UNIVERSITI TEKNOLOGI MARARESEARCH PROPOSAL (MAT 530)FEKETE-SZEGÖ DETERMINANT OF CERTAIN CLASS OF UNIVALENT FUNCTIONSNIK HASHIMAH BINTI RASHIDEE(2016577381)NUR NADIRA ASHYIKIN BINTI MUHAMMAD FUAD TIEW(2016328775)SARAH BINTI ROSLAN(2016728305)Supervisor:Abdullah Bin YahyaBachelor of Science (Hons.) (Mathematics)Center of Mathematical StudiesFaculty of Computer and Mathematical SciencesMay 2018?

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2|

1. INTRODUCTION1.1 Introduction to Geometric Function TheoryGeometric Function Theory is one of the branches of complex analysis. This ?eld of study explores and discusses the geometric properties of analytic functions. This theory of univalent functions had been introduced by Koebe in 1907, then in 1916, the geometric function theory was then being established by a German mathematician, Ludwig Bieberbach. Bieberbach worked on the second coef?cient of a function in the form of Taylor series expansion that isBieberbach had proven that , with equality, if and only if is a rotation of the Koebe function:Furthermore, Bieberbach added that ” is in overall valid” thus this declaration was known as the Bieberbach Conjecture (Bieberbach, 1916). Through the years, many mathematicians had proven this conjecture such as Löwner in 1923 – the author said that .Under the geometric function theory, we can estimate coef?cient bounds for the functions of the classes that are to be discussed in another section. We can apply certain ways to determine these bounds for example by using Fekete-Szegö inequality, and second Hankel determinant.c2f?#(1).

f(z)=z+c2z2+c3z3+c4z4+?+cmzm+?=z+

?

?m=2

cmzm|c2|?2f(2).

k(z)=1

4(

1+z

1?z)

2

?1

=z+2z2+3z3+?=z

(1?z)2=

?

?m=1

mzm|cm|?m|c3|?31

According to Ehrenborg (2000), the Hankel determinant of order is the determinant of the corresponding Hankel matrix, that isFrom the Bieberbach Conjecture, we know that and . In 1933, Fekete and Szegö later generalised the estimation of where is real and . After that, Noonan and Thomas (1976) had outlined the Hankel determinant of for Notice that the Fekete-Szegö functional is in fact Hankel determinant with and whereAs early as 1933, Fekete and Szegö had presented the following form of equation that isand this result is unique (sharp).(m+1).

det(ci+j)0?i,j?m=det

c0c1?cm

c1c2?cm+1

????

cmcm+1?c2m

|c2|?2|c3|?3|c3??c22|?f?#qthfq.

Hq(m)=

cmcm+1?cm+q+1

cm+1cm+2?cm+q+2

????

cm+q?1cm+q?cm+2q?2

q=2m=1,.

H2(1)=c1c2

c2c3=c1c3?c22

(3).

|c3??c22|?

3?4?,if??0

1+2e?2?(1??),if0???1

4??3,if??1

2

1.2 Background TheoryLet be the set of all complex numbers. A function be a complex-valued function of the complex variable is said to be differentiable at if the derivative of at exists. The derivative of is de?ned asThe function is claimed to be analytic (also known as holomorphic, or regular) at given that is differentiable at the point . If exists for every point in the region, then the function is holomorphic everywhere in that region in the complex plane. In short, an analytic function is a one-to-one mapping of one particular region in the complex plane onto another particular region.An analytic function is then said to be univalent or ‘schlicht’ or simple in a domain if it offers a one-to-one mapping onto its image (Goodman, 1983), – that is to say, Brie?y, the arbitrary domain is substituted by a unit disk, .Let indicates the class of normalised analytic functions in the unit disk , and has the the form of (1). The function ful?ls the conditions for normalisation, that is, and . This class of normalised univalent functions is then denoted as in the same form of (1).?f(z)zz0??fz0fz0fz0f?(z0)f(z).

