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 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.

Holomorphic Functions with Positive Real Part

1982 ◽  
Vol 34 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Eric Sawyer
Keyword(s):  
Unit Ball ◽  
Hardy Spaces ◽  
Convex Cone ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Spaces Of Holomorphic Functions ◽  
Extreme Element ◽  
Image Position ◽  
Special Case

The main purpose of this note is to prove a special case of the following conjecture.Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functionsis in Hp(B2) for 0 < p < 3/2.

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 Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1707
Author(s):  
Renata Długosz ◽  
Piotr Liczberski
Keyword(s):  
Power Series ◽  
Power Series Expansion ◽  
Holomorphic Functions ◽  
Positive Real Part ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Homogeneous Polynomials ◽  
Positive Real ◽  
Sharp Estimates ◽  
Several Variables

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families.

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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