The author proves that if \(0\leq \beta <\alpha \leq 1\) and \(f\in H_{\alpha}\), then \(\| E^ q_ n(f)-f\|_{\beta}=O(n^{\beta - \alpha}\log n)\) holds, where \(H_{\alpha}\) denotes the well-known Hölder space of order \(\alpha\), \(\| \cdot \|_{\alpha}\) its norm, and \(E^ q_ n(f)\) are the Euler-means of the Fourier series of f.