We study the asymptotic behavior of the range of the ratio of products of power sums. For x = (x1,…, xn), define Mp = Mp(x) = Σxip. As two representative and explicit results, we show that the maximum and minimum of the function M1M3/M22 are ±3√3/16n1/2 + 5/8 + O(n-1/2) and that n ≤ M1M3/M4 > -n/8, where "1/8" is the best possible constant. We give readily computable, if less explicit, formulas of this kind for Mp1a1 … Mprar/Mqb, Σa1p1 = bq. Applications to integral inequalities are discussed. Our results generalize the classical Hölder and Jensen inequalities. All proofs are elementary.
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