scholarly journals Radius problems for a subclass of close-to-convex univalent functions

1992 ◽  
Vol 15 (4) ◽  
pp. 719-726
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
The Other ◽  
Radius Problems ◽  
Convex Univalent Functions ◽  
Class Of Functions

LetP[A,B],−1≤B<A≤1, be the class of functionspsuch thatp(z)is subordinate to1+Az1+Bz. A functionf, analytic in the unit diskEis said to belong to the classKβ*[A,B]if, and only if, there exists a functiongwithzg′(z)g(z)∈P[A,B]such thatRe(zf′(z))′g′(z)>β,0≤β<1andz∈E. The functions in this class are close-to-convex and hence univalent. We study its relationship with some of the other subclasses of univalent functions. Some radius problems are also solved.

Radius problems for univalent functions

2020 ◽  
Vol 26 (1) ◽  
pp. 111-115
Author(s):  
Janusz Sokół ◽  
Katarzyna Trabka-Wiȩcław
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Radius Problems ◽  
Radius Of Convexity

AbstractThis paper considers the following problem: for what value r, {r<1} a function that is univalent in the unit disk {|z|<1} and convex in the disk {|z|<r} becomes starlike in {|z|<1}. The number r is called the radius of convexity sufficient for starlikeness in the class of univalent functions. Several related problems are also considered.

scholarly journals Univalent functions with univalent Gelfond-Leontev derivatives

1984 ◽  
Vol 29 (3) ◽  
pp. 329-348 ◽  
Author(s):  
O.P. Juneja ◽  
S.M. Shah
Keyword(s):  
Unit Disk ◽  
Shift Operator ◽  
Univalent Functions ◽  
Nondecreasing Sequence ◽  
Growth Properties ◽  
Linear Invariant ◽  
Definition Of ◽  
Image Position ◽  
Linear Invariant Family ◽  
Class Of Functions

Let be a nondecreasing sequence of positive numbers. We consider Gelfond-Leontev derivative Df(z), of a function , defined by for univalence and growth properties, and extend some results of Shah and Trimble. Set en = {d1d2 … dn), n≥l, e0 = 1, . Let r be the radius of convergence of p(z). We state parts of Theorem 1 and Corollaries. Let f and all Dkf, k = 1, 2,…, be analytic and univalent in the unit disk U. Then(iii) if p is entire and of growth (ρ, T) then f must be entire and of growth not exceeding (ρ, 2d2T),(iv) if D corresponds to the shift operator (dn ≡ l), then .Another class of functions is defined by a condition of the form |an+1/an| ≤ bn+1/dn+1, where is a sequence of positive numbers satisfying and inequality, and it is shown that all functions in this class along with all their Gelfond–Leontev successive derivatives are regular and univalent in U. An extension of the definition of a linear invariant family is given and results analogous to (i) and (ii) are stated.

scholarly journals Applications of Certain Operators to the Classes of Analytic Functions Related to the Generalized Janowski Functions

2020 ◽  
pp. 211-225
Author(s):  
Khalida Inayat Noor ◽  
Shujaat Ali Shah
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Linear Operators ◽  
Janowski Functions ◽  
Radius Problems ◽  
Convex Univalent Functions

We introduce certain subclasses of analytic functions related to the class of analytic, convex univalent functions. We discuss some results including inclusion relationships and invariance of the classes under convex convolution in terms of certain linear operators. Applications of these results associated with the generalized Janowski functions and conic domains are considered. Also, several radius problems are investigated.

Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions

1980 ◽  
Vol 32 (6) ◽  
pp. 1311-1324 ◽  
Author(s):  
D. Bshouty ◽  
W. Hengartner ◽  
G. Schober
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Univalent Functions ◽  
The Other ◽  
Open Unit ◽  
Open Unit Disk ◽  
Identity Mapping ◽  
The Best Possible Constant

Let S be the set of all normalized univalent analytic functions ƒ(z) = z + a2z2 + … in the open unit disk U. Then ƒ(U) contains the disk . Here is the best possible constant and is referred to as the Koebe constant for S. On the other extreme, ƒ(U) cannot contain the disk {|w| < 1}; unless ƒ is the identity mapping.In order to interpolate between the class S and the identity mapping, one may introduce the families , of functions ƒ ∈ S such that ƒ(U) contains the disk {|w| < d};. Then S(d1) ⊃ S(d2) for d1 < d2, and S(1) contains only the identity mapping. It is obvious that d is the “Koebe constant” for S(d). The relation between d and the second coefficient a2 has been studied by E. Netanyahu [5, 6].

scholarly journals On an integral operator for convex univalent functions

1986 ◽  
Vol 34 (2) ◽  
pp. 211-218
Author(s):  
Vinod Kumar ◽  
S. L. Shukla
Keyword(s):  
Integral Operator ◽  
Real Number ◽  
Complex Number ◽  
Univalent Functions ◽  
Corrected Version ◽  
Convex Univalent Functions ◽  
Class Of Functions

