Article ID Journal Published Year Pages File Type
4618096 Journal of Mathematical Analysis and Applications 2011 6 Pages PDF
Abstract

Let ϕ(x)=2inf{|x−n|:n∈Z}ϕ(x)=2inf{|x−n|:n∈Z}, and define for α>0α>0 the functionfα(x)=∑j=0∞12αjϕ(2jx). Tabor and Tabor [J. Tabor, J. Tabor, Takagi functions and approximate midconvexity, J. Math. Anal. Appl. 356 (2) (2009) 729–737] recently proved the inequalityfα(x+y2)⩽fα(x)+fα(y)2+|x−y|α, for α∈[1,2]α∈[1,2]. By developing an explicit expression for fαfα at dyadic rational points, it is shown in this paper that the above inequality can be reduced to a simple inequality for weighted sums of binary digits. That inequality, which seems of independent interest, is used to give an alternative proof of the result of Tabor and Tabor, which captures the essential structure of fαfα.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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