We present new structures and results on the set $$ of mean functions on a given symmetric domain in ℝ². First, we construct on $$ a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on $$ a structure of metric space under which $$ is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of $$. Finally, we give two theorems...

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.

We show that any quasi-arithmetic mean $A_{\phi }$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f\left(M(x,y)\right)=A_{\phi }(f\left(x\right),f\left(y\right))$ and $f\left(A_{\phi }(x,y)\right)=M(f\left(x\right),f\left(y\right))$ are the constant ones.

In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y’(x)=a(x)y((x))+b(x)y(x),      xI=[x0,). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z’(x)=b(x)z(x),      xI and a solution of the functional equation |a(x)|((x))=|b(x)|(x),      xI.

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation $$y^{\text{'}}\left(x\right)=ay\left(\tau \left(x\right)\right)+by\left(x\right),x\in [x_0,\infty )$$ of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution $y\left(x\right)$ which behaves in this way.