The generalized mean M(x,y) is defined to be Ψ-1 [pΨ (x) + aΨ (y)] where p, a>0, p + a = 1 and Ψ (x) is monotone and continuous.
This mean is applied to the second difference. Δ2(f: x, h) = f(x+h) + (f-h)-2f(x) to form a generalized second difference Δ2Ψ (f: x, h) = MΨ [f(x+h), f(x-h)] - f(x)..
A study is made of functions whose generalized second differences satisfy certain conditions. Maxima of classes of generalized quasi-smooth functions are examined. It is the purpose of this note to apply the generalized mean to the study of second differences. A generalized second difference will be defined and certain properties of the second difference will be examined under this generalization.
Proceedings of the Iowa Academy of Science
©1959 Iowa Academy of Science, Inc.
Nolte, Sidney D.
"An Application of Generalized Means,"
Proceedings of the Iowa Academy of Science, 66(1), 357-361.
Available at: https://scholarworks.uni.edu/pias/vol66/iss1/48