Abstract
Suppose (T, Σ, μ) is a space with positive measure,f: ℝ → ℝ is a strictly monotone continuous function, and &(T) is the set of real μ-measurable functions onT. Letx(·) ∈ &(T) andf ○x)(·) ∈L 1(T,μ). Comparison theorems are proved for the means\(\mathfrak{M}_{(T,{\mathbf{ }}\mu ,{\mathbf{ }}f)} (x( \cdot ))\) and the mixed means\(\mathfrak{M}_{(T_1 ,{\mathbf{ }}\mu _1 ,{\mathbf{ }}f_1 )} (\mathfrak{M}_{(T_2 ,{\mathbf{ }}\mu _2 ,{\mathbf{ }}f_2 )} (x( \cdot )))\) these inequalities imply analogs and generalizations of some classical inequalities, namely those of Hölder, Minkowski, Bellman, Pearson, Godunova and Levin, Steffensen, Marshall and Olkin, and others. These results are a continuation of the author's studies.
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References
G. Hardy, D. Littlewood, and G. Pólya,Inequalities, Cambridge Univ. Press, Cambridge (1952).
E. Beckenbach and R. Bellman,Inequalities, Berlin (1961).
E. K. Godunova,Inequalities Involving Convex Functions [in Russian], Kandidat thesis in the physico-mathematical sciences, Moscow State Pedagogical Institute, Moscow (1965).
E. K. Godunova and V. I. Levin, “A general class of inequalities containing Steffensen's inequality,”Mat. Zametki [Math. Notes],3, No. 3, 339–344 (1968).
J. Pecarič, “On the Bellman generalization of Steffensen's inequality,”J. Math. Anal. Appl.,88, No. 2, 505–507 (1982).
J. Steffensen, “Bounds of certain trigonometric integrals,”10th Scandinavian Math. Congress, 181–186 (1945).
A. Marshall and I. Olkin,Inequalities: Theory of Majorization and Its Applications, Academic Press, New York-London-Toronto (1979).
R. Kh. Sadikova, “A general class of inequalities containing Steffensen's inequality and its generalizations,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], No. 4, 79–80 (1980).
R. Kh. Sadikova, “Inequalities with generalized means,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], Dep., No. 1456 (1981).
R. Kh. Sadikova, “A general class of inequalities with generalized means,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], Dep., No. 2631-84 (1983).
R. Kh. Sadikova,A General Class of Inequalities with Sharp Constants [in Russian], Kandidat thesis in the physicomathematical sciences, Peoples' Friendship University, Moscow (1990).
R. Kh. Sadikova, “Comparison theorems for mixed generalized means,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], Dep., No. 3439 (1975).
O. Shisha and G. Cargo, “On comparable means,”Pacific J. Math.,4, No. 3, 1053–1059 (1964).
R. Cooper, “Notes on certain inequalities. I,”J. London Math. Soc., No. 2, 17–21 (1927); “Notes on certain inequalities. II,”J. London Math. Soc., No. 2, 156–163 (1927).
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Translated fromMatematicheskie Zametki, Vol. 61, No. 6, pp. 864–872, June, 1997.
Translated by N. K. Kulman
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Sadikova, R.K. A general class of inequalities with mixed means. Math Notes 61, 724–730 (1997). https://doi.org/10.1007/BF02361214
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DOI: https://doi.org/10.1007/BF02361214