Volume 9 (2008), Issue 3, Article 70, 7 pp.
NOTES ON CERTAIN SUBCLASS OF p-VALENTLY BAZILEVIČ FUNCTIONS
ZHI-GANG WANG AND YUE-PING JIANG
S CHOOL OF M ATHEMATICS AND C OMPUTING S CIENCE
C HANGSHA U NIVERSITY OF S CIENCE AND T ECHNOLOGY
C HANGSHA 410076, H UNAN ,
P EOPLE ’ S R EPUBLIC OF C HINA
zhigangwang@foxmail.com
S CHOOL OF M ATHEMATICS AND E CONOMETRICS
H UNAN U NIVERSITY
C HANGSHA 410082, H UNAN ,
P EOPLE ’ S R EPUBLIC OF C HINA
ypjiang731@163.com
Received 04 February, 2007; accepted 07 July, 2008
Communicated by A. Sofo
A BSTRACT. 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.
Key words and phrases: Analytic functions, Multivalent functions, Bazilevič functions, subordination between analytic functions, Briot-Bouquet differential subordination.
2000 Mathematics Subject Classification. Primary 30C45.
1. I NTRODUCTION
Let Ap denote the class of functions of the form:
∞
X
p
f (z) = z +
an z n
(p ∈ N := {1, 2, 3, . . .}),
n=p+1
which are analytic in the open unit disk
U := {z : z ∈ C
and
|z| < 1}.
For simplicity, we write
A1 =: A.
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.
043-07
2
Z HI -G ANG WANG AND Y UE -P ING J IANG
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),
if there exists a Schwarz function ω, which is analytic in U with
ω(0) = 0
|ω(z)| < 1
and
(z ∈ U)
such that
f (z) = g ω(z)
(z ∈ U).
Indeed it is known that
f (z) ≺ g(z)
(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)
(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
+
−
(1
−
µ)
−
µ
≺
p
f 1−µ (z)g µ (z)
f 0 (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 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
+λ 1+ 0
− (1 − µ)
−µ
>α ,
:= f (z) ∈ Ap : <
f 1−µ (z)g µ (z)
f (z)
f (z)
g(z)
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:
Theorem 1.1. Let
− 1 5 B < A 5 1.
µ = 0, λ > 0 and
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
where
Q(z) =
 R
p

 01 s λ −1

 R1
p
1+Bsz
1+Bz
s λ −1 exp
0
and q(z) is the best dominant of (1.2).
p(A−B)
λB
p
(s
λ
ds
(z ∈ U),
(B 6= 0),
− 1)Az ds (B = 0),
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.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 70, 7 pp.
http://jipam.vu.edu.au/
p-VALENTLY BAZILEVI Č F UNCTIONS
3
2. P RELIMINARY R ESULTS
In order to prove our main results, we shall require the following lemmas.
Lemma 2.1. Let
− 1 5 B < A 5 1.
µ = 0, λ = 0 and
Then
Mp (λ, µ, A, B) ⊂ Mp (0, µ, A, B).
Proof. Suppose that f ∈ Mp (λ, µ, A, B). By virtue of (1.2), we know that
zf 0 (z)
1 + Az
≺
p
f 1−µ (z)g µ (z)
1 + Bz
(z ∈ U),
which implies that f ∈ Mp (0, µ, A, B). Therefore, the assertion of Lemma 2.1 holds true.
Lemma 2.2 (see [3]). Let
−1 5 B1 5 B2 < A2 5 A1 5 1.
Then
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
Lemma 2.4 (see [6]). Let
f (z) =
∞
X
(0 5 λ 5 1).
ak z k
k=0
be analytic in U and
g(z) =
∞
X
bk z k
k=0
be analytic and convex in U. If f ≺ g, then
|ak | 5 |b1 |
(k ∈ N).
3. M AIN R ESULTS
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
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
Since
−1 5 B1 5 B2 < A2 5 A1 5 1,
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 70, 7 pp.
http://jipam.vu.edu.au/
4
Z HI -G ANG WANG AND Y UE -P ING J IANG
it follows from Lemma 2.2 that
zf 0 (z)
zg 0 (z)
zf 0 (z)
zf 00 (z)
1 + A1 z
(3.1)
+ λ2 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 (λ2 , µ, A1 , B1 ). Thus, the assertion of Theorem 3.1 holds true for
λ2 = λ1 = 0.
If
λ2 > λ1 = 0,
by virtue of Lemma 2.1 and (3.1), we know that f ∈ Mp (0, µ, A1 , B1 ), that is
zf 0 (z)
1 + A1 z
≺p
.
