Means and equations of means

Abstract

Over the past two decades, means and mean equations have been popular directions of research in the field of functional equations. This paper first introduces the definition and properties of mean as well as the development of the Gauss iteration, invariant equations, and the M-S problem. It then reviews the Bajraktarević mean, the Cauchy mean, and other related means, especially elaborating on the progress of researching corresponding equality and invariance problems. Both the Bajraktarević mean and the Cauchy mean contain two derivative functions, and the corresponding equation problems have been basically solved. However, the invariance of these two symmetric means, due to the presence of four unknown functions, has not been fully solved. Finally, this paper introduces the applications of means in other fields.

Graphical abstract

Keywords

Mean / equality / invariance / Cauchy mean / Bajraktarević mean

E-mail: [e-mail]

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