Journal of Inequalities in Pure and
Applied Mathematics
A CRITERION FOR p-VALENTLY STARLIKENESS
MUHAMMET KAMALI
Ataturk University,
Faculty of Science and Arts,
Department of Mathematics,
25240, Erzurum-TURKEY.
E-Mail: mkamali@atauni.edu.tr
volume 4, issue 2, article 36,
2003.
Received 16 December, 2002;
accepted 8 May, 2003.
Communicated by: A. Sofo
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
148-02
Quit
Abstract
It is the purpose of the present paper to obtain some sufficient conditions for
p- valently starlikeness for a certain class of functions which are analytic in the
open unit disk E.
2000 Mathematics Subject Classification: 30C45, 31A05.
Key words: p−valently starlikeness, Jack Lemma.
A Criterion for p-Valently
Starlikeness
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Criterion for p-Valently Starlikeness . . . . . . . . . . . . . . . . . . .
Muhammet Kamali
3
6
9
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
1.
Introduction
Let A(p) be the class of functions of the form:
f (z) = z p +
∞
X
an z n
(p ∈ N = {1, 2, 3, . . .}) ,
n=p+1
which are analytic in E = {z ∈ C : |z| < 1} .
A function f (z) ∈ A(p) is said to be p-valently starlike if and only if
0 f (z)
Re z
> 0 (z ∈ E).
f (z)
We denote by S(p) the subclass of A(p) consisting of functions which are pvalently in E (see, e.g., Goodman [1]).
Let
(1.1)
f (z) = z +
∞
X
n
an z .
n=2
A function f (z) of the form (1.1) is said to be α−convex in E if it is regular,
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
Contents
JJ
J
II
I
Go Back
Close
f (z) 0
f (z) 6= 0,
z
Page 3 of 12
and
(1.2)
Quit
f 00 (z)
f 0 (z)
Re α 1 + z 0
+(1 − α)z
>0
f (z)
f (z)
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
for all z in E. The set of all such functions is denoted by α − CV , where α
is a real number. Of course, if α = 1, then an α−convex function is convex;
and if α = 0, an α−convex function is starlike. Thus the sets α − CV give
a “continuous” passage from convex functions to starlike functions. Sakaguchi
considers functions of the form
p
f (z) = z +
∞
X
an z n ,
n=p+1
where p is a positive integer, and he imposes the condition
f 0 (z)
zf 00 (z)
(1.3)
Re 1 + 0
+ kz
>0
f (z)
f (z)
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
for z in E. He proved that if k = −1, there is only one function that satisfies
(1.3), namely f (z) ≡ z p . If −1 < k 6 0,then f (z) is p-valent convex; and if
0 < k, then f (z) is p-valent starlike. We can pass from (1.3) back to (1.2) if we
1
divide by 1 + k > 0 and set α = 1+k
[6]. We denote by S(p, k) the subclass
A(p) consisting of functions which satisfy the condition (1.3).
Obradovic and Owa [7] have obtained a sufficient condition for starlikeness
of f (z) ∈ A(1) which satisfies a certain condition for the modulus of
zf 00 (z)
f 0 (z)
zf 0 (z)
f (z)
JJ
J
II
I
Go Back
Close
Quit
1+
we recall their result as:
Contents
,
Page 4 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
Theorem 1.1. If f (z) ∈ A(1) satisfies
0 00
zf
(z)
1 +
< K zf (z) f 0 (z)
f (z) (z ∈ E),
where K = 1.2849..., then f (z) ∈ S(1).
Nunokawa [4] improved Theorem 1.1 by proving
Theorem 1.2. If f (z) ∈ A(p), and if
0 00
zf (z) 1
zf
(z)
1 +
<
log(4ep−1 )
0
f (z)
f (z) p
then f (z) ∈ S(p).
A Criterion for p-Valently
Starlikeness
(z ∈ E),
Muhammet Kamali
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
2.
Preliminaries
In order to obtain our main result, we need the following lemma attributed to
Jack [2] (given also by Miller and Mocanu [3]).
Lemma 2.1. Let w(z) be analytic in E with w(0) = 0. If |w(z)| attains its
maximum value in the circle |z| = r < 1 at a point z0 , then we can write
z0 w0 (z0 ) = kw(z0 ), where k is a real number and k ≥ 1.
