Inequalities for the moduli of circumferentially mean \(p\)-valent functions (Q280887)

The paper deals with normalized families of holomorphic functions multi-valent in the unit disc \({\mathbb U}\). The function \(f\) is said to be circumferentially mean \(p\)-valent if for any \(\rho > 0\) the inequality \[ \frac{1}{2\pi} \int\limits_{0}^{2\pi} n(\rho e^{i\varphi}, f) d\varphi \leq p \] holds, where \(n(w,f)\) is the number of roots of the equation \(f(z) = w\) in \({\mathbb U}\). Denote by \(M_p(\omega)\), \(p > 1\), the class of functions \(f\) which are holomorphic in \({\mathbb U}\), circumferentially mean \(p\)-valent, and normalized by the conditions \(f(0) = 0\) and \(f(\omega) = \omega\), and denote by \(M_p(\omega,\lambda)\) the subclass of all functions of the class \(M_p(\omega)\) mapping the disc \({\mathbb U}\) onto the Riemann surface \({\mathcal R}(f)\), which possesses the following property: for all \(\rho\), \(\lambda \leq \rho < \infty\), every closed Jordan curve on the surface \({\mathcal R}(f)\) covers this circle in a \(p\)-fold manner. Sharp upper and lower bounds on the modulus \(|f(z)|\) for some \(z\in (-1,0)\) for the functions in the class \(M_p(\omega,\lambda)\) are obtained.