An inequality for sums of binary digits, with application to Takagi functions

• Published in: Journal of mathematical analysis and applications Vol. 381; no. 2; pp. 689 - 694
• Main Author: Allaart, Pieter C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.09.2011; Elsevier
• Subjects: Approximate convexity; Digital sum inequality; Exact sciences and technology; Mathematical analysis; Mathematics; Sciences and techniques of general use; Takagi function; Approximate convexity; Digital sum inequality; Takagi function; Convexity; Mathematical analysis
• ISSN: 0022-247X; 1096-0813; 1096-0813
• DOI: 10.1016/j.jmaa.2011.03.039

Let ϕ ( x ) = 2 inf { | x − n | : n ∈ Z } , and define for α > 0 the function f α ( x ) = ∑ j = 0 ∞ 1 2 α j ϕ ( 2 j x ) . Tabor and Tabor [J. Tabor, J. Tabor, Takagi functions and approximate midconvexity, J. Math. Anal. Appl. 356 (2) (2009) 729–737] recently proved the inequality f α ( x + y 2 ) ⩽ f α ( x ) + f α ( y ) 2 + | x − y | α , for α ∈ [ 1 , 2 ] . By developing an explicit expression for f α 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 f α .