Let \(f\) be an entire function with \(f(z) = \sum c_{k} z^{k}\), and let \[ F_{n}(z) = \frac {1} {2 \pi n} \int_{-\pi}^{\pi} f(ze^{iu})\bigg(\frac {\sin{(nu/2)}} {\sin{(u/2)}} \bigg)^{2} du = \sum_{k=0}^{n-1} c_{k} \frac {n - k} {n} z^{k} . \] For \(r \geq 0\), let \(\| f\| _{r} = \sup{\{| f(z)| : | z| \leq r\}}\). It is known that \[ \| F_{n}(f) - f\| _{r} \leq M \omega_{1}(f, 1/n)_{r} \leq \frac {M \| f\| _{r}} {n} \;, \] where \(M > 0\) and \(\omega_{1}(f, 1/n)_{r}\) is the usual modulus of continuity of \(f\) in the disc \(\{z: | z| \leq r \}\). For \(s\) and \(n\) positive integers, define the Riesz–Zygmund mean \[ R_{n,s}(f)(z) = \sum_{k=0}^{n-1} c_{k} \bigg[1 - \bigg(\frac {k} {n} \bigg)^{s} \bigg] z^{k}. \] The inequality above can be stated as \[ \| R_{n,1}(f) - f\| _{r} \leq O(1/n) . \] The author shows a corresponding result for \(p\)–th derivatives, namely, that \[ \| R_{n,s}^{(p)}(f) - f^{(p)} \| _{r} \sim 1/n^{s} , \] whenever \(f\) is not a polynomial of degree less than \(p\). Here, \(a \sim b\) means there exist positive constants \(M_{1}\) and \(M_{2}\) such that \(M_{1}b \leq a \leq M_{2}b\). The proof of the result is basically elementary, making use of a result of I. Bruj and G. Schmieder [J. Approximation Theory 100, No. 1, 157–182 (1999; Zbl 0952.41015)].