Let K (m,M) denote the class of functions regular and satisfying |1 + zf″(z)/f′(z)− m| < M in |z| < 1, where |m−1| < M ≤ m. Recently, R.K. Pandey and G. P. Bhargava have shown that if f ε K (m,M), then the function du also belongs to K (m,M) provided α is a complex number satisfying the inequality |α| ≤ (1−b)/2, where b = (m-1)/M. In this paper we show by a counterexample that their inequality is in general wrong, and prove a corrected version of their result. We show that F ε K (m,M) provided that α is a real number satisfying −φ ≤ α ≤1, φ = (M−|m−1|)/(M + |m−1|), or a complex number satisfying |α| ≤ φ. In both cases the bounds for α are sharp.

scholarly journals A problem on the Riesz-Dunford operator calculus and convex univalent functions

1983 ◽  
Vol 24 (2) ◽  
pp. 129-130 ◽  
Author(s):  
J. S. Hwang
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Univalent Function ◽  
Convex Set ◽  
Univalent Functions ◽  
Operator Calculus ◽  
Convex Univalent Functions ◽  
Ky Fan ◽  
Convex Univalent Function

In his paper [3], Ky Fan asked whether if f is a convex univalent function in the unit disk, with f(0) = 0 and f'(0) = 1, then is it true that the set of f(A) is a convex set of operators, when A runs through all proper contractions on a Hilbert space? We answer this question in the negative.

scholarly journals A note on successive coefficients of spirallike functions

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1199-1207 ◽  
Author(s):  
Ming Li
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Extremal Functions ◽  
Lower And Upper Bounds ◽  
Bieberbach Conjecture ◽  
Precise Form ◽  
Spirallike Functions ◽  
Class Of Functions

Even there were several facts to show that ||an+1(f)|-|an(f)|| ? 1 is not true for the whole class of normalised univalent functions in the unit disk with with the form f(z) = z + ??,k=2 akzk. In 1978, Leung[7] proved ||an+1(f)|-|an(f)|| is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion |an| ? n for starlike functions. Since ||an+1(f)|-|an(f)|| ? 1 implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of ||an+1(f)|-|an(f)|| for the class of spirallike functions as this class of functions is closely related to starlike functions. In this article we prove that this functional is bounded by 1 and equality occurs only for the starlike case. We are also able to give a precise form of extremal functions. Furthermore we also try to find the sharp bound of ||an+1(f)|-|an(f)|| for non-starlike spirallike functions. By using the Carath?odory-Toeplitz theorem, we obtain the sharp lower and upper bounds of |an+1(f)|-|an(f)| for n = 1 and n = 2. These results disprove the expected inequality ||an+1(f)|-|an(f)||? cos ? for ?-spirallike functions.

scholarly journals Hankel Determinants for Univalent Functions Related to the Exponential Function

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1211
Author(s):  
Paweł Zaprawa
Keyword(s):  
Unit Disk ◽  
Exponential Function ◽  
Univalent Functions ◽  
Main Idea ◽  
The Other ◽  
Hankel Determinants ◽  
Other Hand ◽  
Almost All

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.

scholarly journals Inequalities of harmonic univalent functions with connections of hypergeometric functions

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Janusz Sokół ◽  
Rabha W. Ibrahim ◽  
M. Z. Ahmad ◽  
Hiba F. Al-Janaby
Keyword(s):  
Unit Disk ◽  
Hypergeometric Functions ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Class Of Functions

AbstractLet SH be the class of functions f = h+g that are harmonic univalent and sense-preserving in the open unit disk U = { z : |z| < 1} for which f (0) = f'(0)-1=0. In this paper, we introduce and study a subclass H( α, β) of the class SH and the subclass NH( α, β) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H( α, β) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.

scholarly journals Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Davood Alimohammadi ◽  
Ebrahim Analouei Adegani ◽  
Teodor Bulboacă ◽  
Nak Eun Cho
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Current Paper ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Bounds ◽  
Logarithmic Coefficients ◽  
Class Of Functions ◽  
Logarithmic Coefficient

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f z = z + ∑ n = 2 ∞ a n z n analytic and univalent in the open unit disk U , then the logarithmic coefficients γ n f of the function f ∈ S are defined by log f z / z = 2 ∑ n = 1 ∞ γ n f z n . In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like C 1 + α z for α ∈ 0 , 1 and C V hpl 1 / 2 were estimated. Further, conjectures for the logarithmic coefficients γ n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f ∈ C 1 + α z , then the logarithmic coefficients of f satisfy the inequalities γ n ≤ α / 2 n n + 1 , n ∈ ℕ . Equality is attained for the function L α , n , that is, log L α , n z / z = 2 ∑ n = 1 ∞ γ n L α , n z n = α / n n + 1 z n + ⋯ , z ∈ U .

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