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

TL;DR

This paper presents a simplified proof of the (1,p)-midconvexity of a family of generalized Takagi functions by deriving an explicit expression at dyadic rationals and linking it to binary digit sums.

Contribution

It introduces a new, straightforward proof technique for the midconvexity of f_p, connecting it to an inequality involving binary digit sums.

Findings

f_p is (1,p)-midconvex for p in [1,2]

Explicit expression for f_p at dyadic rationals

Midconvexity reduces to an inequality on binary digits

Abstract

This paper considers a parametrized family of generalized Takagi functions f_p with parameter p. Tabor and Tabor [J. Math. Anal. Appl. 356 (2009), 729-737] recently proved that for p in [1,2], f_p is (1,p)-midconvex. We give a simpler proof of this result by developing an explicit expression for f_p at dyadic rational points and showing that (1,p)-midconvexity of f_p reduces to a simple inequality for weighted sums of binary digits.