Making use of Lemma 2.1,we first prove
Lemma 2.2. Let q(z) be analytic in E with q(0) = p and suppose that
0
zq (z)
1
(2.1)
Re
<
(z ∈ E, 0 6 λ 6 1),
2
p(λ + 1)
[q(z)]
q(z) = p
Title Page
JJ
J
Proof. Let us put
1 1
+ λ
2 2
Muhammet Kamali
Contents
then Re{q(z)} > 0 in E.
A Criterion for p-Valently
Starlikeness
1 + w(z)
+
1 − w(z)
1 1
− λ
2 2
1 − w(z)
1 + w(z)
,
II
I
Go Back
Close
where 0 6 λ 6 1.
Then w(z) is analytic in E with w(0) = 0 and by an easy calculation, we
have
q 0 (z)
2 (λw2 (z) + 2w(z) + λ)zw0 (z)
1+z
=1+ ·
.
p
(w2 (z) + 2λw(z) + 1)2
[q(z)]2
Quit
Page 6 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
If we suppose that there exists a point z0 ∈ E such that max|z|6|z0 | |w(z)| =
|w(z0 )| = 1, then, from Lemma 2.1, we have z0 w0 (z0 ) = kw(z0 ), (k > 1).
Putting w(z0 ) = eiθ , we find that
z0
q 0 (z0 )
2 λw2 (z0 )w0 (z0 )z0 + 2w(z0 )w0 (z0 )z0 + λw0 (z0 )z0
=
·
p
[q(z0 )]2
[w2 (z0 ) + 2λw(z0 ) + 1]2
2k λe3iθ + 2e2iθ + λeiθ
=
·
p (e2iθ + 2λeiθ + 1)2
2
e−2iθ + 2λe−iθ + 1
2k λe3iθ + 2e2iθ + λeiθ
=
·
·
p
(e2iθ + 2λeiθ + 1)2
(e−2iθ + 2λe−iθ + 1)2
k λ cos 3θ + (4λ2 + 2) cos 2θ + (11λ + 4λ3 ) cos θ + (8λ2 + 2)
= ·
p
4 (λ + cos θ)4
k (1 + λ cos θ) (λ + cos θ)2
= ·
p
(λ + cos θ)4
k 1 + λ cos θ
= ·
,
p (λ + cos θ)2
so that
q 0 (z0 )
k 1 + λ cos θ
k λ2 + λ cos θ + 1 − λ2
Re z0
=
·
=
·
p (λ + cos θ)2
p
[q(z0 )]2
(λ + cos θ)2
k
λ
1 − λ2
=
+
p (λ + cos θ) (λ + cos θ)2
1
1
>
.
p λ+1
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
This contradicts (2.1). Therefore, we have |w(z)| < 1 in E, and it follows that
Re {q(z)} > 0 in E. This completes our proof of Lemma 2.2.
If we take λ = 1 in Lemma 2.2, then we have the following Lemma 2.3 by
Nunokawa [5].
Lemma 2.3. Let q(z) be analytic in E with q(0) = p and suppose that
0
zq (z)
1
<
(z ∈ E) .
Re
2
2p
[q(z)]
Then Re {q(z)} > 0 in E.
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
3.
A Criterion for p-Valently Starlikeness
Theorem 3.1. Let f (z) ∈ A(p), f (z) 6= 0, in 0 < |z| < 1 and suppose that

h
00
i0 
f (z)
f 0 (z)




1 + z f 0 (z) + k f (z)
1
1
(3.1) Re 1 + z h
(z ∈ E) .
00
i2 < 1 +
0


k + 1 2p
(z)
(z)


1 + z ff 0 (z)
+ k ff (z)
Then f (z) ∈ S(p, k).
A Criterion for p-Valently
Starlikeness
Proof. Let us put
1
q(z) =
k+1
f 00 (z)
f 0 (z)
1+z 0
+ kz
f (z)
f (z)
Muhammet Kamali
(k > 0) .
Then, q(z) is analytic in E with q(0) = p, q(z) 6= 0 in E. We have
00 0 0
00 0
0 0
f (z)
f 0 (z)
f (z)
f 0 (z)
(z)
f 00 (z)
z f 0 (z) + kz f (z)
+ z f 0 (z) + k f (z) + kz ff (z)
f 0 (z)
q 0 (z)
=
=
.
