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On Jessen’s inequality for convex functions. (English) Zbl 0591.26009

Let E be a nonempty set and let L be a linear space of real-valued functions on E; suppose that the constant function 1 belongs to L. Let \(\phi\) be a convex function on an interval \(I\subset R\) and let A be a positive linear functional on L, with \(A(1)=1.\) Then for all \(g\in L\) such that \(\phi (g)\in L\) we have \(A(g)\in I\) and \(\phi (A(g))\leq A(\phi (g))\) (Jessen’s inequality). The authors give a short proof of this inequality and prove some general complementary inequalities. They obtain Hölder’s and Minkowski’s inequality for positive linear functionals, as well as certain complementary Hölder and Minkowski inequalities. Other known results are generalized and some examples and applications are given.
Reviewer: I.Raşa

MSC:

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
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References:

[1] Jessen, B., Bemaerkinger om konvekse Funktioner og Uligheder imellem Middelvaerdier I, Mat. Tidsskrift B, 17-28 (1931) · JFM 57.0310.01
[2] Popoviciu, T., Les fonctions convexes, (Actualités Sc. Ind. No. 992 (1944), Hermann: Hermann Paris) · JFM 64.0202.02
[3] Lah, P.; Ribarić, M., Converse of Jensen’s inequality for convex functions, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 412-460, 201-205 (1973) · Zbl 0265.26013
[4] Mitrinović, D. S.; Vasić, P. M., The centroid method in inequalities, Univ. Beograd Elektrotehn. Fak. Ser. Mat. Fiz., No. 498-541, 3-16 (1975) · Zbl 0319.26010
[5] Beesack, P. R., On inequalities complementary to Jensen’s, Canad. J. Math., 35, 324-338 (1983) · Zbl 0488.26009
[6] Vasić, P. M.; Pečarić, J. E., On the Jensen inequality, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 634-677, 50-54 (1979) · Zbl 0451.26013
[7] Lupas, A., A generalization of Hadamard inequalities for convex functions, Univ. Beograd Elektrotehn. Fak. Ser. Mat. Fiz., No. 544-576, 115-121 (1976) · Zbl 0346.26009
[8] Mulholland, H. P., The generalization of certain inequality theorems involving powers, (Proc. London Math. Soc. (2), 33 (1932)), 481-516 · Zbl 0004.25101
[9] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0047.05302
[10] Mitrinović, D. S.; Bullen, P. S.; Vasić, P. M., Sredine i sa njima povezane nejednakosti I, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 600, 1-232 (1977) · Zbl 0422.26009
[11] Mitrinović, D. S., (Analytic Inequalities (1970), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), (in cooperation with P.M. Vasić) · Zbl 0199.38101
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