f?(z0)=limz?z0

f(z)?f(z0)

z?z0

f(z)f(D) where .f(z1)?f(z2)?z1?z2z1,z2?DD/:={z??:|z|=1}0/f(z)f(0)=0f(0)=1#3

Koebe function, in (2) is an example of the class of normalised function . This function maps to the unit disk onto the whole complex plane but the part of negative real axis from to . In the meantime, a function is the class of the functions in the form ofwhere is regular in , and such for , . Therefore, a function with positive real part in contains any functions in .Geometrically, a set in the plane is convex (denoted by K) if for each pair of points and whose in the interior of , the line segment connecting both and is also in the interior of (Goodman, 1983). Hence, the function is said to be convex function if and meets the condition below The condition in (5) was being ?rst revealed and stated by Study back in 1913 (Duren, 1983). On the other hand, a set is starlike, S with respect to if every ray with initial point intersects the interior of in the set that is either a line segment or a ray. Thus, is a starlike function provided that if and only ifThis condition in (6) is based on Nevanlinna in 1921.k(z)#k(z)/?1

4

?P(4).

p(z)=1+p1z+p2z2+?+pmzm+?=1+

?

?m=1

pmzmP/z?/Re(p(z))>0/PDr1r2Dr1r2Df(z)f?#f(5).

Re(1+zf??(z)

f?(z))>0,z?/Dr0r0Dff?#(6).

Re(1+zf?(z)

f?(z))>0,z?/

4

A set in the plane is close-to-convex, C if whenever increases, the tangent direction ought to never decrease by as much as from any previous value. If such situation happens, it does have the chance to overlap and meet at more than one point thus will cause the function to be not univalent. If and there exists a real number where and a function K, such that then is said to be a close-to-convex function and was discovered by Kaplan (1952). The condition in (7) can also be written asThe latter condition in (8) is owing to a relationship between convex and starlike functions mentioned by Alexander in 1916. According to Alexander, if S then K (Goodman, 1983). As a result, it portrays a relationship between convex function and starlike function in which both functions are actually close-to-convex and this relationship can be summarised as K S C (Goodman, 1983).D?arg{

?

??f(rei?)}

?ff?#???

20,z?/,h(z)?h(z)?h(z)=zg?(z)????#

5

2. LITERATURE REVIEWFekete-Szegö determinant is one of the renowned coef?cient inequalities in the Geometric Function Theory. This coef?cient inequality is widely being used to ?nd the solutions and submit proofs to the inequalities problems. The classic result of Fekete-Szegö inequality can be seen in (3) (Fekete and Szegö, 1933).2.1 Subclass of Starlike Functions with Respect to Symmetric PointsAccording to Shanmugam and Sivasubramanian (2005), let be a univalent starlike function with respect to 1 in which it maps onto the right half plane of a region which is symmetric to the real axis, and . The function is in the class of ifFor ?xed , the authors introduced that the class to be the class of functions for which .Then let . If is given by (1) and belongs to , then?(z)/?(0)?1=0??(0)>0f?#M?(?).

zf?(z)+?z2f??(z)

(1??)f(z)+?zf?(z)??(z),??0

g?#Mga(?)f?#(f*g)?Ma(?)?(z)=1+B1z+B2z2+?f(z)M?(?).

|c3??c22|?

B2

2(1+2?)??

(1+2?)2B21+1

2(1+2?)(1+2?)B21,if???1

B1

2(1+2?),if?1????2

?B2

2(1+2?)+?

(1+2?)2B21?1

2(1+2?)(1+2?)B21,if???2

6

where 2.2 Class of Parabolic Starlike FunctionsSrivastava and Mishra (2000) carried out a study on the class of parabolic starlike functions which was denoted by SP. A function is in the class of parabolic starlike functions if it satisfy where is the parabolic region, and . Then, let , given by (1), be in the class of SP. Hence, the Fekete-Szegö result iswhere,.

?1=(1+?)2(B2?B1)+(1+?2)B21

2(1+2?)B21

?2=(1+?)2(B2+B1)+(1+?2)B21

2(1+2?)B21

f?#, ,zf?(z)

f(z)??z?/??:={w:w=u+ivv2