Digital sum inequalities and approximate convexity of Takagi-type functions

TL;DR

This paper establishes inequalities related to the sum of digits in different bases and applies these results to derive approximate convexity inequalities for Takagi-like functions, including a new inequality.

Contribution

It introduces new inequalities for digit sums and applies them to establish a novel approximate convexity inequality for Takagi-type functions.

Findings

Derived inequalities for sum of base-b digits over integer ranges.

Applied inequalities to establish approximate convexity of Takagi-like functions.

Discovered a new inequality related to Takagi functions.

Abstract

For an integer b>=2, let s_b(n) be the sum of the digits of the integer n when written in base b, and let S_b(N) be the sum of s_b(n) over n=0,...,N-1, so that S_b(N) is the sum of all b-ary digits needed to write the numbers 0,1,...,N-1. Several inequalities are derived for S_b(N). Some of the inequalities can be interpreted as comparing the average value of s_b(n) over integer intervals of certain lengths to the average value of a beginning subinterval. Two of the main results are applied to derive a pair of "approximate convexity" inequalities for a sequence of Takagi-like functions. One of these inequalities was discovered recently via a different method by V. Lev; the other is new.