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

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 .