Let f belong to the space \(L_p\) on the real line or on the period if \(p\in [1,+\infty )\), or to the space \(UCB(\mathbb {R})\) of uniformly continuous bounded on \(\mathbb {R}\) functions if \(p=+\infty\). For \(N>0\), denote by \(\sigma _Nf\) the Nth-order Fejér average of f, and let \(A_u(f)_p\) be the best approximation of f by entire functions of exponential type not greater than u. We prove the inequality $$\begin{aligned} \Vert f-\sigma _Nf\Vert _p\leqslant \frac{D}{N}\int \limits _{0}^{N}A_u(f)_p~du \end{aligned}$$with the constant \(D<3.219206\). In the space \(UCB(\mathbb {R})\), the sharp constant in this estimate is not less than 2. This result sharpens the previous estimates of S. B. Stechkin and G. I. Natanson.