Abstract
Beckenbacg–Gini–Lehmer type means and mean-type mappings generated by functions of several variables, for which the arithmetic mean is invariant, are introduced. Equality of means of that type, their homogeneity, and convergence of the iterates of the respective mean-type mappings are considered. An application to solving a functional equation is given.
Highlights
Introduction onIf I is an interval and φ is a positive function defined on I, the two-variable function M[ φ] defined by xφ( x ) + yφ(y), x, y ∈ I, M[ φ] ( x, y) = φ( x ) + φ(y)I2 maps I 2 into I, and even more: it is a mean in I, that is, min( x, y) ≤ M[ φ] ( x, y) ≤ max( x, y) for all x, y ∈ I, and it is called a Gini mean, Beckenbach–Gini mean, or Lehmer mean ([1,2,3]) of a generator φ
M[ φ], M[ φ] : I → I 2, i.e., x +y is invariant with respect to the mean-type mapping
In this paper we show that these properties remain true for some broader classes of means which generalize the two-variable B–G–L means
Summary
If I is an interval and φ is a positive function defined on I, the two-variable function M[ φ] defined by xφ( x ) + yφ(y). M[ φ] , M[∗φ] converges to the mean-type mapping ( A2 , A2 ) , uniformly on compact sets ([4,5]). For this reason M[∗φ] is called complementary to M[ φ] (and vice-versa) with respect to the arithmetic mean A2 ([6]). K −times single-variable function φ is replaced by a suitable generator f of k − 1 variables, which leads to a mean-type mapping M[ f ],k (Theorem 3). In the last section we give an application of Theorem 2 in solving some functional equations
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