On an Inequality of Diananda
نویسنده
چکیده
where Pn,0(x) denotes the limit of Pn,r (x) as r → 0+ and where qi > 0, 1≤ i≤n, are positive real numbers with ∑n i=1qi = 1 and x = (x1,x2, . . . ,xn). In this note, we let q =minqi and always assume n≥ 2 and 0≤ x1 <x2 < ···<xn. We define An(x) = Pn,1(x), Gn(x) = Pn,0(x), and Hn(x) = Pn,−1(x) and we will write Pn,r for Pn,r (x), An for An(x), and similarly for other means when there is no risk of confusion. For mutually distinct numbers r , s, and t and any real number α and β, we define
منابع مشابه
On an Inequality of Diananda, Iii
We extend the results in part I, II on certain inequalities involving the generalized power means.
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Let Mn,r(x) be the generalized weighted means Mn,r(x) = ( ∑n i=1 qix r i ) 1/r , where Mn,0(x) denotes the limit of Mn,r(x) as r → 0+, x = (x1,x2, . . . ,xn), and qi > 0 (1≤ i≤ n) are positive real numbers with ∑n i=1 qi = 1. In this paper, we let q =minqi and always assume n≥ 2, 0≤ x1 < x2 < ··· < xn. We define An(x)=Mn,1(x), Gn(x)=Mn,0(x),Hn(x)=Mn,−1(x) and we will writeMn,r forMn,r(x),An for...
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Let Pn,r(x) be the generalized weighted means: Pn,r(x) = ( ∑n i=1 qix r i ) 1/r , where Pn,0(x) denotes the limit of Pn,r(x) as r → 0+, x = (x1,x2, . . . ,xn) and qi > 0 (1 ≤ i ≤ n) are positive real numbers with ∑n i=1 qi = 1. In this paper, we let q = minqi and always assume n≥ 2, 0 ≤ x1 < x2 < ··· < xn. We defineAn(x) = Pn,1(x),Gn(x) = Pn,0(x),Hn(x) = Pn,−1(x), and we will write Pn,r for Pn,...
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