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The aim of this note is to characterize the real coefficients p₁,...,pₙ and q₁,...,qₖ so that
be valid whenever the vectors x₁,...,xₙ, y₁,...,yₖ satisfy
y₁,...,yₖ ⊆ convx₁,...,xₙ.
Using this characterization, a class of generalized weighted quasi-arithmetic means is introduced and several open problems are formulated.
In this work we apply the method of a unique partition of a complex function of complex variables into symmetrical functions to solving a certain type of functional equations.
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