Academic literature on the topic 'FEKETE-SZEGO COEFFICIENT FUNCTIONAL'

The purpose of this paper is to study the coefficient estimates of the class of functions in ()consisting of starlike functions. The sharp upper bounds for the initial coefficients and the Fekete-Szego functional of the functions in the class were established using the Opoola Differential Operator

Making use of Gegenbauer polynomials, we initiate and explore two sets of normalized regular and bi-univalent (or bi-Schlicht) functions in the unit disc linked with Gegenbauer polynomials. We investigate certain coefficients bounds and the Fekete-Szego functional for functions in these families. We also present few interesting observations and provide relevant connections of the results investigated.

The main aim of this work is to find some coefficient inequalities and sufficient condition for some subclasses of meromorphic starlike functions by using q-difference operator. Here we also define the extended Ruscheweyh differential operator for meromorphic functions by using q-difference operator. Several properties such as coefficient inequalities and Fekete-Szego functional of a family of functions are investigated.

In this paper, a new subclass of analytic functions ML_{\lambda}^{*} associated with the right half of the lemniscate of Bernoulli is introduced. The sharp upper bound for the Fekete-Szego functional |a_{3}-\mu a_{2}^{2}| for both real and complex \mu are considered. Further, the sharp upper bound to the second Hankel determinant |H_{2}(1)| for the function f in the class ML_{\lambda}^{*} using Toeplitz determinant is studied. Relevances of the main results are also briefly indicated.

In this study, by making the use of q-analogous of the hyperbolic tangent function and a Sălăgean q-differential operator, a new class of q-starlike functions is introduced. The prime contribution of this study covers the derivation of sharp coefficient bounds in open unit disk U, especially the first three coefficient bounds, Fekete–Szego type functional, and upper bounds of second- and third-order Hankel determinant for the functions to this class. We also use Zalcman and generalized Zalcman conjectures to investigate the coefficient bounds of a newly defined class of functions. Furthermore, some known corollaries are highlighted based on the unique choices of the involved parameters l and q.

Abstract This paper deals with the class S containing functions which are analytic and univalent in the open unit disc U = {z ∈ ℂ : |z| < 1}. Functions f in S are normalized by f(0) = 0 and f′(0) = 1 and has the Taylor series expansion of the form f ( z ) = z + ∑ n = 2 ∞ a n z n f\left( z \right) = z + \sum\limits_{n = 2}^\infty {{a_n}{z^n}} . In this paper we investigate on the subclass of S of close-to-convex functions denoted as C gα (λ, δ) where function f ∈ C gα (λ, δ) satisfies Re { e i λ z f ′ ( z ) g α ( z ) } {\mathop{\rm Re}\nolimits} \left\{ {{e^{i\lambda }}{{zf'\left( z \right)} \over {g\alpha \left( z \right)}}} \right\} for | λ | < π 2 \left| \lambda \right| < {\pi \over 2} , cos(λ) > δ, 0 ≤ δ < 1, 0 ≤ α ≤ 1 and g α = z ( 1 − α z ) 2 {g_\alpha } = {z \over {{{\left( {1 - \alpha z} \right)}^2}}} . The aim of the present paper is to find the upper bound of the Fekete-Szego functional |a 3 − µa 2 2| for the class C g α (λ, δ). The results obtained in this paper is significant in the sense that it can be used in future research in this field, particularly in solving coefficient inequalities such as the Hankel determinant problems and also the Fekete-Szego problems for other subclasses of univalent functions.

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].

We discuss the sharpness of the bound of the Fekete-Szego functional for close-to-convex functions with respect to convex functions. We also briefly consider other related developments involving the Fekete-Szego functional |a3 ??a22| (0 ? ? ? 1) as well as the corresponding Hankel determinant for the Taylor-Maclaurin coefficients {an}n?N\{1} of normalized univalent functions in the open unit disk D, N being the set of positive integers.

The main objective of this paper is to study a new family of analytic functions that are q-starlike with respect to m-symmetrical points and subordinate to the q-Janowski function. We investigate inclusion results, sufficient conditions, coefficients estimates, bounds for Fekete–Szego functional |a3−μa22| and convolution properties for the functions belonging to this new class. Several consequences of main results are also obtained.

