Inequalities for differences of powers. (English) Zbl 0649.26014

Generalizing a result by E. Leach and M. Sholander [same Journal 92, 207-223 (1983; Zbl 0517.26007)] the author presents necessary conditions for \[ \prod^{n}_{k=1}| x^{a_ k}-y^{a_ k}|^{q_ k}\quad \geq \prod^{n}_{k=1}| a_ k|^{q_ k}\quad (\sum^{n}_{k=1}q_ k=0) \] and shows that in the case \(k=4\), \(a_ 1\leq a_ 2\leq a_ 3\leq a_ 4\), \((a_ 1+a_ 4)(a_ 2+a_ 3)\geq 0\), they are also sufficient.
{Note: It seems that in (i) and (ii), on page 272, \(\leq\) should stand rather than =.}
Reviewer: J.Aczél


26D15 Inequalities for sums, series and integrals
41A10 Approximation by polynomials
26A51 Convexity of real functions in one variable, generalizations
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
Full Text: DOI


[1] Brenner, J. L., A unified treatment and extension of some means of classical analysis. I. Comparison theorems, J. Combin. Inform. System Sci., 3, 175-199 (1978) · Zbl 0408.26008
[2] Burk, F., By all means, Amer. Math. Monthly, 92, 50 (1985)
[3] Carlson, B. C., The logarithmic mean, Amer. Math. Monthly, 79, 615-618 (1972) · Zbl 0241.33001
[4] Daróczy, Z.; Losonczi, L., Über den Vergleich von Mittelwerten, Publ. Math. Debrecen, 17, 289-297 (1970) · Zbl 0227.26010
[5] Dodd, E. L., Some generalizations of the logarithmic mean and of similar means of two variates which become indeterminate when the two variates are equal, Ann. Math. Statist., 12, 422-428 (1971) · Zbl 0063.01124
[6] Leach, E.; Sholander, M., Extended mean values, Amer. Math. Monthly, 85, 84-90 (1978) · Zbl 0379.26012
[7] Leach, E.; Sholander, M., Extended mean values II, J. Math. Anal. Appl., 92, 207-223 (1983) · Zbl 0517.26007
[8] Lin, T. P., The power mean and the logarithmic mean, Amer. Math. Monthly, 81, 879-883 (1974) · Zbl 0292.26015
[9] Losonczi, L., Inequalities for integral mean values, J. Math. Anal. Appl., 61, 586-606 (1977) · Zbl 0378.26008
[10] Páles, Zs., On inequalities for products of power sums, Monatsh. Math., 100, 137-144 (1985) · Zbl 0569.26014
[11] Zs. Páles; Zs. Páles
[12] Pittinger, A. O., Inequalities between arithmetic and logarithmic means, Univ. Beograd Publ. Elektrotechn. Fak. Ser. Mat. Fiz., 680, 15-18 (1980) · Zbl 0469.26009
[13] Pittinger, A. O., The logarithmic mean in \(n\) variables, Amer. Math. Monthly, 92, 99-104 (1985) · Zbl 0597.26027
[14] Stolarsky, K. B., Generalization of the logarithmic mean, Math. Mag., 48, 87-92 (1975) · Zbl 0302.26003
[15] Székely, G., A classification of means, Ann. Univ. Sci. Budapest Eötvös Sect. Math., 18, 129-133 (1975) · Zbl 0336.60004
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