INTRODUCTION
A single valued function f is said to be univalent
schlicht in a domain D c C if it never takes
twice; that is, if
f(z^)
the
same
or
value
* f(z2) for all points z1 and zg in
D
with z1 * Zg. The function f is said to be locally univalent at
a point zQ e D if it is univalent in some neighbourhood of zQ.
The theory of
univalent
complicated that some simplifying
functions
is
assumptions
so
vast
are
The most obvious one is to replace the arbitrary
and
necessary.
domain
D
by
the unit disk U = {z:|z| < 1}.
Let g(z) be regular in U then
it
has
a
Haclaurin’s
expansion
(D
g(z) = bQ ♦ b z ♦ b2 z2 + ... =
I b zn
n=0
which is convergent in U.
Observe that if g(z) is univalent in U then g(z) is also univalent in U.
that
if
g'(zQ) = 0, then g(z) is not univalent in any neighbourhood
of
zQ.
Hence if
g(z)
is
Consider g(z) - bQ,
univalent
in
U,
observe
b
then
b =g* (0) ?= 0.
Therefore we may divide by b^ and consider f(z) = (g(z)-b )/b .
2
We see that if g(z) is univalent in D then f(z)=(g(z)-bo)/b1 is
also univalent in the same domain
and
conversly
if
f(z)
is
univalent in D, then g(z) is also univalent. Setting b^/^ = »n
in (1) we have the normalized form
(2)
, v
2
3
f(z) = z + a z + a z + ... = z +
Definition :
to be normalized.
CD
_
n
E anz
n=2
A function f(z) of the form (2) is
If f(z) is univalent and has the
form
then it is called a normalized univalent function.
of all normalized functions that are regular and
said
The
(2),
class
univalent
in
U, is denoted by S.
The Koebe function
z
2
3
n
k(z) = ----- — = z + 2z + 3z + . . . + nz + ...
(1-zT
00
(3)
=
E nzn
n=1
is a member of S.
The
size
of
the
suggests the conjecture : If f(z) is
coefficients
in
S
has
the
of
k(z)
Taylor’s
series (2), then for each n > 2
U)
|an| < n
This conjecture is often referred
conjecture [2].
This conjecture is
also
as
known
the
as
Bieberbach
"central-
3
conjecture" of the theory of
univalent
functions
coefficient conjecture and has challenged
the field for about 70 years.
A great
the
or
as
the
researchers
number
of
in
researchers
have devoted their time to solve it.
In 1984 Louis de Branges [3] from
the
University
of
A set D c £ is said to be starlike with respect to
a
Purdue has resolved Bieberbach conjecture.
point wQ e D if the linear segment joining wQ
point w e D lies entirely in D.
to
every
other
The set D is said to be convex
if it is starlike with respect to each of its points;
that
is
if the linear segment joining any two points of D lies entirely
in D.
A function
oo
(5)
f(z) =
[a/
n=1
regular in U is said to be starlike there if it maps U
one-to-
one onto a domain starshaped with respect to the origin.
is equivalent to the analytic condition that
in U.
Moreover,
if
there
exists
This
Re(zf'(z )/f(z ))>0
a, 0 S' a < 1
such
that
Re(zf'(z)/f(z))>« in U, then we say that f is starlike of order
a
and
is
identified
denoted
with
by
#
S ,
f € S (a).
the
class
Observe
of
starlike
analytic function of the form (5) is said to be
that
S (0)
is
functions.
An
convex
it
if
4
maps U one-to-one onto a convex domain.
Analytically
this
is
equivalent to the condition that Re (1+zf"(z)/f'(z)) > 0 for
e u.
Moreover, if there
(1+zf"(z)/f' (z)) > a for
exists
a,
0 * a < 1
z e U, we say that
f
such that
Re
is
of
convex
order a and is denoted by f e K(a). Observe that K(0) = K,
K c S
class of convex functions. Thus
*
c s.
z
Note
the
that
the
Koebe function is starlike but not convex.
A function f regular in the unit disk is
close-to-convex if there
is a
convex
Re {f' (z)/g# (z)} > 0 for
all z e U,
function
said
g such
We shall denote
class of close-to-convex functions f normalized
conditions f(0) = 0 and f'(0) = 1.
Note
function g need not be normalized.
Every
by
that
the
convex
by
to
be
that
C
the
the
usual
associated
function
is
obviously close-to-convex and it is known that close-to- convex
function
is
univalent.
