scholarly journals A radius of convexity problem

1981 ◽  
Vol 24 (3) ◽  
pp. 381-388 ◽  
Author(s):  
M.L. Mogra ◽  
O.P. Juneja
Keyword(s):  
Power Series ◽  
Unit Disc ◽  
Series Representation ◽  
Radius Of Convexity ◽  
Power Series Representation

The authors determine the sharp radius of convexity for functions analytic and starlike in the unit disc having power series representation of the form where an+1 is fixed. The estimate obtained is an improvement over the corresponding fixed second coefficient result. It is expected that this approach will lead to sharpening and improvement of a number of earlier known results.

scholarly journals A radius of convexity problem

1983 ◽  
Vol 28 (3) ◽  
pp. 433-439
Author(s):  
K.S. Padmanabhan ◽  
M.S. Ganesan
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Unit Disc ◽  
Series Representation ◽  
Radius Of Convexity ◽  
Schwarz’S Lemma ◽  
Power Series Representation

The authors determine the sharp radius of convexity for functions analytic in the unit disc having power series representation of the form f(z) = z + an+1zn+1 + an+2zn+2 + … where an+1 is fixed and such that zf′(z)/f(x) = (1 + Aw(z))/(1 + Bw(z)), −1 ≤ B < 0 < A ≤ 1 where w(z) is an analytic function satisfying the conditions of Schwarz's lemma, in the case A + B ≥ 0. The estimate obtained is an improvement over the corresponding result obtained by Mogra and Juneja for functions analytic and starlike in the unit disc, with missing coefficients where the initial non-vanishing coefficient is fixed.

scholarly journals A unified approach to radius of convexity problems for certain classes of univalent analytic functions

1984 ◽  
Vol 7 (3) ◽  
pp. 443-454
Author(s):  
M. L. Mogra ◽  
O. P. Juneja
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Unit Disc ◽  
Series Representation ◽  
Unified Approach ◽  
Special Cases ◽  
Radius Of Convexity ◽  
Power Series Representation

We consider functionsfanalytic in the unit disc and assume the power series representation of the formf(z)=z+an+1zn+1+an+2zn+2+…wherean+1is fixed throughout. We provide a unified approach to radius convexity problems for different subclasses of univalent analytic functions. Numerous earlier estimates concerning the radius of convexity such as those involving fixed second coefficient,ninitial gaps,n+1symmetric gaps, etc. are discussed. It is shown that several known results, follow as special cases of those presented in this paper.

scholarly journals A class of gap series with small growth in the unit disc

2002 ◽  
Vol 32 (1) ◽  
pp. 29-40
Author(s):  
L. R. Sons ◽  
Zhuan Ye
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Unit Disc ◽  
Series Representation ◽  
Gap Series ◽  
Small Growth ◽  
Power Series Representation

Letβ>0and letαbe an integer which is at least2. Iffis an analytic function in the unit discDwhich has power series representationf(z)=∑k=0∞ak zkα,limsupk→∞ (log+|ak|/logk)=α(1+β), then the first author has proved thatfis unbounded in every sector{z∈D:Φ−ϵ<argz<Φ+ϵ, forϵ>0}. A natural conjecture concerning these functions is thatlimsupr→1−(logL(r)/logM(r))>0, whereL(r)is the minimum of|f(z)|on|z|=randM(r)is the maximum of|f(z)|on|z|=r. In this paper, investigations concerning this conjecture are discussed. For example, we prove thatlimsupr→1−(logL(r)/logM(r))=1andlimsupr→1−(L(r)/M(r))=0whenak=kα(1+β).

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