Let \(n\) be a positive integer. Let \(\mathbf{U }\) be the unit disk, \(p\ge 1\) and let \(h^p(\mathbf{U })\) be the Hardy space of harmonic functions. Kresin and Maz’ya in a recent paper found a representation for the function \(H_{n,p}(z)\) in the inequality $$\begin{aligned}|f^{(n)} (z)|\le H_{n,p}(z)\Vert \mathfrak R (f-\mathcal{P }_{l})\Vert _{h^p(\mathbf{U })}, \quad \mathfrak R f\in h^p(\mathbf{U }), z\in \mathbf{U },\end{aligned}$$ where \(\mathcal{P }_{l}\) is a polynomial of degree \(l\le n-1\). We determine the sharp constant \(C_{p,n}\) in the inequality \(H_{n,p}(z)\le \frac{C_{p,n}}{(1-|z|^2)^{1/p+n}}\). This extends a recent result of Kalaj and Markovic, where only the case \(n=1\) was considered. As a corollary, an inequality for the modulus of \(n\)-th derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves a recent result of Kresin and Maz’ya.