TY - JOUR
T1 - Some inequalities for products of power sums
AU - Reznick, Bruce
PY - 1983/2
Y1 - 1983/2
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84972569566&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84972569566&partnerID=8YFLogxK
U2 - 10.2140/pjm.1983.104.443
DO - 10.2140/pjm.1983.104.443
M3 - Article
AN - SCOPUS:84972569566
SN - 0030-8730
VL - 104
SP - 443
EP - 463
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -