NOTES ON CERTAIN SUBCLASS OF p-VALENTLY
BAZILEVIČ FUNCTIONS
ZHI-GANG WANG
YUE-PING JIANG
School of Mathematics and Computing Science
Changsha University of Science and Technology
Changsha 410076, Hunan,
People’s Republic of China
EMail: zhigangwang@foxmail.com
School of Mathematics and Econometrics
Hunan University
Changsha 410082, Hunan,
People’s Republic of China
EMail: ypjiang731@163.com
Received:
04 February, 2007
Accepted:
07 July, 2008
Communicated by:
A. Sofo
2000 AMS Sub. Class.:
Primary 30C45.
Key words:
Analytic functions, Multivalent functions, Bazilevič functions, subordination between analytic functions, Briot-Bouquet differential subordination.
Abstract:
Acknowledgements:
In the present paper, we discuss a subclass Mp (λ, µ, A, B) of p-valently Bazilevič functions, which was introduced and investigated recently by Patel [5]. Such
results as inclusion relationship, coefficient inequality and radius of convexity for
this class are proved. The results presented here generalize and improve some
earlier results. Several other new results are also obtained.
The present investigation was supported by the National Natural Science Foundation under Grant 10671059 of People’s Republic of China. The first-named
author would like to thank Professors Chun-Yi Gao and Ming-Sheng Liu for their
continuous support and encouragement. The authors would also like to thank the
referee for his careful reading and making some valuable comments which have
essentially improved the presentation of this paper.
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
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1
Introduction
3
2
Preliminary Results
6
3
Main Results
8
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
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1.
Introduction
Let Ap denote the class of functions of the form:
f (z) = z p +
∞
X
an z n
(p ∈ N := {1, 2, 3, . . .}),
n=p+1
p-Valently Bazilevič Functions
which are analytic in the open unit disk
Zhi-Gang Wang and Yue-Ping Jiang
U := {z : z ∈ C
and
|z| < 1}.
vol. 9, iss. 3, art. 70, 2008
For simplicity, we write
A1 =: A.
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For two functions f and g, analytic in U, we say that the function f is subordinate
to g in U, and write
f (z) ≺ g(z)
(z ∈ U),
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if there exists a Schwarz function ω, which is analytic in U with
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ω(0) = 0
and
|ω(z)| < 1
(z ∈ U)
such that
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f (z) = g ω(z)
(z ∈ U).
Indeed it is known that
f (z) ≺ g(z)
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(z ∈ U) =⇒ f (0) = g(0)
and f (U) ⊂ g(U).
Furthermore, if the function g is univalent in U, then we have the following equivalence:
f (z) ≺ g(z)
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(z ∈ U) ⇐⇒ f (0) = g(0)
and f (U) ⊂ g(U).
Let Mp (λ, µ, A, B) denote the class of functions in Ap satisfying the following
subordination condition:
zf 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
1 + Az
(1.1)
+λ 1+ 0
− (1 − µ)
−µ
≺p
1−µ
µ
f (z)g (z)
f (z)
f (z)
g(z)
1 + Bz
(−1 5 B < A 5 1; z ∈ U)
for some real µ (µ = 0), λ (λ = 0) and g ∈ Sp∗ , where Sp∗ denotes the usual class of
p-valently starlike functions in U.
For simplicity, we write
2α
Mp λ, µ, 1 −
, −1 = Mp (λ, µ, α)
p
zf 00 (z)
zf 0 (z)
zg 0 (z)
zf 0 (z)
:= f (z) ∈ Ap : <
+λ 1+ 0
− (1 − µ)
−µ
>α ,
f 1−µ (z)g µ (z)
f (z)
f (z)
g(z)
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
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for some α (0 5 α < p) and z ∈ U.
The class Mp (λ, µ, A, B) was introduced and investigated recently by Patel [5].
The author obtained some interesting properties for this class in the case λ > 0, he
also proved the following result:
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Theorem 1.1. Let
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µ = 0, λ > 0 and
− 1 5 B < A 5 1.
If f ∈ Mp (λ, µ, A, B), then
(1.2)
zf 0 (z)
λ
1 + Az
≺
= q(z) ≺
1−µ
µ
pf (z)g (z)
pQ(z)
1 + Bz
(z ∈ U),
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where
Q(z) =
 R
p

 01 s λ −1

 R1
0
1+Bsz
1+Bz
p
s λ −1 exp
p(A−B)
λB
p
(s
λ
ds
(B 6= 0),
− 1)Az ds (B = 0),
and q(z) is the best dominant of (1.2).
In the present paper, we shall derive such results as inclusion relationship, coefficient inequality and radius of convexity for the class Mp (λ, µ, A, B) by making use
of the techniques of Briot-Bouquet differential subordination. The results presented
here generalize and improve some known results. Several other new results are also
obtained.
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
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2.
Preliminary Results
In order to prove our main results, we shall require the following lemmas.
Lemma 2.1. Let
µ = 0, λ = 0 and
− 1 5 B < A 5 1.
p-Valently Bazilevič Functions
Then
Mp (λ, µ, A, B) ⊂ Mp (0, µ, A, B).
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
Proof. Suppose that f ∈ Mp (λ, µ, A, B). By virtue of (1.2), we know that
1 + Az
zf 0 (z)
≺
p
f 1−µ (z)g µ (z)
1 + Bz
(z ∈ U),
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which implies that f ∈ Mp (0, µ, A, B). Therefore, the assertion of Lemma 2.1
holds true.
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Lemma 2.2 (see [3]). Let
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−1 5 B1 5 B2 < A2 5 A1 5 1.
Then
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1 + A2 z
1 + A1 z
≺
.
1 + B2 z
1 + B1 z
Lemma 2.3 (see [4]). Let F be analytic and convex in U. If
f, g ∈ A
and
f, g ≺ F,
then
λf + (1 − λ)g ≺ F
(0 5 λ 5 1).
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Lemma 2.4 (see [6]). Let
f (z) =
∞
X
ak z k
k=0
be analytic in U and
g(z) =
∞
X
bk z k
k=0
p-Valently Bazilevič Functions
be analytic and convex in U. If f ≺ g, then
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
|ak | 5 |b1 |
(k ∈ N).
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3.
Main Results
We begin by stating our first inclusion relationship given by Theorem 3.1 below.
Theorem 3.1. Let
µ = 0, λ2 = λ1 = 0 and
− 1 5 B1 5 B2 < A2 5 A1 5 1.
Then
p-Valently Bazilevič Functions
Mp (λ2 , µ, A2 , B2 ) ⊂ Mp (λ1 , µ, A1 , B1 ).
Proof. Suppose that f ∈ Mp (λ2 , µ, A2 , B2 ). We know that
zf 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
1 + A2 z
+
λ
1
+
−
(1
−
µ)
−
µ
≺
p
.
2
f 1−µ (z)g µ (z)
f 0 (z)
f (z)
g(z)
1 + B2 z
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
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Since
−1 5 B1 5 B2 < A2 5 A1 5 1,
it follows from Lemma 2.2 that
zf 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
1 + A1 z
(3.1) 1−µ
+λ2 1 + 0
− (1 − µ)
−µ
≺p
.
µ
f (z)g (z)
f (z)
f (z)
g(z)
1 + B1 z
that is, that f ∈ Mp (λ2 , µ, A1 , B1 ). Thus, the assertion of Theorem 3.1 holds true
for
λ2 = λ1 = 0.
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If
λ2 > λ1 = 0,
by virtue of Lemma 2.1 and (3.1), we know that f ∈ Mp (0, µ, A1 , B1 ), that is
(3.2)
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zf 0 (z)
1 + A1 z
≺p
.
1−µ
µ
f (z)g (z)
1 + B1 z
At the same time, we have
zf 0 (z)
zf 0 (z)
zg 0 (z)
zf 00 (z)
(3.3)
+ λ1 1 + 0
− (1 − µ)
−µ
f 1−µ (z)g µ (z)
f (z)
f (z)
g(z)
0
0
λ1
zf (z)
λ1
zf (z)
= 1−
+
1−µ
µ
1−µ
λ2 f (z)g (z) λ2 f (z)g µ (z)
zf 0 (z)
zg 0 (z)
zf 00 (z)
− (1 − µ)
−µ
.
+λ2 1 + 0
f (z)
f (z)
g(z)
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
It is obvious that
1 + A1 z
h1 (z) =
1 + B1 z
is analytic and convex in U. Thus, we find from Lemma 2.3, (3.1), (3.2) and (3.3)
that
zf 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
1 + A1 z
+ λ1 1 + 0
− (1 − µ)
−µ
≺p
,
1−µ
µ
f (z)g (z)
f (z)
f (z)
g(z)
1 + B1 z
that is, that f ∈ Mp (λ1 , µ, A1 , B1 ). This implies that
Mp (λ2 , µ, A2 , B2 ) ⊂ Mp (λ1 , µ, A1 , B1 ).
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Remark 1. Setting A1 = A2 = A and B1 = B2 = B in Theorem 3.1, we get the
corresponding result obtained by Guo and Liu [2].
Corollary 3.2. Let
µ = 0, λ2 = λ1 = 0 and
p > α2 = α1 = 0.
Then
Mp (λ2 , µ, α2 ) ⊂ Mp (λ1 , µ, α1 ).
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Theorem 3.3. If f ∈ Ap satisfies the following conditions:
f (z)
zf 0 (z)
1
<
> 0 and 1−µ
− p < νp
0 5 µ < ; 0 < ν 5 1; z ∈ U
zp
f (z)g µ (z)
2
for g ∈ Sp∗ , then f is p-valently convex (univalent) in |z| < R(p, µ, ν), where
p
2pµ + 2µ − ν − 2 + (2pµ + 2µ − ν − 2)2 + 4(ν + p)(p − 2pµ)
.
R(p, µ, ν) =
2(ν + p)
Proof. Suppose that
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
h(z) :=
zf 0 (z)
−1
pf 1−µ (z)g µ (z)
(z ∈ U).
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Then h is analytic in U with
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h(0) = 0
and
|h(z)| < 1
(z ∈ U).
Thus, by applying Schwarz’s Lemma, we get
h(z) = νzψ(z)
(0 < ν 5 1),
where ψ is analytic in U with
(z ∈ U).
Therefore, we have
zf (z) = pf
1+
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0
1−µ
µ
(z)g (z)(1 + νzψ(z)).
Differentiating both sides of (3.4) with respect to z logarithmically, we obtain
(3.5)
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|ψ(z)| 5 1
(3.4)
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zf 0 (z)
zg 0 (z) νz(ψ(z) + zψ 0 (z))
zf 00 (z)
=
(1
−
µ)
+
µ
+
.
f 0 (z)
f (z)
g(z)
1 + νzψ(z)
We now suppose that
(3.6)
φ(z) :=
f (z)
= 1 + c1 z + c2 z 2 + · · · ,
zp
by hypothesis, we know that
<(φ(z)) > 0
(3.7)
(z ∈ U).
p-Valently Bazilevič Functions
It follows from (3.6) that
Zhi-Gang Wang and Yue-Ping Jiang
zf 0 (z)
zφ0 (z)
=p+
.
f (z)
φ(z)
(3.8)
Upon substituting (3.8) into (3.5), we get
(3.9) 1 +
vol. 9, iss. 3, art. 70, 2008
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zφ0 (z)
zg 0 (z) νz(ψ(z) + zψ 0 (z))
zf 00 (z)
=
(1
−
µ)p
+
(1
−
µ)
+
µ
+
.
f 0 (z)
φ(z)
g(z)
1 + νzψ(z)
Now, by using the following well known estimates (see [1]):
0 0 2r
zg (z)
1−r
zφ (z)
<
=−
,
<
= −p
,
2
φ(z)
1−r
g(z)
1+r
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and
0
z(ψ(z) + zψ (z))
r
=−
1 + zψ(z)
1−r
for |z| = r < 1 in (3.9), we obtain
zf 00 (z)
2(1 − µ)r pµ(1 − r)
νr
< 1+ 0
= (1 − µ)p −
−
−
2
f (z)
1−r
1+r
1 − νr
2(1 − µ)r pµ(1 − r)
νr
−
−
= (1 − µ)p −
2
1−r
1+r
1−r
<
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(ν + p)r2 − (2pµ + 2µ − ν − 2)r − (p − 2pµ)
=−
,
1 − r2
which is certainly positive if r < R(p, µ, ν).
