Let \(\mathcal H(\mathbb D)\) be the space of all analytic functions on the unit disk \(\mathbb D \subset \mathbb C\) with the topology of uniform convergence on compact subsets of \(\mathbb D\). For \(f\in \mathcal H(\mathbb D)\) denote \((P_{r}f)(z)=f(rz)\), \(0\leq r<1\), \((R_{t}f)(z)=f(e^{it}z)\), \(t\in\mathbb R\), \((s_{n}f)(z)=\sum_{k=0}^{n}a_{k}z^{k}\) for \(f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\) and \(n\in\mathbb N\). Let \(A(\mathbb D)\) be the disk algebra. A Banach space \(X\) is called a Banach space of analytic functions if \(A(\mathbb D)\subset X \subset \mathcal H(\mathbb D)\) with continuous inclusions. Such space \(X\) is called homogeneous if it is invariant for all operators \(P_{r}\) and \(R_{t}\), \(R_{t}\) are isometric on \(X\) and \(P_{r}\) are uniformly bounded on \(X\). Let \(\omega:[0,\pi]\to\mathbb R^{+}\) be a continuous and non-decreasing weight function such that \(\omega(0)=0\) satisfying some additional conditions. The main result of the paper establishes the equivalence of some statements for the functions from a homogeneous Banach space \(X\) of analytic functions satisfying some additional conditions. We mention here three of these properties: 1) \(\| R_{t}f - f \|_{X}=O(\omega(t))\), \(t \to 0^{+}\); 2) \(\| P_{r}f - f \|_{X}=O(\omega(1-r))\), \(r \to 1^{-}\); 3) \(\| f - s_{n}f \|_{X}=O(\omega(n^{-1}))\), \(n\to\infty\).