Successive coefficients for spirallike and related functions

Abstract

We consider the family of all analytic and univalent functions in the unit disk of the form f (z) = z+a 2 z 2 +a 3 z 3 +· · · . Our objective in this paper is to estimate the difference of the moduli of successive coefficients, that is |a n+1 | − |a n | , for f belonging to the family of γ-spirallike functions of order α. Our particular results include the case of starlike and convex functions of order α and other related class of functions.2010 Mathematics Subject Classification. 30D30, 30C45, 30C50 30C55.

On Some Classes of Spirallike Functions Defined by the Salagean Operator

“…Using convolution properties, we find the necessary and sufficient condition, coefficient estimates and inclusion properties for these classes. More recent works can be found on [2,5,10,12,16].…”

“…In this section, we study some of the properties of foresaid convolution. Unless otherwise mentioned, we assume throughout this paper that −1 ≤ B < A ≤ 1, |α| < π 2 , |ξ| = 1 and D n f (z) is defined by (2). To prove our convolution properties, we shall need the following lemmas due to Silverman and Silvia [14].…”

Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction

“…Coefficient problems of analytic functions have always been of the great interest to researchers. Let A be a class of functions of the form f (z) = z + ∞ ∑ n=2 a n z n (1) which are analytic in the open unit disk ∆ = {z ∈ C : |z| < 1}. There are many papers in which the n-th coefficient a n has been estimated in various subclasses of analytic functions.…”

“…There are many papers in which the n-th coefficient a n has been estimated in various subclasses of analytic functions. The difference of the moduli of successive coefficients |a n+1 | − |a n | of a certain class of functions was also estimated (see, for example, [1][2][3][4]). The idea of estimating the difference of successive coefficients |a n+1 − a n | follows from the obvious inequality ||a n+1 | − |a n || ≤ |a n+1 − a n | .…”

On the Difference of Coefficients of Ozaki Close-to-Convex Functions

“…Let S denote the subclass of all univalent (that is, one-to-one) functions in A. In 1985, de Branges [4] solved the famous Bieberbach conjecture by showing that if f ∈ S, then |a n | ≤ n for n ≥ 2 with equality when f (z) = k(z) := z/(1 − z) 2 or a rotation of it. It was therefore natural to ask if for f ∈ S, the inequality ||a n+1 | − |a n || ≤ 1 is true when n ≥ 2.…”