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

作者：Allaart Pieter C^*

来源：Journal of Mathematical Analysis and Applications, 2011, 381(2): 689-694.

DOI：10.1016/j.jmaa.2011.03.039

• Takagi function
• Approximate convexity
• Digital sum inequality

Let phi(x) = 2 inf{|x - n|: n is an element of Z}, and define for alpha > 0 the function
f(alpha) (x) = Sigma(infinity)(j=0) 1/2(alpha j) phi (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(alpha)(x + y/2) <= f(alpha)(x) + f(alpha)(y)/2 + |x - y|(alpha),
for alpha is an element of [1, 2]. By developing an explicit expression for f(alpha), 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(alpha).

• 出版日期2011-9-15