The estimates are given for the
radius of the largest disk about the origin on which all the functions of the form
a∕z + ϕ(z) are p-valent, where ϕ(z) is an analytic function defined in the unit disk
and of modulus less than unity there. Similar results are obtained concerning the
starlikeness of the image of circles about the origin.
Suppose a polynomial of degree n assumes p-times a certain value in the center of
a circle and does not take this value elsewhere in this circle. The author determines
the largest concentric circle in which the polynomial is p-valent. The same
problem is then considered in a more general setting and similar results are
obtained.
