NISHIWAKI, JUNICHI

OWA, SHIGEYOSHI

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ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.1, 16
ISSN 1857-8365, UDC: 517.546
AN EXTREME FUNCTION FOR A CERTAIN CLASS OF ANALYTIC
FUNCTIONS
JUNICHI NISHIWAKI AND SHIGEYOSHI OWA
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. Let A be the class of analytic functions f(z) in the open unit disk U.
Furthermore, the subclass B of A concerned with the class of uniformly convex func-
tions or the class Sp is dened. By virtue of some properties of uniformly convex
functions and the class Sp, an extreme function of the class B and its power series are
considered.
1. Introduction
Let A be the class of functions f(z) of the form
f(z) = z +
1X
n=2
anz
n
which are analytic in the open unit disk U = fz 2 C : jzj < 1g. A function f(z) 2 A
is said to be in the class of uniformly convex (or starlike) functions denoted by UCV (or
UST ) if f(z) is convex (or starlike) in U and maps every circle or circular arc in U with
center at  in U onto the convex arc (or the starlike arc) with respect to f(). These
classes are introduced by Goodman [1] (see also [2]). For the class UCV, it is dened as
the one variable characterization by Rønning [4] and [5], that is, a function f(z) 2 A is
said to be in the class UCV if it satises
Re

1 +
zf 00(z)
f 0(z)

>
zf 00(z)f 0(z)
 (z 2 U):
It is independently studied by Ma and Minda [3]. Further, a function f(z) 2 A is said to
be the corresponding class denoted by Sp if it satises
Re

zf 0(z)
f(z)

>
zf 0(z)f(z)   1
 (z 2 U):
2010 Mathematics Subject Classication. 30C45.
Key words and phrases. Analytic function, uniformly convex function, extreme function,power series.
1
2 J. NISHIWAKI AND S. OWA
This class Sp was introduced by Rønning [4]. We easily know that the relation f(z) 2 UCV
if and only if zf 0(z) 2 Sp. In view of these classes, we introduce the subclass B of A
consisting of all functions f(z) which satisfy
Re

z
f(z)

>
 zf(z)   1
 (z 2 U):
We try to derive some properties of functions f(z) belonging to the class B.
Remark 1.1. For f(z) 2 B, we write w(z) = f(z)
z
= u+ iv, then w lies in the domain
which is the part of the complex plane which contains w = 1 and is bounded by a kind
of teardrop-shape domain such that
u4   2u3 + 2u2v2   2uv2 + v4 + v2 < 0:
2. An extreme function for the class B
In this section, we would like to exhibit an extreme function of the class B and its
power series. For our results, we need to recall here some properties of the class Sp.
Lemma 2.1. (Rønning [4]). The extremal function f(z) for the class Sp is given
by
zf 0(z)
f(z)
= 1 +
2
2

log

1 +
p
z
1 pz
2
:
By using the expansion of logarithmic part of
zf 0(z)
f(z)
in Lemma 2.1, we get
Lemma 2.2. (Rønning [4]). The power series of
zf 0(z)
f(z)
is following
zf 0(z)
f(z)
= 1 +
2
2

log

1 +
p
z
1 pz
2
= 1 +
8
2
1X
n=1
 
nX
k=1
1
2k   1
1
2n+ 1  2k
!
zn:
From Remark 1.1 and Lemma 2.1, we have the rst result for the class B.
AN EXTREME FUNCTION FOR A CERTAIN CLASS OF ANALYTIC FUNCTIONS 3
Theorem 2.1. The extreme function f(z) for the class B is given by
f(z) =
z
1 +
2
2

log

1 +
p
z
1 pz
2 :
Proof. Let us consider the function
f(z)
z
as given by
f(z)
z
=
1
1 +
2
2

log

1 +
p
z
1 pz
2 :
It sucies to show that
f(z)
z
maps U onto the interior of the domain such that
u4   2u3 + 2u2v2   2uv2 + v4 + v2 < 0;
implying that
f(z)
z
maps the unit circle onto the boundary of the domain. Taking z = ei,
we obtain that
1
1 +
2
2

log

1 +
p
z
1 pz
2 = 1
1 +
2
2
 
log
 
1 + ei

2
1  ei 2
!!2
=
1
1 +
2
2

log i  log

tan

4
2
=
1
1
2
+
2
2

log

tan

4
2
  i 2

log

tan

4

=
1
2
+
2
2

log

tan

4
2
1
4
+
6
2

log

tan

4
2
+
4
4

log

tan

4
4
+ i
2

log

tan

4

1
4
+
6
2

log

tan

4
2
+
4
4

log

tan

4
4 :
Writing
f(z)
z
= u+ iv, we see that
log

tan

4

=
(upu2   v2)
2v
:
4 J. NISHIWAKI AND S. OWA
Thus we have
v =
2

log

tan

4

1
4
+
6
2

log

tan

4
2
+
4
4

log

tan

4
4
=
2

(upu2   v2)
2v
1
4
+
6
2
 
(upu2   v2)
2v
!2
+
4
4
 
(upu2   v2)
2v
!4 :
Therefore, we arrive that
u4   2u3 + 2u2v2   2uv2 + v4 + v2 = 0:
This completes the proof of the theorem. 
Considering the power series of the function f(z) in Theorem 2.1, we derive
Theorem 2.2. The power series of the extreme function for the class B is given by
f(z) =
z
1 +
2
2

log

1 +
p
z
1 pz
2
= z +
1X
n=2
n 1X
p=1
( 1)p

8
2
p n pX
m1=1
 
m1X
k=1
1
2k   1
1
2m1 + 1  2k
!

