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

Abstract

Let Open image in new window denote the set of normalized analytic functions f(z) = z + Σk=2∞akzk in the unit disk |z| < 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 \(|\frac{{s_n (z)}} {{f(z)}} - 1| \) when f ∈ Open image in new window is univalent in ⅅ. Let Open image in new window denote the set of all f ∈ Open image in new window in ⅅ satisfying the condition $$\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^2 - 1} \right| < 1$$ for |z| 1. In case f″ (0) = 0, we find that all corresponding sections sn of f ∈ Open image in new window are in Open image in new window in the disk \(|z| 1/2 in the disk \( \left| z \right| < \sqrt {\sqrt 5 - 2}\). Finally, we establish a necessary coefficient condition for functions in Open image in new window and the related radius problem for an associated subclass of Open image in new window. In result, we see that if f ∈ Open image in new window then for n ≥ 3 we have $$\left| {\frac{{f(z)}} {{s_n (z)}} - \frac{4} {3}} \right| < \frac{2} {3}for|z| < r_n : = 1 - \frac{{2\log n}} {n}$$