NAKAMURA, YAYOI

OWA, SHIGEYOSHI

SHIMODA, YUTAKA

English

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ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 2 (2013), no.2, 4347
ISSN 1857-8365, UDC: 517.546/.547
SOME PROPERTIES OF FUNCTIONS CONCERNED WITH OZAKI
AND NUNOKAWA RESULT
YUTAKA SHIMODA,YAYOI NAKAMURA AND SHIGEYOSHI OWA
Abstract. For analytic functions f(z) with f(0) = f 0(0) =    = f(n 1)(0) = 0 and
f(n)(0) 6= 0 in the open unit disk U, a new class An() concerned with Ozaki and
Nunokawa result in 1972 is introduced. The object of the present paper is to discuss
some properties of functions f(z) in the class An().
1. Introduction
Let An be the class of functions f(z) of the form
(1.1) f(z) = z +
1X
k=n
akz
k (n = 2; 3; 4;    )
with an 6= 0 which are analytic in the open unit disk U = fz 2 C : jzj < 1 g.
For f(z) 2 An, we dene the subclass An() of An consisting of f(z) which satisfy
(1.2)
z
2f 0(z)
f(z)2
  1
 <  (z 2 U)
for some real  > 0. We note that f(z) 2 A2(1) is univalent in U by Ozaki and Nunokawa
[5].
The class A2() was dened and discussed by Obradovi¢ and Ponnusamy [4]. The
class A3() with 0 <  < 1 was considered by Singh [7].
Resently, Shimoda, Nakamura and Owa [6] discussed some radius problems for f(z) 2
A2() with a3   a
2
2 = 0.
In order to discuss our problems for f(z) 2 An(), we have to recall here the following
lemmas.
Lemma 1.1. Let the function w(z) dened by
(1.3) w(z) = bnz
n + bn+1z
n+1 + bn+2z
n+2 +    (bn 6= 0)
2010 Mathematics Subject Classication. 30C45.
Key words and phrases. Analytic, Jack's lemma, Carathéodory function.
43
44 Y. SHIMODA,Y. NAKAMURA AND S. OWA
be analytic in U . If jw(z)j attains its maximum value at a point z0 2 U on the circle
jzj = r, then there exists a real number k  n such that
(1.4) z0w
0(z0) = kw(z0)
and
(1.5) Re

z0w
00(z0)
w0(z0)

 k   1:
This lemma was given by Jack [2] (also, due to Miller and Mocanu [3]).
Lemma 1.2. Let the function p(z) given by
(1.6) p(z) = 1 + c1z + c2z
2 + c3z
3 +   
be analytic in U. If p(z) satises Rep(z) > 0 (z 2 U), then
(1.7) jcnj  2 (n = 1; 2; 3;    ):
The equality in (1.7) is attained for p(z) given by
(1.8) p(z) =
1 + z
1  z
:
This lemma is called as Carathéodory's lemma [1].
2. Some properties for the class An()
First result of our problems is contained in:
Theorem 2.1. If f(z) 2 An(n  2) with n  3, then
(2.1)
 zf(z)   1
 < 1 (z 2 U)
and
(2.2) Re

f(z)
z

>
1
2
(z 2 U):
Proof. For An(n  2), we dene the function w(z) by
w(z) =
z
f(z)
  1 = bnz
n 1 + bn+1z
n + bn+2z
n+1 +    :
Then w(z) is analytic in U and w(0) = 0.
Further, we see that
(2.3)
z
2f 0(z)
f(z)2
  1
 = jw(z)  zw0(z)j < n  2 (z 2 U):
We suppose that there exists a point z0 2 U such that
(2.4) max
jzj=r
jw(z)j = jw(z0)j = 1:
Then, Lemma 1.1 gives us that
(2.5) z0w
0(z0) = kw(z0) (k  n  1):
SOME PROPERTIES OF FUNCTIONS . . . 45
It follows from (2.3) , (2.4), and (2.5) that
z
2
0f
0(z0)
f(z0)2
  1
 =
w(z0)

1 
z0w
0(z0)
w(z0)

= jk   1j  n  2:
This contradicts that f(z) 2 An(n  2). This means that there is no z0 2 U such that
jw(z0)j = 1. Therefore, we conclude that jw(z)j < 1 (z 2 U), and prove that equation
(2.1) Furthermore, it is easy to see that (2.1) gives equation (2.2), proving the theorem.