Univalent function theory is a branch of geometric function theory which comprises of the various geometric properties of analytic functions. The first milestone in the field of univalent functions theory was achieved by Bieberbach in the year 1916, wherein he proved the second coefficient bound for a function f ∈ S of normalised analytic univalent functions. He also proposed a conjecture for the nth coefficient of the function in the class S in the same year. Bieberbach’s Conjecture states that the coefficients of thethefunction f ∈S satisfy|an|≤nforn = 2,3,4,··· withequalityifandonlyifholds if f is some rotation of the famous Koebe function. Bieberbach’s conjecture paved way for many mathematicians to work in the area of univalent functions and a vast literature is available now. The present research work focusses on investigating the various types of coefficient estimate problems in geometric function theory such as computing the bounds on the second andthe thirdHankel determinants, theFekete- Szeg¨o coefficientfunctional. The thesis also aims at computing the sharp radius estimates and various inclusion relationships between certain classes of analytic functions. To begin with, Chapter 1 introduces some basic concepts and results in the theory of univalent functions which will be required later in our investigations. Chapter 2, entitled “Initial Coefficients of Starlike Functions w.r.t. Symmetric Points” aims at studying the functions which are starlike with respect to symmetric points. It is well known that the class of analytic functions f defined on the unit disk satisfying vii viii Preface Re(zf0(z)/(f(z)− f(−z))) > 0 is a subclass of close-to-convex functions and the nth Taylorcoefficientofthesefunctionsareboundedbyone. However,noboundsareknown for the nth coefficients of functions f ∈ S∗ s (ϕ) satisfying 2zf0(z)/(f(z)− f(−z)) ≺ ϕ(z), except for n = 2,3. Thus, the sharp bound for the fourth coefficient of analytic univalent functions f satisfying the following subordination 2zf0(z)/(f(z)− f(−z)) ≺ ϕ(z) has been obtained. The bound for the fifth coefficient has also been obtained in certain special cases of ϕ including ez and√1+z. Chapter3, entitled“Fekete-Szeg¨oCoefficientFunctional”,dealswithobtainingthebound for the Fekete-Szeg¨o coefficient functional. Let ϕ be an analytic function with the positive real part satisfying ϕ(0) = 1 and ϕ0(0) > 0. Let f(z) = z + a2z2 + a3z3 +··· be an analytic function satisfying the subordination αf0(z) + (1−α)zf0(z)/f(z) ≺ ϕ(z), (f0(z))α(zf0(z)/f(z))(1−α) ≺ ϕ(z), (f0(z))α(1 + zf00(z)/f0(z))(1−α) ≺ ϕ(z), (f(z)/z)α(zf0(z)/ f(z))(1−α) ≺ ϕ(z) or (f(z)/z)α(1 + zf00(z)/f0(z))(1−α) ≺ ϕ(z). Forfunctionssatisfyingtheabovesubordination,theboundsofFekete-Szeg¨ocoefficient functional have been obtained. In Chapter 4 entitled “Hankel Determinant of Certain Analytic Functions”, we have obtainedtheboundsforthesecondHankeldeterminant H2(2) = a2a4−a2 3 forthefunction f satisfying αf0(z) + (1−α)zf0(z)/f(z) ≺ ϕ(z), (f0(z))α(zf0(z)/f(z))(1−α) ≺ ϕ(z), (f0(z))α(1 + zf00(z)/f0(z))(1−α) ≺ ϕ(z), (f(z)/z)α(zf0(z)/f(z))(1−α) ≺ ϕ(z) or (f(z)/z)α (1+zf00(z)/f0(z))(1−α) ≺ ϕ(z). Here ϕ is an analytic function with the positiverealpart, ϕ(0) = 1and ϕ0(0) > 0. WehavealsodeterminedthethirdHankeldeterminant H3(1) = a3(a2a4−a2 3)−a4(a4−a2a3) + a5(a3−a2 2) for an analytic function f of the form f(z) = z+∑anzn satisfying either Re(f0(z))α(zf0(z)/f(z))(1−α) > 0 or Re(f0(z))α(1+zf00(z)/f0(z))(1−α) > 0. Our results include some previously known results. In Chapter 5, entitled “Janowski Starlikeness and Convexity”, certain necessary and sufficientconditionshavebeendeterminedforthefunctions f(z) = z−∑∞ n=2 anzn ∈T, an ≥0,definedonD,tobelongtorenownedsubclassesofJanowskistarlikeandconvex functions. In the same chapter, we have also discussed certain sufficient conditions for Preface ix the normalised analytic functions f satisfying (z/f(z))µ = 1+∑∞ n=1 bnzn, µ ∈C to be in the classS∗[A,B] of Janowski starlike functions. In Chapter 6, named “The classes S∗ α,e and SL∗(α)”, we have attempted to study the function f defined on D, with normalisations f(0) = 0 = f0(0)−1, satisfying the subordinations zf0(z)/f(z) ≺ α + (1−α)ez or zf0(z)/f(z) ≺ α + (1−α)√1+z respectively, where 0 ≤ α < 1. The sharp radii has been determined for these functions to belong to several known subclasses of analytic functions. In addition, some inclusion relations and coefficient problems including the bounds for the first four coefficient estimates and the Fekete-Szeg¨o functional have also been obtained.