequalities”, Cambridge, 1934, 1952, 1967.
HAYASHI, T., On curves with monotonous curvature, Tôhoku Math. J. 15 (1919), 236–239.
HAYASHI, T., On certain inequalities,Tôhoku Math. J. 18 (1920) 75–89.
MEIDELL, B., Note sur quelques inégalités et formules d’approximation, Skand. Aktuarietids. 1918, 180–198.
STEFFENSEN, J. F., On certain inequalities and methods of approximation, J. Inst. Actuaries 51 (1919), 247–297.
APÉRY, R., Une inégalité sur les fonctions de variable réelle,Atti del Quarto Congresso dell’Unione Matematica Italiana 1951, vol. 2 (1953), 3–4, Roma.
STEFFENSEN, J. F., Bounds of certain trigonometrical integrals, C. R. Dixième Congrès Math. Scandinaves 1946, 181–196, Copenhagen, 1947.
BELLMAN, R., On inequalities with alternating signs, Proc. Amer. Math. Soc. 10 (1959), 807–809.
OSTROWSKI, A., “Aufgabensammlung zur Infinitesimalrechnung”, vol. 1, Basel-Stutgart, 1964.
DieudonnÉ, J., “Calcul infinitésimal”, Paris, 1968.
BOURBAKI, N., “Fonctions d’une variable réelle”, Chap. 2, §3.6, exerc. 2, Paris.
GODUNOVA, E. K., V. I. LEVIN and I. V. CEBAEVSKAJA, Novye issledovanija po funkcional’nym neravenstvam, Materialysestoi mez. fiz.mat. nauc. konf. Dal’nego Vostoka. Diff. i Int. Uravnenija. Tom 3. Habarovsk, 1967, pp. 70–77.
GODUNOVA, E. K. and V. I. Levin, Obicii klass neravensiv, soderiaiciâ neravenstvo Steffensena, Mat. Zametki 3 (1968), 339–344.
MARJANOVIC, M., Some inequalities with convex functions, Publ. Inst. Math. (Beograd) 8 (22) (1968), 66–68.
MITRINOVIC, D. S. The Steffens en inequality, Univ. Beograd. Publ. Elektrotehn. Fak. S.r. Mat. Fiz. No. 247–273 (1969), 1–14.
VOLKOV, V. N., Inequalities connected with an inequality of Gauss (Russian), Volz. Matem. Sb. 1969, No. 7, 11–13.
BULLEN, P. S., The Steffensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. S.r. Mat. Fiz. No. 320–328 (1970), 59–63.
PECARIC, J. E., On the Jensen-Steffensen inequality, Univ. Beograd. Publ. E.ektrotehn. Fak. Ser. Mat. Fiz. No. 634–677 (1979), 101–107.
BERGH, J., A. generalizationof Steffensen’s inequality, J. Math. A.al. Appl. 41 (1973), 187–191.
MILOVANOVIC, G. V. and J. E. PECARIC, The Steffensen inequality for convex function of order n, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 634–677 (1979), 97–100.
SADIKOVA, R. H., Obscii klass neravenstv soderzaäc’ii neravenstovo Steffensena i ego obobJcenija, Izv. Vis. Uceb. Zaved. Matematika No. 4 (215) (1980), 79–80.
PECARIC, J. E., A generalization of an inequality of Ky Fan, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 678–715 (1980), 75–84.
PECARIC, J. E. and B. D. CRSTICI, A further generalization of an inequality of Ky Fan, Itin. sem. func. equat. approx. convexity. Cluj-Napoca, 1981.
VASIC, P. M. and J. E. PECARIC, Note on the Steffens en inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 716–734 (1981), 80–82.
FINK, A. M., Steffensen type inequalities, Rocky Mountain J. Math. 12 (1982), 785–793.
PECARIC, J. E., Inverse of Steffensen’s inequality, Glasnik Matematicki 17 (27) (1982), 265–270.
Notes on some general inequalities,Publ. Inst. Math. (Beograd) N.S. 32 (46) (1982), 131–135.
On the Bellman generaliztion of Steffensen’s inequal-ity,J. Math. Anal. Appl. 88 (1982), 505–507.
On the Bellman generalization of Steffens en’s inequal-ity II,J. Math. Anal. Appl. 104 (1984), 432–434.
VASIC, P. M. and J. E. PECARIC, Sur une inégalité de Jensen-Steffensen. Pecaric, Sur une inégalité de Jensen-Steffensen, “General Inequalities 4”, Basel-Boston-Stuttgard, 1984, 87–92.
MITRINOVIC, D. S. and J. E. PECARIC, On the Bellman generalization of Steffensen inequality, Iii, J. Math. Anal. Appl. 135 (1988), 342–345.
PECARIC, J. E., Connection between some inequalities of Gauss, Steffensen and Ostrowski, Southeast Asian Bull. Math. 13 (1989), 89–91.
PECARIC, J. E. and R. R. JANIC, A remark on the Steffensen inequality,to appear.
PECARIC, J. E. and S. S. DRAGOMIR, On an inequality of GodunovaLevin and some refinements of Jensen integral inequality, Babe§-Bolyai Univ. Fac. Math. Res. Sem. Prep. 1989, 263–268.
IMORU, C. O., Some extensions of Steffensen’s inequality,to appear.
PETSCHKE, M., Extremalstrahlen konvexen Kegel und komplementäre Ungleichungen, Dissertation, Darmstadt, 1989.
CAO, Y., Correction of an inequality of Bellman,manuscript.
BECKENBACH, E. F. and R. BELLMAN, “Inequalities”, Berlin-HeidelbergNew York, 1965.
ALZER, H., On an inequality of Gauss, Revista Matematica 4 (1991), 179–183.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1993). Steffensen’s Inequality. In: Classical and New Inequalities in Analysis. Mathematics and Its Applications (East European Series), vol 61. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1043-5_11
Download citation
DOI: https://doi.org/10.1007/978-94-017-1043-5_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4225-5
Online ISBN: 978-94-017-1043-5
eBook Packages: Springer Book Archive