Partial sums and the radius problem for some class of conformal mappings

Let A denote the set of normalized analytic functions f(z) = z + Σk=2∞akzk in the unit disk {pipe}z{pipe} < 1, and let sn(z) represent the nth partial sum of f(z). Our first objective of this note is to obtain a bound for, when f ∈ A is univalent in D. Let U denote the set of all f ∈ A in D satisfying the condition, for {pipe}z{pipe} < 1. In case f″ (0) = 0, we find that all corresponding sections sn of f ∈ U are in A in the disk,. We also show that Re (f(z)/sn(z)) > 1/2 in the disk,. Finally, we establish a necessary coefficient condition for functions in U and the related radius problem for an associated subclass of U. In result, we see that if f ∈ U then for n ≥ 3 we have,. © 2011 Pleiades Publishing, Ltd.