Let $f$ be a continuous periodic function with Fourier sums $S_n(f)$, $E_n(f)=E_n$ be the best approximation to $f$ by trigonometric polynomials of order $n$. The following estimate is proved: $$ ||f-S_n(f)||\leqslant c\sum_{\nu=n}^{2n}\frac{E_\nu}{\nu-n+1}. $$ (Here $c$ is an absolute constant.) This estimate sharpens Lebesgue's classical inequality for “fast” decreasing $E_\nu$. The sharpness of this estimate is proved for an arbitrary class of functions having a given majorant of best approximations. Also investigated is the sharpness of the corresponding estimate for the rate of convergence of a Fourier series almost everywhere.