n+1 p m1X
m2=1
 
m2X
k=1
1
2k   1
1
2m2 + 1  2k
!
   

n 2 Ap 2X
mp 1=1
 
mp 1X
k=1
1
2k   1
1
2mp 1 + 1  2k
!0
@n 1 Ap 1X
k=1
1
2k   1
1
2(n  Ap 1)  1  2k
1
A zn;
where Ap =
pP
l=1
ml.
Proof. Let us suppose that
f(z)
z
=
1
1 +
2
2

log

1 +
p
z
1 pz
2
as the proof of Theorem 2.1. Then from Lemma 2.2, we have
f(z)
z
=
1
1 +
8
2
1P
n=1

nP
k=1
1
2k   1
1
2n+ 1  2k

zn
AN EXTREME FUNCTION FOR A CERTAIN CLASS OF ANALYTIC FUNCTIONS 5
= 1  8
2
 
1X
n=1
 
nX
k=1
1
2k   1
1
2n+ 1  2k
!
zn
!
+

8
2
2 1X
n=1
 
nX
k=1
1
2k   1
1
2n+ 1  2k
!
zn
!2
 

8
2
3 1X
n=1
 
nX
k=1
1
2k   1
1
2n+ 1  2k
!
zn
!3
+   
+( 1)n

8
2
n 1X
n=1
 
nX
k=1
1
2k   1
1
2n+ 1  2k
!
zn
!n
+   
= 1  8
2
1X
k=1
1
2k   1
1
3  2kz
+
(
  8
2
2X
k=1
1
2k   1
1
5  2k +

8
2
2 1X
k=1
1
2k   1
1
3  2k
! 
1X
k=1
1
2k   1
1
3  2k
!)
z2
+
"
  8
2
3X
k=1
1
2k   1
1
7  2k +
(
8
2
2 1X
k=1
1
2k   1
1
3  2k
! 
2X
k=1
1
2k   1
1
5  2k
!
+
 
2X
k=1
1
2k   1
1
5  2k
! 
1X
k=1
1
2k   1
1
3  2k
!)
 

8
2
3 1X
k=1
1
2k   1
1
3  2k
!335 z3
+   
+
(
  8
2
nX
k=1
1
2k   1
1
2n+ 1  2k
+

8
2
2 n 1X
m1=1
 
m1X
k=1
1
2k   1
1
2m1 + 1  2k
! 
n m1X
k=1
1
2k   1
1
2n  2m1 + 1  2k
!
 

8
2
3 n 2X
m1=1
 
m1X
k=1
1
2k   1
1
2m1 + 1  2k
!
n 1 m1X
m2=1
 
m2X
k=1
1
2k   1
1
2m2 + 1  2k
!

 
n m1 m2X
k=1
1
2k   1
1
2n  2m1   2m2 + 1  2k
!
+   
+( 1)p

8
2
p n+1 pX
m1=1
 
m1X
k=1
1
2k   1
1
2m1 + 1  2k
!
n+2 p m1X
m2=1
 
m2X
k=1
1
2k   1
1
2m2 + 1  2k
!
   
n 1 Ap 2X
mp 1=1
 
mp 1X
k=1
1
2k   1
1
2mp 1 + 1  2k
!0
@n Ap 1X
k=1
1
2k   1
1
2(n  Ap 1) + 1  2k
1
A
+   +
 
1X
k=1
1
2k   1
1
3  2k
!n)
zn
+   
 
Ap =
pX
l=1
ml
!
6 J. NISHIWAKI AND S. OWA
= 1 +
1X
n=2
nX
p=1
( 1)p

8
2
p n+1 pX
m1=1
 
m1X
k=1
1
2k   1
1
2m1 + 1  2k
!

n+2 p m1X
m2=1
 
m2X
k=1
1
2k   1
1
2m2 + 1  2k
!
   

n 1 Ap 2X
mp 1=1
 
mp 1X
k=1
1
2k   1
1
2mp 1 + 1  2k
!0@n Ap 1X
k=1
1
2k   1
1
2(n  Ap 1) + 1  2k
1
A zn:
This completes the proof of the theorem. 
References
[1] A. W. Goodman: On uniformly convex functions, Annal. Polon. Math. 56(1991), 87  92.
[2] A. W. Goodman: On uniformly starlike functions, J. Math. Anal. Appl. 155(1991), 364  370.
[3] W. Ma, D. Minda: Uniformly convex functions, Annal. Polon. Math. 57(1992), 165  175.
[4] F. Rønning: Uniformly convex functions and a corresponding class of starlike functions, Proc.
Amer. Math. Soc. 118(1993), 189  196.
[5] F. Rønning: On uniform starlikeness and related properties of univalent functions, Complex
Variables 24(1994), 233  239.
Department of Mathematics and physics
Setsunan University
Neyagawa, Osaka 572-8508 Japan
E-mail address: jerjun2002@yahoo.co.jp
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502 Japan
E-mail address: shige21@ican.zaq.ne.jp


AN EXTREME FUNCTION FOR A CERTAIN CLASS OF ANALYTIC FUNCTIONS

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AN EXTREME FUNCTION FOR A CERTAIN CLASS OF ANALYTIC FUNCTIONS

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