00
0
00
0
q(z)
1 + z f 0 (z) + kz f (z)
1 + z f 0 (z) + kz f (z)
f (z)
f (z)
f (z)
f (z)
Then, we obtain
1
q 0 (z)
z
=
q(z)
Title Page
Contents
JJ
J
II
I
Go Back
00 (z)
0 (z)
+ z ff 0 (z)
+ kz ff (z)
−
00
0
(z)
(z)
1 + z ff 0 (z)
+ kz ff (z)
z2
=1+
f 00 (z)
f 0 (z)
0
00
+k
kz
1
+z
f 0 (z)
f (z)
0
f 0 (z)
f (z)
+z
f 00 (z)
f 00 (z)
f 0 (z)
f 0 (z)
1 + z f 0 (z) + kz f (z)
0
(z)
(z)
1 + z ff 0 (z)
+ kz ff (z)
0
0
Close
Quit
Page 9 of 12
−1
,
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
or
q 0 (z)
(k + 1)q(z) + z
q(z)
0
0 0 f 00 (z)
(z)
2
z
+ k ff (z)
−1
f 0 (z)
+ (k + 1)q(z)
=1+
00 (z)
0 (z)
1 + z ff 0 (z)
+ kz ff (z)
00
00
00
0 (z) 0
0 (z)
0 (z) 2
(z)
(z)
(z)
+ 2z ff 0 (z)
+ z 2 ff 0 (z)
z 2 ff 0 (z)
+ k ff (z)
+ k ff (z)
+ k ff (z)
=1+
0 (z)
00 (z)
1 + z ff 0 (z)
+ kz ff (z)
00
0 00
f (z)
f 0 (z)
f (z)
f 0 (z)
00
z
+
k
+
+
k
0
f 0 (z)
f (z)
f 0 (z)
f (z)
f (z)
f (z)
.
=1+z
+
k
+
z
0
00
f 0 (z)
f (z)
1 + z f (z) + kz f (z)
f 0 (z)
f (z)
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
Contents
Thus,
0
1
q (z)
1+
z
=1+z
k + 1 [q(z)]2
z
f 00 (z)
f 0 (z)
0
(z)
+ k ff (z)
00
0
+
f 00 (z)
f 0 (z)
0
(z)
(z)
1 + z ff 0 (z)
+ kz ff (z)
i0
h
00
f 0 (z)
f (z)
1 + z f 0 (z) + k f (z)
=1+z .
00 (z)
0 (z) 2
1 + z ff 0 (z)
+ kz ff (z)
From Lemma 2.3 and (3.1), we thus find that
f 00 (z)
f 0 (z)
Re 1 + z 0
+ kz
>0
f (z)
f (z)
0
(z)
+ k ff (z)
2
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 12
(z ∈ E, k > 0) .
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
This completes our proof of Theorem 3.1.
If we take α = 0, after writing
Nunokawa’s theorem as follows.
1
k+1
= α in (3.1), then we obtain M.
Theorem 3.2. Let f (z) ∈ A(p), f (z) 6= 0, in 0 < |z| < 1 and suppose that
(
Re
Then f (z) ∈ S(p).
zf 00 (z)
f 0 (z)
zf 0 (z)
f (z)
1+
)
<1+
1
, z ∈ E.
2p
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au
References
[1] A.W. GOODMAN, On the Schwarz-Christoffel transformation and p-valent
functions, Trans. Amer. Math. Soc., 68 (1950), 204–223.
[2] I.S. JACK, Functions starlike and convex of order α, J. London Math. Soc.,
2(3) (1971), 469–474.
[3] S.S. MILLER AND P.T. MOCANU, Second order differential inequalities
in the complex plane, J. Math. Anal. Appl., 65 (1978), 289–305.
[4] M. NUNOKAWA, On certain multivalent functions, Math. Japon., 36
(1991), 67–70.
[5] M. NUNOKAWA, A certain class of starlike functions, in Current Topics in
Analytic Function Theory, H.M. Srivastava and S. Owa (Eds.), Singapore,
New Jersey, London, Hong Kong, 1992, p. 206–211.
[6] A.W. GOODMAN, Univalent Functions, Volume I, Florida, 1983, p.142–
143.
[7] M. OBRADOVIC AND S. OWA, A criterion for starlikeness, Math. Nachr.,
140 (1989), 97–102.
A Criterion for p-Valently
Starlikeness
Muhammet Kamali
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 12
J. Ineq. Pure and Appl. Math. 4(2) Art. 36, 2003
http://jipam.vu.edu.au