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′.

scholarly journals Some characteristic properties of analytic functions

2016 ◽  
Vol 24 (1) ◽  
pp. 353-369
Author(s):  
R. K. Raina ◽  
Poonam Sharma ◽  
G. S. Sălăgean
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Integral Representations ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Bilinear Mapping ◽  
Initial Coefficient

AbstractIn this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that,where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

scholarly journals Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hameed Ur Rehman ◽  
Maslina Darus ◽  
Jamal Salah
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Positive Real Part ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Real ◽  
Faber Polynomial ◽  
Coefficient Problems

In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. This formula is applied to a certain class of bi-univalent functions and solve the n -th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n -th term.

scholarly journals On Coefficient Functionals for Functions with Coefficients Bounded by 1

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 491
Author(s):  
Paweł Zaprawa ◽  
Anna Futa ◽  
Magdalena Jastrzębska
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Extremal Functions ◽  
Hankel Determinant ◽  
Schwarz Function ◽  
Positive Real ◽  
Special Case

In this paper, we discuss two well-known coefficient functionals a 2 a 4 - a 3 2 and a 4 - a 2 a 3 . The first one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class Q R ( 1 2 ) of analytic functions with real coefficients which satisfy the condition ( ) f ( z ) z > 1 2 for z in the unit disk Δ . It is known that all coefficients of f ∈ Q R ( 1 2 ) are bounded by 1. We find the upper bound of a 2 a 4 - a 3 2 and the bound of | a 4 - a 2 a 3 | . We also consider a few subclasses of Q R ( 1 2 ) and we estimate the above mentioned functionals. In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates significantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefficient functionals considered in the case of real coefficients and arbitrary coefficients.

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

An Inequality Concerning Analytic Functions with a Positive Real Part

1969 ◽  
Vol 21 ◽  
pp. 1172-1177 ◽  
Author(s):  
Thomas H. MacGregor
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Image Position

This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for |z| < 1 and has values lying in a strip of width δ. This result is known and was first proved by Tammi (1).Our second theorem is a generalization of this. Namely, ifis analytic for |z| < 1 and satisfies Re{zmf(m>(z)} ≧ A andthenconverges.Another application of our fundamental inequality is the following. Let be analytic for |z| < 1 and satisfy Re p(z) ≧ 0 and set and .

Second hankel determinat for certain analytic functions satisfying subordinate condition

2018 ◽  
Vol 68 (2) ◽  
pp. 463-471
Author(s):  
Erhan Deniz ◽  
Levent Budak
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Real Axis ◽  
Upper Bound ◽  
Analytic Functions ◽  
Positive Real Part ◽  
Hankel Determinant ◽  
Positive Real ◽  
Toeplitz Determinants ◽  
Second Hankel Determinant

Abstract In this paper, we introduce and investigate the following subclass $$\begin{array}{} \displaystyle 1+\frac{1}{\gamma }\left( \frac{zf'(z)+\lambda z^{2}f''(z)}{\lambda zf'(z)+(1-\lambda )f(z)}-1\right) \prec \varphi (z)\qquad\left( 0\leq \lambda \leq 1,\gamma \in \mathbb{C} \smallsetminus \{0\}\right) \end{array} $$ of analytic functions, φ is an analytic function with positive real part in the unit disk 𝔻, satisfying φ (0) = 1, φ '(0) > 0, and φ (𝔻) is symmetric with respect to the real axis. We obtain the upper bound of the second Hankel determinant | a2a4− $\begin{array}{} a^2_3 \end{array} $ | for functions belonging to the this class is studied using Toeplitz determinants. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

scholarly journals On the univalency for certain subclass of analytic functions involving Ruscheweyh derivatives

2002 ◽  
Vol 31 (9) ◽  
pp. 567-575 ◽  
Author(s):  
Liu Mingsheng
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Covering Theorem ◽  
Inclusion Relations ◽  
Special Case ◽  
Class Of Functions

LetHbe the class of functionsf(z)of the formf(z)=z+∑ K=2 + ∞a k z k, which are analytic in the unit diskU={z;|z|<1}. In this paper, we introduce a new subclassBλ(μ,α,ρ)ofHand study its inclusion relations, the condition of univalency, and covering theorem. The results obtained include the related results of some authors as their special case. We also get some new results.

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 .

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