Chebyshev's inequalities for functions monotone in the mean

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 + ? + w_i. Givenf ∈ F, define (bar f) ∈F by $$bar fleft( i right) = frac{{left{ {w_i fleft( 1 right) + ldots + w_i fleft( i right)} right}}}{{W_i }}.$$ Callf mean increasing if (bar f) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u min left{ {w_i^{u - 1} W_i^{t - u} } right}.$$ Then $$kmathop sum limits_{i = 1}^n w_i f_1 left( i right) ldots f_{t + u} left( i right) leqslant mathop prod limits_{j = 1}^{t + u} (mathop sum limits_{i = 1}^n w_i f_1 (i)).$$