Next our result is:
Theorem 2.2. If f(z) 2 An

n  3
4

with n  4, then
(2.6)
f(z)z   1
 < 1 (z 2 U):
Proof. Let us dene the function w(z) by
w(z) =
f(z)
z
  1 ;
for f(z) 2 An

n  3
4

. It follows that:
(2.7)
z
2f 0(z)
f(z)2
  1
 =
 1w(z) + 1

zw0(z)
w(z) + 1
  w(z)
 < n  34 (z 2 U):
Since, w(z) is analytic in U and w(0) = 0, we suppose that there exists a point z0 2 U
such that
max
jzj=r
jw(z)j = jw(z0)j = 1:
Then, we can write w(z0) = e
i and
z0w
0(z0) = kw(z0) (k  n  1):
Therefore, we have that
z
2
0f
0(z0)
f(z0)2
  1
 =
 w(z0)w(z0) + 1

k
w(z0) + 1
  1


jk   1  w(z0)j
jw(z0) + 1j2

n  3
4
;
which contradicts our condition (2.7). This shows that there is no z0 2 U such that
jw(z0)j = 1 for z0 2 U. Thus we have that jw(z)j < 1 for all z 2 U, proving (2.6) . 
46 Y. SHIMODA,Y. NAKAMURA AND S. OWA
3. Coefficient inequalities for the class A2()
In this section we consider some coecient inequalities for f(z) in the class A2().
Theorem 3.1. If f(z) 2 A2() with 0 <  < 1, then
(3.1) ja3   a
2
2j  2;
(3.2) ja4   2a2a3 + a
3
2j  ;
and
(3.3) ja5   2a2a4 + 3a
2
2a3   a
2
3   a
4
2j 
2
3
:
Proof. Let us dene the function p(z) by
p(z) =
z2f 0(z)
f(z)2
  (1  )

(z 2 U):
This gives us that p(z) is analytic in U, Rep(z) > 0 (z 2 U), and
p(z) = 1 + c1z + c2z
2 + c3z
3 +    :
Noting that
z2f 0(z)
f(z)2
  1 +  = + (a3   a
2
2)z
2 + 2(a4   2a2a3 + a
3
2)z
3
+3(a5   2a2a4 + 3a
2
2a3   a
2
3   a
4
2)z
4 +    ;
we have that
a3   a
2
2 = c2;
2(a4   2a2a3 + a
3
2) = c3;
and
3(a5   2a2a4 + 3a
2
2a3   a
2
3   a
4
2) = c4:
Therefore, applying Lemma 1.2, we prove (3.1), (3.2) and (3.3). 
If a2 = 0 in Theorem 2.2, then we have:
Corollary 3.1. If f(z) 2 A3() with 0 <  < 1, then
ja3j  2;
ja4j  ;
and
ja5j 
8
3
:
SOME PROPERTIES OF FUNCTIONS . . . 47
References
[1] C. Carathéodory: "Uber den Variabilitätsbereich der Fourier'schen Konstanten von positiven
harmonischen Funktionen, Rend. Circ. Mat. Palermo 32(1911), 193  217.
[2] I. S. Jack: Functions starlike and convex of order , J. London Math. Soc., 3(1971), 469  474.
[3] S. S. Miller, P. T. Mocanu: Second-order dierential inequalities in the complex plane, J.
Math. Anal. Appl., 65(1978), 289  305.
[4] M. Obradovi¢, S. Ponnusamy: Radius properties for subclasses of univalent functions, Analy-
sis, 25(2005), 183  188.
[5] S. Ozaki, M. Nunokawa: The Schwarzian derivative and univalernt functions, Proc. Amer.
Math. Soc., 33 (1972), 392  394.
[6] Y. Shimoda, Y. Nakamura, S. Owa: Radius problems for certain strongly close-to-convex
functions, PanAmer. Math. Journal, 23(2013), 53  58.
[7] V. Singh: On a class of univalent functions, Internet. J. Math. Math. Sci., 23(2000), 855  857.
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
E-mail address: yutakashimoda119@gmail.com
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
E-mail address: yayoi@math.kindai.ac.jp
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502, Japan
E-mail address: shige21@ican.zaq.ne.jp


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