Generalized Classical Weighted Means, the Invariance, Complementarity and Convergence of Iterates of the Mean-Type Mappings
Tóm tắt
Under some simple conditions on real function f defined on an interval I, the bivariable functions given by the following formulas
$$\begin{aligned} A_{f}\left( x,y\right):= & {} f\left( x\right) +y-f\left( y\right) , \\ G_{f}\left( x,y\right):= & {} \frac{f\left( x\right) }{f\left( y\right) }\,y, \\ \text{ and } \quad H_{f}\left( x,y\right):= & {} \frac{xy}{f\left( x\right) +y-f\left( y\right) }, \end{aligned}$$
for all
$$x,y\in I$$
, generalize, respectively, the classical weighted arithmetic, geometric and harmonic means. The invariance equations
$$\begin{aligned} A_{f}\circ \left( G_{g},H_{h}\right) =A_{f}, \quad G_{g}\circ \left( A_{f},H_{h}\right) =G_{g} \quad \text{ and } \quad H_{h}\circ \left( A_{f},G_{g}\right) =H_{h}, \end{aligned}$$
where f, g, h are the unknown functions are, in some special cases, solved. The convergence of iterates of the relevant mean-type mappings is considered. As an application the solutions of some functional equations are determined.
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