The
close-to-convex
function
is
introduced by Kaplan [9] in 1952.
Let 51 denote the class of functions of the form
oo
f(z) = -1- + E anzn
(6)
z
which are regular in
the
with a
at
simple
pole
„ n
n=1
punctured
z = 0.
(0 ^ a < 1) denote the subclasses
disk
Let
of
Z
E = {z:0 < jzj < 1}
Z ,
s
*
Z (a)
that
are
and
Z. (l*)
k
univalent,
s
meromorphlc starlike of order a and meromorphic convex of order
*
a respectively.
and
only
if
Analytically f of the form (6), is in I (a) if
Re
z e li.
(-zf'(z)/f(z)) > a,
Similarly,
f e Z^(a), if and only if, f is of the form (6)
Re(-(1+zf"(z)/f' (z))) >
a,
z e U.
Observe
identified with Z^, the class of meromorphic
3k
*
and I (0) = Z , the class of
meromorphic
and
satisfies
that
Ik(0)
convex
starlike
is
functions
functions.
*
The class Z (a) and similar other classes have been extensively
studied by Pommerenke[21], Clunie[4],
Miller[14],
Royster[24]
and others.
A function f(z) given by (6) is said to
be
close-to-
convex in the punctured disk E relative to F(z) if there exists
a meromorphic starlike (i.e. if Re(zF' (z)/F(z)) < 0 for z e
schlicht function F(z) in U with a simple pole
at
the
u)
origin
given by
b_,
F(z) = —— + b„ + btz + ... (b_t r 0)
(7)
such that Re(zf' (z)/F(z)) > 0 for z e U.
Let
f(z)
be
gien
be
starlike
by
(6)
and
F(z) =
b-1
— + bQ + b^z + —
for
z € U,
satisfying
Re(zf* (z)/F(z)) > a, 0 i a < 1 for z e U, then f(z) is said
to
6
be
close-to-convex
of
order
cx
and
denoted
by
f = I v> (os).
E (0) = £ , the class of meromorphic close-to-convex functions,
c
c
A meromorphic close- to-convex function need not be univalent.
In the
first
chapter
of
the
thesis
integrals
of
univalent functions with negative and fixed second coefficients
and
meromorphic
functions
with
positive
and
fixed
second
coefficients are considered.
In the second chapter univalent functions of the
form
oo
f(z) = a.z +
1
7 a z
“ n
n=2
(a
n
>0)
f* (zQ) = 1 where zQ is real
which
are
satisfy
considered.
f(zn) = z_
oo
In
or
particular
extreme points for these subclasses are considered.
00
00
Let f(z) = z + £ a z11, a > 0
„ n
n
n=2
b
In
the
g(z) belongs to
first
section
g(z) = z + £ b z11,
„ n
n=2
the third chapter some
oo
properties of h(z) = f(z) * g(z) = z + T a b z where f(z) and
“ n n
n=2
n
> 0.
and
certain
of
subclasses
of
starlike
functions are obtained. In the second section
some
and
convex
distortion
theorems for the fractional calculus for functions of the
form
oo
f(z)=z+
obtained.
F a z , a >0 that are regular and
n
n=2« n
In
the
third
section
standard
univalent
properties
coefficient inequalities, distortion and covering theorems
are
like
and
7
extreme points are obtained for multivalent
form f(z) = z
d
oo
i
i
+ £ |ap+n|z
n=1
D+n
functions
E = {z: 0 < |z| < 1}
and
where * dentoes the Hadamard
greater than -p.
the
.
co a
1
k-1
Let f(z)= —- + £ —- be regular in the
zP
k"l zp-k
disk
of
on+P 1f(z) =
product
For -1 * B < A < 1
and
let
n
punctured
1
*f (z)
zP(1-z)n+p
is
any
integer
6
U
. (A,B)
n.p.A,
denotes
the class of functions f(z) satisfying
-2P+1(Dn+P‘ 1 (f(z))' < p
1 + (JZ
|z| < 1
1 + Bz
where p = B + X(A-B) cos£e
In the fourth chapter the
class
Further meromorphic starlike functions
of
U
x
n.p.X
complex
is
studied.
order
are
conditions
for
also considered.
In the last chapter
some
sufficient
meromorphic close-to-convex functions
to-star functions are obtained.
and
meromorphic
close-