Assume $\mu = \sum_{i \in \Lambda} p_i \cdot \mu \circ \phi_i^{-1}$ is a self-similar measure on $\mathbb{R}$ associated to an iterated function system $\Phi = \lbrace \phi_i \rbrace_{i \in \Lambda}$, $\phi_i(x) = r_i x + a_i$. I want to prove the following inequality; $$\dim \mu \leq \min \lbrace 1, \text{s-}\dim \mu \rbrace,$$ where $$ \text{s-}\dim \mu = \dfrac{\sum_{i \in \Lambda} p_i \log p_i}{\sum_{i \in \Lambda} p_i \log r_i} $$ is the similarity dimension of $\mu$. According to the author of the paper I am reading, this inequality is trivial. An identical one exists for the dimension of the attractor of $\Phi$ and that is quite easy to prove.
The $\text{s-}\dim \mu \leq 1$ part is trivial, $\text{s-}\dim \mu \leq \dim X \leq 1$ since $\mu$ is supported on the attractor $X$ but what about $\dim \mu \leq \text{s-}\dim \mu$?