Abstract:
The following inequalities are shown to hold for the least uniform rational deviations $R_n(f)$ of a function $f(x)$, continuous and convex in the interval $[a,b]$:
$$
R_n(f)\le C(\nu)\Omega(f)n^{-1}\overbrace{\ln\dots\ln}^{\nu\text{ times}}n
$$
($\nu$ is an integer, $C(\nu)$ depends only on $\nu$, and $\Omega(f)$ is the total oscillation of $f$);
$$
R_n(f)\le C_1(\nu)n^{-1}\overbrace{\ln\dots\ln}^{\nu\text{ times}}n\inf\limits_{(b-a)\varkappa_n\le\lambda<b-a}\Bigl\{\omega(\lambda,f)+M(f)n^{-1}\ln\frac{b-a}\lambda\Bigr\}
$$
($\nu$ is an integer, $C_1(\nu)$ depends only on $\nu$, $\varkappa_n=\exp(-n/(500\ln^2n))$), $\omega(\delta,f)$ is the modulus of continuity of $f$, and $M(f)=\max|f(x)|$.
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