An Application of Generalized Means

The generalized mean M(x,y) is defined to be Ψ-1 [pΨ (x) + aΨ (y)] where p, a\u3e0, 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