1−µ
µ
f (z)g (z)
1 + B1 z
(3.2)
At the same time, we have
(3.3)
zf 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
+ λ1 1 + 0
− (1 − µ)
−µ
f 1−µ (z)g µ (z)
f (z)
f (z)
g(z)
0
zf (z)
λ1
= 1−
1−µ
λ2 f (z)g µ (z)
λ1
zf 0 (z)
zf 00 (z)
zf 0 (z)
zg 0 (z)
+ λ2 1 + 0
− (1 − µ)
−µ
.
+
λ2 f 1−µ (z)g µ (z)
f (z)
f (z)
g(z)
It is obvious that
1 + A1 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
+
λ
1
+
−
(1
−
µ)
−
µ
≺
p
,
1
f 1−µ (z)g µ (z)
f 0 (z)
f (z)
g(z)
1 + B1 z
h1 (z) =
that is, that f ∈ Mp (λ1 , µ, A1 , B1 ). This implies that
Mp (λ2 , µ, A2 , B2 ) ⊂ Mp (λ1 , µ, A1 , B1 ).
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 ).
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)
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 70, 7 pp.
http://jipam.vu.edu.au/
p-VALENTLY BAZILEVI Č F UNCTIONS
5
Proof. Suppose that
h(z) :=
zf 0 (z)
−1
pf 1−µ (z)g µ (z)
(z ∈ U).
Then h is analytic in U with
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
|ψ(z)| 5 1
(z ∈ U).
Therefore, we have
zf 0 (z) = pf 1−µ (z)g µ (z)(1 + νzψ(z)).
(3.4)
Differentiating both sides of (3.4) with respect to z logarithmically, we obtain
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
(3.5)
1+
φ(z) :=
(3.6)
f (z)
= 1 + c1 z + c2 z 2 + · · · ,
zp
by hypothesis, we know that
<(φ(z)) > 0
(3.7)
(z ∈ U).
It follows from (3.6) that
zφ0 (z)
zf 0 (z)
=p+
.
f (z)
φ(z)
Upon substituting (3.8) into (3.5), we get
(3.8)
zf 00 (z)
zφ0 (z)
zg 0 (z) νz(ψ(z) + zψ 0 (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 zφ (z)
2r
zg (z)
1−r
<
=−
,
<
= −p
,
2
φ(z)
1−r
g(z)
1+r
and
z(ψ(z) + zψ 0 (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 −
−
−
1 − r2
1+r
1−r
2
(ν + p)r − (2pµ + 2µ − ν − 2)r − (p − 2pµ)
=−
,
1 − r2
which is certainly positive if r < R(p, µ, ν).
(3.9)
1+
Putting ν = 1 in Theorem 3.3, we get the following result.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 70, 7 pp.
http://jipam.vu.edu.au/
6
Z HI -G ANG WANG AND Y UE -P ING J IANG
Corollary 3.4. If f ∈ Ap satisfies the following conditions:
0
zf
(z)
f (z)
<p
>
0
and
−
p
<
f 1−µ (z)g µ (z)
zp
1
05µ< ; z∈U
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
µ = 0, λ = 0 and
If
f (z) = z +
∞
X
− 1 5 B < A 5 1.
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
then
A−B
(3.11)
|an+1 | 5
(n ∈ N).
(1 + nλ)(n + µ)
Proof. Suppose that
f (z) = z +
∞
X
ak z k
(z ∈ U)
k=n+1
satisfies (3.10). It follows that
µ−1
00
f (z)
zf (z)
zf 0 (z)
0
(3.12) f (z)
+λ
+ (1 − µ) 1 −
z
f 0 (z)
f (z)
= 1 + (1 + nλ)(n + µ)an+1 z n + · · · ≺
1 + Az
1 + Bz
(z ∈ U).
Therefore, we find from Lemma 2.4, (3.12) and −1 5 B < A 5 1 that
(3.13)
|(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.
Corollary 3.6. Let
µ = 0, λ = 0 and
If
f (z) = z +
0 5 α < 1.
∞
X
ak z k
k=n+1
satisfies the following inequality:
µ−1
00
!
0
f
(z)
zf
(z)
zf
(z)
< f 0 (z)
+λ
+ (1 − µ) 1 −
> α,
z
f 0 (z)
f (z)
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 70, 7 pp.
http://jipam.vu.edu.au/
p-VALENTLY BAZILEVI Č F UNCTIONS
then
|an+1 | 5
2(1 − α)
(1 + nλ)(n + µ)
7
(n ∈ N).
Remark 3. Corollary 3.6 provides an extension of the corresponding result obtained by Guo
and Liu [2].
R EFERENCES
[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).
[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].
[6] W. ROGOSINSKI, On the coefficients of subordinate functions, Proc. London Math. Soc. (Ser. 2),
48 (1943), 48–82.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 70, 7 pp.
http://jipam.vu.edu.au/