scholarly journals On the Fekete-Szegö Problem for a Class of Analytic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Zhigang Peng
Keyword(s):  
Power Series ◽  
Real Number ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Upper Bounds ◽  
The Family ◽  
Class Of Functions

Let 𝒜 denote the class of functions which are analytic in the unit disk D={z:|z|<1} and given by the power series f(z)=z+∑n=2∞‍anzn. Let C be the class of convex functions. In this paper, we give the upper bounds of |a3-μa22| for all real number μ and for any f(z) in the family 𝒱={f(z):f∈𝒜, Re(f(z)/g(z))>0 for  some g∈C}.

scholarly journals Hankel Determinant of Second Order for Some Classes of Analytic Functions

2021 ◽  
Author(s):  
Milutin Obradović ◽  
Nikola Tuneski
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Second Order ◽  
Hankel Determinant

Let ƒ be analytic in the unit disk B and normalized so that ƒ (z) = z + a2z2 + a3z3 +܁܁܁. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order α, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.

scholarly journals A Generalized Class of Close-to-Convex Functions

2020 ◽  
Vol 44 (4) ◽  
pp. 533-538
Author(s):  
PARDEEP KAUR ◽  
SUKHWINDER SINGH BILLING
Keyword(s):  
Real Number ◽  
Unit Disk ◽  
Convex Functions ◽  
Starlike Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Real Numbers ◽  
Special Cases ◽  
Class Of Functions

Let ℋαϕ(β) denote the class of functions f, analytic in the open unit disk ???? which satisfy the condition ( ( ) ) zf-′(z-)- zf-′′(z-) ℜ (1 − α) + α 1 + ′ > β, z ∈ ????, ϕ(z ) f (z ) where α, β are pre-assigned real numbers and ϕ(z) is a starlike function. The special cases of the class ℋαϕ(β) have been studied in literature by different authors. In 2007, Singh et al. [?] studied the class ℋαz(β) and they established that functions in ℋαz(β) are univalent for all real numbers α, β satisfying the condition α ≤ β < 1 and the result is sharp in the sense that constant β cannot be replaced by a real number smaller than α. Singh et al. [?] in 2005, proved that for 0 < α < 1 functions in class ℋαz(α) are univalent. In 1975, Al-Amiri and Reade [?] showed that functions in class ℋαz(0) are univalent for all α ≤ 0 and also for α = 1 in ????. In the present paper, we prove that members of the class ℋαϕ(β) are close-to-convex and hence univalent for real numbers α, β and for a starlike function ϕ satisfying the condition β + α − 1 < αℜ( ) zϕ′(z) ϕ(z)≤ β < 1.

Third Hankel determinants for two classes of analytic functions with real coefficients

2021 ◽  
Vol 33 (4) ◽  
pp. 973-986
Author(s):  
Young Jae Sim ◽  
Paweł Zaprawa
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Lower And Upper Bounds ◽  
Hankel Determinants ◽  
The Third ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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 On Certain Classes of Convex Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Young Jae Sim ◽  
Oh Sang Kwon
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Real Numbers ◽  
Inverse Functions

For real numbersαandβsuch that0≤α<1<β, we denote by𝒦α,βthe class of normalized analytic functions which satisfy the following two sided-inequality:α<ℜ1+zf′′z/f′z<β  z∈𝕌,where𝕌denotes the open unit disk. We find some relationships involving functions in the class𝒦(α,β). And we estimate the bounds of coefficients and solve the Fekete-Szegö problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or biunivalent functions.

scholarly journals Subclasses of close-to-convex functions

1983 ◽  
Vol 6 (3) ◽  
pp. 449-458 ◽  
Author(s):  
E. M. Silvia
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Integral Operators ◽  
Coefficient Bounds ◽  
Class Of Functions

Let𝒦[C,D],−1≤D<C≤1, denote the class of functionsg(z),g(0)=g′(0)−1=0, analytic in the unit diskU={z:|z|<1}such that1+(zg″(z)/g′(z))is subordinate to(1+Cz)/(1+Dz),z ϵ U. We investigate the subclasses of close-to-convex functionsf(z),f(0)=f′(0)−1=0, for which there existsg ϵ 𝒦[C,D]such thatf′/g′is subordinate to(1+Az)/(1+Bz),−1≤B<A≤1. Distortion and rotation theorems and coefficient bounds are obtained. It is also shown that these classes are preserved under certain integral operators.

scholarly journals Some Connections between Class𝒰- andα-Convex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Edmond Aliaga ◽  
Nikola Tuneski
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk

The class𝒰(λ,μ)of normalized analytic functions that satisfy|(z/f(z))1+μ·f′(z)−1|<λfor allzin the open unit disk is studied and sufficient conditions for anα-convex function to be in𝒰(λ,μ)are given.

scholarly journals On the generalised Ruscheweyh class of analytic functions of complex order

1993 ◽  
Vol 47 (2) ◽  
pp. 247-257 ◽  
Author(s):  
O.P. Ahuja
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Order Complex ◽  
Sharp Estimates ◽  
Complex Order ◽  
Special Cases ◽  
The Family ◽  
Uniform Treatment

The main object of this paper is to identify various classes of analytic functions which are starlike, convex, pre-starlike, Ruscheweyh class of order α, β-spiral-like, β-convex-spiral-like, starlike of complex order, complex of complex order, and others as special cases of a family of analytic functions of complex order in the unit disk. This makes a uniform treatment possible. Finally, we derive sharp estimates for all coefficients of the functions from the family.

scholarly journals On the second hankel determinant for the kth-root transform of analytic functions

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 227-245 ◽  
Author(s):  
Najla Alarifi ◽  
Rosihan Ali ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
The Given ◽  
Logarithmically Convex Functions

Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of ?.

scholarly journals Coefficient Estimates of Bi-Starlike and Bi-Convex Functions with Respect to Symmetrical Points Associated with the Second Kind Chebyshev Polynomials

2020 ◽  
pp. 191-198
Author(s):  
Abbas Kareem Wanas
Keyword(s):  
Unit Disk ◽  
Chebyshev Polynomials ◽  
Convex Functions ◽  
Upper Bounds ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Symmetrical Points

In this paper, by making use the second kind Chebyshev polynomials, we introduce and study a certain class of bi-starlike and bi-convex functions with respect to symmetrical points defined in the open unit disk. We find upper bounds for the second and third coefficients of functions belong to this class.

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