Inequalities for moduli of the circumferentially mean $p$-valent functions

Let $f$ be a circumferentially mean $p$-valent function in the disk $|z|<1$ with Montel's normalization: $f(0)=0$, $f(\omega)=\omega$ $(0<\omega<1)$. Under an additional constraint on the covering of the concentric circles by $f$, precise lower and upper bounds of modulus $|f(z)|$ for some $z\in(-1,0)$ are established. The necessity of such constraint for the non-trivial estimates to be true is shown.