Putting ν = 1 in Theorem 3.3, we get the following result.
Corollary 3.4. If f ∈ Ap satisfies the following conditions:
zf 0 (z)
1
f (z)
<
> 0 and 1−µ
− p < p
05µ< ; z∈U
zp
f (z)g µ (z)
2
for g ∈ Sp∗ , then f is p-valently convex (univalent) in |z| < R(p, µ), where
p
2pµ + 2µ − 3 + (2pµ + 2µ − 3)2 + 4(1 + p)(p − 2pµ)
R(p, µ) =
.
2(1 + p)
Remark 2. Corollary 3.4 corrects the mistakes of Theorem 3.8 which was obtained
by Patel [5].
Theorem 3.5. Let
p-Valently Bazilevič Functions
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vol. 9, iss. 3, art. 70, 2008
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µ = 0, λ = 0 and
− 1 5 B < A 5 1.
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If
f (z) = z +
∞
X
ak z k
(z ∈ U)
k=n+1
satisfies the following subordination condition:
µ−1
00
zf (z)
zf 0 (z)
1 + Az
f (z)
0
(3.10) f (z)
+λ
+ (1 − µ) 1 −
≺
,
0
z
f (z)
f (z)
1 + Bz
Close
then
(3.11)
|an+1 | 5
A−B
(1 + nλ)(n + µ)
(n ∈ N).
Proof. Suppose that
f (z) = z +
∞
X
ak z k
(z ∈ U)
p-Valently Bazilevič Functions
k=n+1
Zhi-Gang Wang and Yue-Ping Jiang
satisfies (3.10). It follows that
00
µ−1
f (z)
zf (z)
zf 0 (z)
0
(3.12) f (z)
+λ
+ (1 − µ) 1 −
z
f 0 (z)
f (z)
1 + Az
= 1 + (1 + nλ)(n + µ)an+1 z n + · · · ≺
1 + Bz
vol. 9, iss. 3, art. 70, 2008
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Therefore, we find from Lemma 2.4, (3.12) and −1 5 B < A 5 1 that
(3.13)
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(z ∈ U).
|(1 + nλ)(n + µ)an+1 | 5 A − B.
The assertion (3.11) of Theorem 3.5 can now easily be derived from (3.13).
Taking A = 1 − 2α (0 5 α < 1) and B = −1 in Theorem 3.5, we get the
following result.
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Corollary 3.6. Let
µ = 0, λ = 0 and
If
f (z) = z +
0 5 α < 1.
∞
X
k=n+1
ak z k
satisfies the following inequality:
00
µ−1
!
0
zf
(z)
f
(z)
zf
(z)
< f 0 (z)
+λ
+ (1 − µ) 1 −
> α,
z
f 0 (z)
f (z)
then
|an+1 | 5
2(1 − α)
(1 + nλ)(n + µ)
(n ∈ N).
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
Remark 3. Corollary 3.6 provides an extension of the corresponding result obtained
by Guo and Liu [2].
vol. 9, iss. 3, art. 70, 2008
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References
[1] W.M. CAUSEY AND E.P. MERKES, Radii of starlikeness for certain classes of
analytic functions, J. Math. Anal. Appl., 31 (1970), 579–586.
[2] D. GUO AND M.-S. LIU, On certain subclass of Bazilevič functions, J. Inequal. Pure Appl. Math., 8(1) (2007), Art. 12. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=825].
[3] M.-S. LIU, On a subclass of p-valent close-to-convex functions of order β and
type α, J. Math. Study, 30 (1997), 102–104 (in Chinese).
[4] M.-S. LIU, On certain subclass of analytic functions, J. South China Normal
Univ., 4 (2002), 15–20 (in Chinese).
p-Valently Bazilevič Functions
Zhi-Gang Wang and Yue-Ping Jiang
vol. 9, iss. 3, art. 70, 2008
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[5] J. PATEL, On certain subclass of p-valently Bazilevič functions, J. Inequal. Pure
Appl. Math., 6(1) (2005), Art. 16. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=485].
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[6] W. ROGOSINSKI, On the coefficients of subordinate functions, Proc. London
Math. Soc. (Ser. 2), 48 (1943), 48–82.
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