Ranges of values of systems of coefficients in the class of functions with positive real part in an annulus

1977 ◽  
Vol 8 (6) ◽  
pp. 642-649 ◽  
Author(s):  
E. G. Goluzina
Keyword(s):  
Positive Real Part ◽  
Positive Real ◽  
Class Of Functions

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 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.

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 On radii of convexity and starlikeness of some classes of analytic functions

1991 ◽  
Vol 14 (4) ◽  
pp. 741-746 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Analytic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Class Of Functions

LetP[A,B],−1≤B<A≤1, be the class of functionspsuch thatp(z)is subordinate to1+Az1+Bz. LetP(α1)be the class of functions with positive real part greater thanα1,0≤α1≤1. It is clear thatp[A,B]⊂P(1−A1−B)⊂P[1,−1]. The principal results in this paper are the determination of the radius ofβ-starlikeness andβ-convexity off(z)withβ=1−A1−B, whenf(z)is restricted to certain classes of univalent and analytic functions related vithP[A,B].

scholarly journals Radius of univalence and starlikeness of a class of analytic functions

1972 ◽  
Vol 14 (1) ◽  
pp. 1-8
Author(s):  
R. S. Gupta
Keyword(s):  
Analytic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
The Family ◽  
Image Position ◽  
Class Of Functions ◽  
Radius Of Univalence

Letp denote the family of functions regular in E{z:|z| < 1} and with positive real part there. We propose to study, in this article, the subclass p2a1 of p whose functions P(z) have pre-assigned second coefficient 2a1. In what follows we may assume, without loss in generality, that a1 is real and non-negative. This assumption will be made throughout. As is well known [2], 0 ≦ a1 ≦ 1. In Theorem 1 we derive a generalization of Zmorovic's theorem 1, [3]. determine the radius of univalence and starlikeness of the class of functions

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