ISSN:
1588-2829
Keywords:
Primary 26D15
;
Secondary 26A48
;
Chebyshev
;
cone
;
inequality
;
mean
;
monotone
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ⋯,n} →R. Choosew ∈ F withw i = w(i) 〉 0. PutW i = w1 + ⋯ + wi. Givenf ∈ F, define $$\bar f$$ ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if $$\bar f$$ is increasing. Letf 1, ⋯, ft be mean decreasing andf t+1,⋯: ft+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02023580
Permalink
