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Means of power type and their inequalities. (English) Zbl 0955.26013

The author proves some general results on means of power type. In particular, he introduces a new mean, and compares it with the well-known power means. He also proves some results connected to compositions of power means.
The results obtained are used to establish upper and lower bounds for the homogenized integrand \(f_{\text{hom}}\) of a family of integrands \({f_\varepsilon}\), where \(f_\varepsilon(x,z)=C({x\over \varepsilon},|z|)|z|^p\) for a.e. \(x\) in \({\mathbb R}^n\) and every \(z\) in \({\mathbb R}^n\), \(C\) is \(]0,1[^n\)-periodic in the \(x\) variable, and \(p>1\).

MSC:

26E60 Means
26D15 Inequalities for sums, series and integrals
35J20 Variational methods for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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References:

[1] Beckenbach, Convexity Properties of Generalized Mean Value Functions, Ann. Math. Statist. 13 pp 88– (1942) · Zbl 0061.11601
[2] Braides , A. Lukkassen , D.
[3] Bullen, Means and Their Inequalities (1988) · doi:10.1007/978-94-017-2226-1
[4] Dal Maso, An Introduction to I’-Convergence (1993) · Zbl 0816.49001 · doi:10.1007/978-1-4612-0327-8
[5] Hardy, Inequalities (1934)
[6] Lukkassen , D. 1996
[7] Lukkassen, On Some Sharp Bounds for the Off-Diagonal Elements of the Homogenized Tensor, Applications of Math. 40 pp 401– (1995) · Zbl 0847.35011
[8] Lukkassen, Some Sharp Estimates Connected to the Homogenized p-Laplacian Equation, ZAMM-Z. angew. Math. Mech. 76 (Suppl. 2) pp 603– (1996) · Zbl 1126.35303
[9] Lukkassen, On Estimates of the Effective Energy for the Poisson Equation with a p-Laplacian, Russian Math. Surveys 51 pp 739– (1996) · Zbl 0871.35035
[10] Lukkassen, Sharp Inequalities Connected to the Homogenized p-Poisson Equation, Math. Ineq. Appl. 2 (2) pp 243– (1999) · Zbl 0927.35012
[11] Peetre, A General Beckenbach’s Inequality with Applications. In: Functions Spaces Operator and Nonlinear Analysis, Pitman Research Notes in Mathematics 211 pp 125– (1989)
[12] Persson, The Homogenization Method and Some of Its Applications, Mathematica Balka-nika, New Series 7 pp 179– (1993) · Zbl 0837.73008
[13] Persson , L. E. Persson , L. Svanstedt , N. Wyller , J. 1993
[14] Rudin, Real and Complex Analysis (1987)
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