ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)

#### Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.25.202302.22-36

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 26A15, 26A16, 26A27

#### On global extrema of power Takagi functions

O. E. Galkin, S. Yu. Galkina, A. A. Tronov

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. By construction, power Takagi functions $S_p$ are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions $S_p(x)$ have one real parameter $p>0$. They are defined on the real axis $\mathbb R$ by the series $S_p(x)=\sum_{n=0}^\infty (S_0(2^nx)/2^n)^p$, where $S_0(x)$ is the distance from real number $x$ to the nearest integer number. We show that for every $p>0$, the functions $S_p$ are everywhere continuous, but nowhere differentiable on $\mathbb R$. Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values $S_p(x)$ at rational points $x$. In addition, for all values of the parameter $p$ from the interval $(0;1)$, we find the global extrema of the functions $S_p$, as well as the points where they are reached. It turns out that the global maximum of $S_p$ equals to $2^p/(3^p(2^p-1))$ and is reached only at points $q+1/3$ and $q+2/3$, where $q$ is an arbitrary integer. The global minimum of the functions $S_p$ equals to $0$ and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions $S_p$ and find the points at which these estimates are reached.

Key Words: power Takagi function, continuity, nowhere differentiability, functional equations, global extrema

For citation: O. E. Galkin, S. Yu. Galkina, A. A. Tronov. On global extrema of power Takagi functions. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:2(2023), 22–36. DOI: https://doi.org/10.15507/2079-6900.25.202302.22-36

Submitted: 05.02.2023; Revised: 10.04.2023; Accepted: 25.05.2023 This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.