Sharp Nagy type inequalities for the classes of functions with given quotient of the uniform norms of positive and negative parts of a function

V.A. Kofanov (Oles Honchar Dnipro National University),
I.V. Popovich (Oles Honchar Dnipro National University)


For any $$$p\in (0, \infty],$$$ $$$\omega > 0,$$$ $$$d \ge 2 \omega,$$$ we obtain the sharp inequality of Nagy type
\|x_{\pm}\|_\infty \le
)}} \left\|x \right\|_{L_{p} \left(I_d  \right)}
on the set $$$S_{\varphi}(\omega)$$$ of $$$d$$$-periodic functions $$$x$$$ having zeros with given the sine-shaped $$$2\omega$$$-periodic
comparison function $$$\varphi$$$, where $$$c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$$$ is such that
 \|x_{+}\|_\infty \cdot
\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot
\|(\varphi+c)_{-}\|^{-1}_\infty .

In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $$$\|x_{+}\|_\infty / \|x_-\|_\infty$$$.


Nagy type inequality; a class of functions with given comparison function; Sobolev class of functions; polynomial; spline

Full Text:



Bojanov B., Naidenov N. "An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös", J. d'Analyse Mathematique, 1999; 78: pp. 263-280.

Kofanov V.A. "Exact upper bounds of norms of functions and their derivatives on the classes of functions with given comparison function", Ukrainian Math. J., 2011; 63(7): pp. 969-984. (in Russian)

Babenko V.F., Kofanov V.A., Pichugov S.A. "Inequalities of Kolmogorov Type and Some Their Applications in Approximation Theory", Rendiconti del Circolo Matematico di Palermo. Serie II, Suppl., 1998; 52: pp. 223-237.

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Naukova dumka, Kyiv, 1992; 304 p. (in Russian)

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", Izbr. tr. Matematika, mekhanika, Nauka, Moscow, 1985; pp. 252-263. (in Russian)

Babenko V.F., Kofanov V.A., Pichugov S.A. "Comparison of rearrangement and Kolmogorov-Nagy type inequalities for periodic functions", Approximation theory: A volume dedicated to Blagovest Sendov (B. Bojanov, ed.), Darba, Sofia, 2002; pp. 24-53.

Korneichuk N.P., Babenko V.F., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Nauk. dumka, Kyiv, 2003; 590 p. (in Russian)

Tikhomirov V.M. "Set widths in functional spaces and theory of the best approximations", Uspekhi mat. nauk, 1960; 15(3): pp. 81-120. (in Russian)




  • There are currently no refbacks.

Copyright (c) 2020 V.A. Kofanov, I.V. Popovich

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in

ISSN (Online): 2664-5009
ISSN (Print): 2664-4991