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 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 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..

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 A differential inequality involving Ruscheweyh operator and univalent functions

2020 ◽  
Vol 28 (1) ◽  
pp. 115-123
Author(s):  
Pardeep Kaur ◽  
Sukhwinder Singh Billing
Keyword(s):  
Convex Functions ◽  
Differential Inequality ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Starlike And Convex Functions

AbstractIn the present paper, we find certain results on Ruscheweyh operator using differential inequality. In particular, we find sufficient conditions for starlike and convex functions.

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 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 Study on new integral operators defined using confluent hypergeometric function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Georgia Irina Oros
Keyword(s):  
Hypergeometric Function ◽  
Complex Plane ◽  
Convex Functions ◽  
Univalent Functions ◽  
Integral Operators ◽  
Confluent Hypergeometric Function ◽  
Differential Inequalities ◽  
Starlike And Convex Functions ◽  
Inclusion Properties ◽  
Admissible Functions

AbstractTwo new integral operators are defined in this paper using the classical Bernardi and Libera integral operators and the confluent (or Kummer) hypergeometric function. It is proved that the new operators preserve certain classes of univalent functions, such as classes of starlike and convex functions, and that they extend starlikeness of order $\frac{1}{2}$ 1 2 and convexity of order $\frac{1}{2}$ 1 2 to starlikeness and convexity, respectively. For obtaining the original results, the method of admissible functions is used, and the results are also written as differential inequalities and interpreted using inclusion properties for certain subsets of the complex plane. The example provided shows an application of the original results.

scholarly journals Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Khalil Ullah ◽  
Saira Zainab ◽  
Muhammad Arif ◽  
Maslina Darus ◽  
Meshal Shutaywi
Keyword(s):  
Holomorphic Function ◽  
Convex Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Open Unit ◽  
Hyperbolic Function ◽  
Open Unit Disc ◽  
Starlike And Convex Functions ◽  
Radius Problems

The aim of this particular article is at studying a holomorphic function f defined on the open-unit disc D = z ∈ ℂ : z < 1 for which the below subordination relation holds z f ′ z / f z ≺ q 0 z = 1 + tan h z . The class of such functions is denoted by S tan h ∗ . The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order β and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.

scholarly journals LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS

2019 ◽  
Vol 109 (2) ◽  
pp. 230-249 ◽  
Author(s):  
SAMINATHAN PONNUSAMY ◽  
NAVNEET LAL SHARMA ◽  
KARL-JOACHIM WIRTHS
Keyword(s):  
Upper Bound ◽  
Convex Functions ◽  
Univalent Functions ◽  
List Type ◽  
Starlike And Convex Functions ◽  
The Family ◽  
Logarithmic Coefficients ◽  
Dilogarithm Function ◽  
Image Position ◽  
Analytic And Univalent Functions

Let${\mathcal{S}}$be the family of analytic and univalent functions$f$in the unit disk$\mathbb{D}$with the normalization$f(0)=f^{\prime }(0)-1=0$, and let$\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$denote the logarithmic coefficients of$f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families${\mathcal{F}}(c)$and${\mathcal{G}}(c)$of functions$f\in {\mathcal{S}}$defined by$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)<1+{\displaystyle \frac{c}{2}},\quad z\in \mathbb{D},\end{eqnarray}$$for some$c\in (0,3]$and$c\in (0,1]$, respectively. We obtain the sharp upper bound for$|\unicode[STIX]{x1D6FE}_{n}|$when$n=1,2,3$and$f$belongs to the classes${\mathcal{F}}(c)$and${\mathcal{G}}(c)$, respectively. The paper concludes with the following two conjectures:∙If$f\in {\mathcal{F}}(-1/2)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$for$n\geq 1$, and$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$where$\text{Li}_{2}(x)$denotes the dilogarithm function.∙If$f\in {\mathcal{G}}(c)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$for$n\geq 1$.

scholarly journals Bounds for the Coefficient of Faber Polynomial of Meromorphic Starlike and Convex Functions

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1368
Author(s):  
Oh Sang Kwon ◽  
Shahid Khan ◽  
Young Jae Sim ◽  
Saqib Hussain
Keyword(s):  
Real Axis ◽  
Convex Functions ◽  
Meromorphic Functions ◽  
Starlike Functions ◽  
Monic Polynomial ◽  
Faber Polynomial ◽  
Strongly Convex ◽  
Starlike And Convex Functions ◽  
Strongly Starlike ◽  
Meromorphic Starlike Functions

Let Σ be the class of meromorphic functions f of the form f ( ζ ) = ζ + ∑ n = 0 ∞ a n ζ − n which are analytic in Δ : = { ζ ∈ C : | ζ | > 1 } . For n ∈ N 0 : = N ∪ { 0 } , the nth Faber polynomial Φ n ( w ) of f ∈ Σ is a monic polynomial of degree n that is generated by a function ζ f ′ ( ζ ) / ( f ( ζ ) − w ) . For given f ∈ Σ , by F n , i ( f ) , we denote the ith coefficient of Φ n ( w ) . For given 0 ≤ α < 1 and 0 < β ≤ 1 , let us consider domains H α and S β ⊂ C defined by H α = { w ∈ C : Re ( w ) > α } and S β = { w ∈ C : | arg ( w ) | < β } , which are symmetric with respect to the real axis. A function f ∈ Σ is called meromorphic starlike of order α if ζ f ′ ( ζ ) / f ( ζ ) ∈ H α for all ζ ∈ Δ . Another function f ∈ Σ is called meromorphic strongly starlike of order β if ζ f ′ ( ζ ) / f ( ζ ) ∈ S β for all ζ ∈ Δ . In this paper we investigate the sharp bounds of F n , n − i ( f ) , n ∈ N 0 , i ∈ { 2 , 3 , 4 } , for meromorphic starlike functions of order α and meromorphic strongly starlike of order β . Similar estimates for meromorphic convex functions of order α ( 0 ≤ α < 1 ) and meromorphic strongly convex of order β ( 0 < β ≤ 1 ) are also discussed.

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