The aim of this note is to characterize the real coefficients p₁,...,pₙ and q₁,...,qₖ so that $∑_{i=1}^{n} p_ix_i + ∑_{j=1}^{k} q_jy_j ∈ conv{x₁,...,xₙ}$ be valid whenever the vectors x₁,...,xₙ, y₁,...,yₖ satisfy {y₁,...,yₖ} ⊆ conv{x₁,...,xₙ}. Using this characterization, a class of generalized weighted quasi-arithmetic means is introduced and several open problems are formulated.
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Let I be an interval, 0 < λ < 1 be a fixed constant and A(x,y) = λx + (1-λ)y, x,y ∈ I, be the weighted arithmetic mean on I. A pair of strict means M and N is complementary with respect to A if A(M(x,y),N(x,y)) = A(x,y) for all x, y ∈ I. For such a pair we give results on the functional equation f(M(x,y)) = f(N(x,y)). The equation is motivated by and applied to the Matkowski-Sutô problem on complementary weighted quasi-arithmetic means M and N.
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