scholarly journals On a certain class of functions whose derivatives have a positive real part in the unit disc

1970 ◽  
Vol 23 (1) ◽  
pp. 73-81 ◽  
Author(s):  
K. S. Padmanabhan
Keyword(s):  
Unit Disc ◽  
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 A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

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 Subordination of functions in subclass of Bazilevič Functions B 1(α, β)

2021 ◽  
Vol 2106 (1) ◽  
pp. 012026
Author(s):  
Marjono
Keyword(s):  
Unit Disc ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Sharp Estimates ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Coefficient Problems ◽  
Bazilevic Functions

Abstract Let f be analytic in the unit disc D = {z : |z| < 1} with f ( z ) = z + ∑ n = 2 ∞ a n z n , and for α ≥ 0 and 0 < β ≤ 1, let B 1(α, ß), denote for the class of Bazilevič functions satisfying the expression | arg z 1 − α f ′ ( z ) f ( z ) 1 − α | < β π 2 . We give sharp estimates for various coefficient problems for functions in B 1(α, β), which unify and extend well-known results for starlike functions, strongly starlike functions and functions whose derivative has positive real part in domain D.

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