scholarly journals Subclasses of univalent functions subordinate to convex functions

1997 ◽  
Vol 20 (2) ◽  
pp. 243-247
Author(s):  
Yong Chan Kim ◽  
Il Bong Jung
Keyword(s):  
Convex Functions ◽  
General Class ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Characterization Theorems

In this paper, we define a new subclassℳα(A,B)of univalent functions and investigate several interesting characterization theorems involving a general classS*A,Bof starlike functions.

scholarly journals On a subclass ofn-starlike functions

2005 ◽  
Vol 2005 (17) ◽  
pp. 2841-2846 ◽  
Author(s):  
Mugur Acu ◽  
Shigeyoshi Owa
Keyword(s):  
Fixed Point ◽  
Convex Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Starlike And Convex Functions

In 1999, Kanas and Rønning introduced the classes of starlike and convex functions, which are normalized withf(w)=f'(w)−1=0andwa fixed point inU. In 2005, the authors introduced the classes of functions close to convex andα-convex, which are normalized in the same way. All these definitions are somewhat similar to the ones for the uniform-type functions and it is easy to see that forw=0, the well-known classes of starlike, convex, close-to-convex, andα-convex functions are obtained. In this paper, we continue the investigation of the univalent functions normalized withf(w)=f'(w)−1=0andw, wherewis a fixed point inU.

scholarly journals Integral operators of certain univalent functions

1985 ◽  
Vol 8 (4) ◽  
pp. 653-662 ◽  
Author(s):  
O. P. Ahuja
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Univalent Functions ◽  
Integral Operators ◽  
Starlike Functions ◽  
The Family

A functionf, analytic in the unit discΔ, is said to be in the familyRn(α)ifRe{(znf(z))(n+1)/(zn−1f(z))(n)}>(n+α)/(n+1)for someα(0≤α<1)and for allzinΔ, wheren ϵ No,No={0,1,2,…}. The The classRn(α)contains the starlike functions of orderαforn≥0and the convex functions of orderαforn≥1. We study a class of integral operators defined onRn(α). Finally an argument theorem is proved.

scholarly journals On the Difference of Coefficients of Starlike and Convex Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1521
Author(s):  
Young Jae Sim ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Convex Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper And Lower Bounds ◽  
Starlike And Convex Functions ◽  
The Difference

Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions given by f(z)=z+∑n=2∞anzn for z∈D. Let S*⊂S be the subset of starlike functions in D and C⊂S the subset of convex functions in D. We give sharp upper and lower bounds for |a3|−|a2| for some important subclasses of S* and C.

scholarly journals On the Difference of Inverse Coefficients of Univalent Functions

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2040
Author(s):  
Young Jae Sim ◽  
Derek Keith Thomas
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Upper Bound ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Starlike Functions ◽  
Upper And Lower Bounds ◽  
The Difference ◽  
Bazilevic Functions

Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions with f(0)=0, and f′(0)=1. Let F be the inverse function of f, given by F(z)=ω+∑n=2∞Anωn for some |ω|≤r0(f). Let S*⊂S be the subset of starlike functions in D, and C the subset of convex functions in D. We show that −1≤|A3|−|A2|≤3 for f∈S, the upper bound being sharp, and sharp upper and lower bounds for |A3|−|A2| for the more important subclasses of S* and C, and for some related classes of Bazilevič functions.

scholarly journals On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1497 ◽  
Author(s):  
Katarzyna Tra̧bka-Wiȩcław ◽  
Paweł Zaprawa ◽  
Magdalena Gregorczyk ◽  
Andrzej Rysak
Keyword(s):  
Analytic Function ◽  
Real Number ◽  
Convex Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Positive Real Part ◽  
Positive Real ◽  
Hankel Determinants ◽  
Coefficient Problems ◽  
Theoretical Results

In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( μ ) = a 4 − μ a 2 a 3 and Φ f ( μ ) = a 2 a 4 − μ a 3 2 for a real number μ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + … , | z | < 1 . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re f ( z ) / z > α or Re f ′ ( z ) > α , α ∈ [ 0 , 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( μ ) and Φ f ( μ ) for close-to-convex functions such that f ′ ( z ) = h ( z ) / ( 1 − z ) 2 , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of Θ f ( μ ) , Φ f ( μ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions.

scholarly journals Convolution conditions for bounded \(\alpha\)-starlike functions of complex order

2017 ◽  
Vol 71 (1) ◽  
pp. 65
Author(s):  
A. Y. Lashin
Keyword(s):  
Complex Plane ◽  
Convex Functions ◽  
Analytic Functions ◽  
Unit Disc ◽  
General Class ◽  
Starlike Functions ◽  
Starlike And Convex Functions ◽  
Complex Order ◽  
Sălăgean Operator ◽  
Salagean Operator

Let \(A\) be the class of analytic functions in the unit disc \(U\) of the complex plane \(\mathbb{C}\) with the normalization \(f(0)=f^{^{\prime }}(0)-1=0\). We introduce a subclass \(S_{M}^{\ast }(\alpha ,b)\) of \(A\), which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class \(S_{M}^{\ast }(n,\alpha ,b)\) (\(n\geq 0\)) related to \(S_{M}^{\ast }(\alpha ,b)\) is also considered under the same conditions. Among other things, we find convolution conditions for a function \(f\in A\) to belong to the class \(S_{M}^{\ast }(\alpha ,b)\). Several properties of the class \(S_{M}^{\ast }(n,\alpha ,b)\) are investigated.

scholarly journals Coefficient problems for classes Hq (φ) and Lq (φ)

2021 ◽  
Vol 36 ◽  
pp. 04003
Author(s):  
Aini Janteng ◽  
Andy Pik Hern Liew
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Derivative Operator ◽  
Coefficient Problems

A class of analytic functions is denoted by M. Furthermore, S⸦M includes of analytic, normalized and univalent functions. The main -subclasses of S are starlike functions, S and convex functions, S* . Recently, many mathematicians studied about the q-derivative operator. Inspired by the ideas from some previous works, we introduce another two new subclasses of M . The coefficient problems in particular the upper bounds of the Fekete-Szegö (F-S) functional for these subclasses were obtained.

scholarly journals Bounds for the Second Hankel Determinant of Certain Univalent Functions

2019 ◽  
Vol 8 (10S) ◽  
pp. 262-266
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Hankel Determinant ◽  
Starlike And Convex Functions ◽  
Second Hankel Determinant

We study the estimates for the Second Hankel determinant of analytic functions. Our class includes (j,k)-convex, (j,k)-starlike functions and Ma-Minda starlike and convex functions..

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

scholarly journals The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

2021 ◽  
Author(s):  
Young Jae Sim ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

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