Partial sums of certain analytic and univalent functions. (English) Zbl 1092.30019

Frasin, B. A.

Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21, 135-145 (2005).

Summary: We study the ratio of a function of the form \[ f(z) = z + \sum^\infty_{k=2} a_k z^k \] to its sequence of partial sums of the form \(f_n(z) = z + \sum^n_{k=2} a_k z^k\). Also, we will determine sharp lower bounds for \(\text{Re} \{f(z)/f_n(z)\}\), \(\text{Re} \{f_n(z)/f(z)\}\), \(\text{Re}\{f^\prime(z)/f^\prime_n(z)\}\) and \(\text{Re}\{f^\prime_n(z)/f^\prime (z)\}\).

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

analytic functions; univalent functions; Hadamard product; partial sums