COEFFICIENT ESTIMATES AND SUBORDINATION FOR UNIVALENT FUNCTIONS

Abstract

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.