The extreme points of a class of functions with positive real part

1973 ◽  
Vol 202 (1) ◽  
pp. 85-87 ◽  
Author(s):  
Finbarr Holland
Keyword(s):  
Extreme Points ◽  
Positive Real Part ◽  
Positive Real ◽  
Class Of Functions

scholarly journals The extreme points of a class of functions with positive real part

1991 ◽  
Vol 44 (2) ◽  
pp. 253-261
Author(s):  
N. Samaris
Keyword(s):  
Extreme Point ◽  
Unit Disc ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Greatest Common Divisor ◽  
Positive Real ◽  
Taylor Coefficients ◽  
Class Of Functions

Let P1 be the class of holomorphic functions on the unit disc U = {z: |z| < 1} for which f(0) = 1 and Re f > 0. Let also Pn be the corresponding class on the unit disc Un. The inequality |ak| ≤ 2 is known for the Taylor coefficients in the class P1. In this paper, it is generalised for the class Pn. If ρ = (ρ1, ρ2, …, ρn), with ρ1, ρ2, …, ρn nonegative integers whose greatest common divisor is equal to 1, we describe the form of the functions f ∈ Pn under the restriction |aρ| = 2. Under the same restriction, we give conditions for a function to be an extreme point of the class Pn.

scholarly journals Extremal problems for a class of functions of positive real part and applications

1986 ◽  
Vol 41 (2) ◽  
pp. 152-164
Author(s):  
V. V. Anh ◽  
P. D. Tuan
Keyword(s):  
Lower Bound ◽  
Extremal Problems ◽  
Positive Real Part ◽  
Positive Real ◽  
Sharp Bounds ◽  
The Family ◽  
Radius Of Convexity ◽  
Class Of Functions

AbstractIn this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

scholarly journals The Extreme and Support Points of a New Class of Analytic Functions with Positive Real Part

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Zhigang Peng
Keyword(s):  
Analytic Functions ◽  
Extreme Points ◽  
Positive Real Part ◽  
Positive Real ◽  
New Class ◽  
Support Points

Suppose that 0<α<β<+∞. Let 𝒫(α,β) denote the set of functions p(z) that are analytic in D= {z:|z|<1}  and satisfy Rep(z)>0(|z|<1) and α≤p(0)≤β. In this paper, we investigate the extreme points and support points of 𝒫(α,β).

scholarly journals On some classes of holomorphic functions whose derivatives have positive real part

2021 ◽  
Vol 66 (3) ◽  
pp. 479-490
Author(s):  
Eduard Stefan Grigoriciuc
Keyword(s):  
Holomorphic Functions ◽  
Upper Bounds ◽  
Positive Real Part ◽  
Positive Real ◽  
Class Of Functions

"In this paper we discuss about normalized holomorphic functions whose derivatives have positive real part. For this class of functions, denoted R, we present a general distortion result (some upper bounds for the modulus of the k- th derivative of a function). We present also some remarks on the functions whose derivatives have positive real part of order , 2 (0; 1). More details about these classes of functions can be found in [6], [8], [7, Chapter 4] and [4]. In the last part of this paper we present two new subclasses of normalized holomorphic functions whose derivatives have positive real part which generalize the classes R and R(alfa). For these classes we present some general results and examples."

scholarly journals A note on neighborhoods of analytic functions having positive real part

1990 ◽  
Vol 13 (3) ◽  
pp. 425-429 ◽  
Author(s):  
Janice B. Walker
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Class Of Functions

LetPdenote the set of all functions analytic in the unit diskD={z||z|<1}having the formp(z)=1+∑k=1∞pkzkwithRe{p(z)}>0. Forδ≥0, letNδ(p)be those functionsq(z)=1+∑k=1∞qkzkanalytic inDwith∑k=1∞|pk−qk|≤δ. We denote byP′the class of functions analytic inDhaving the formp(z)=1+∑k=1∞pkzkwithRe{[zp(z)]′}>0. We show thatP′is a subclass ofPand detemineδso thatNδ(p)⊂Pforp∈P′.

Integral Means of Functions with Positive Real Part

1980 ◽  
Vol 32 (4) ◽  
pp. 1008-1020 ◽  
Author(s):  
F. Holland ◽  
J. B. Twomev
Keyword(s):  
Positive Real Part ◽  
Positive Real ◽  
Integral Means ◽  
Image Position ◽  
Class Of Functions

We denote by the class of functions of the formthat are regular in Δ = {z:|;z| < 1} and satisfy Re h(z) > 0 there. For 0 ≦ r < 1, we writeWe note that, for , the inequalityis classical.

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