Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 982321, 10 pages
doi:10.1155/2012/982321
Research Article
On Generalized Bazilevic Functions Related with
Conic Regions
Khalida Inayat Noor and Kamran Yousaf
Department of Mathematics, COMSATS Institute of Information Technology, Park Road,
Islamabad, Pakistan
Correspondence should be addressed to Khalida Inayat Noor, khalidanoor@hotmail.com
Received 12 March 2012; Accepted 18 March 2012
Academic Editor: Yonghong Yao
Copyright q 2012 K. I. Noor and K. Yousaf. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We define and study some generalized classes of Bazilevic functions associated with convex
domains. These convex domains are formed by conic regions which are included in the right half
plane. Such results as inclusion relationships and integral-preserving properties are proved. Some
interesting special cases of the main results are also pointed out.
1. Introduction
Let A denote the class of analytic functions fz defined in the unit disc E {z : |z| < 1}
and satisfying the conditions f0 0, f 0 1. Let S denote the subclass of A consisting of
univalent functions in E, and let S∗ and C be the subclasses of S which contains, respectively,
star-like and convex in Bazilevič 1 introduced the class Bα, β, h, g as follows.
Let f ∈ A. Then, f ∈ Bα, β, h, g, α, β real and α > 0 if
1/αiβ
z
hzg α ttiβ−1 dt
,
1.1
fz α iβ
0
for some g ∈ S∗ and Re hz > 0, z ∈ E.
The powers appearing in 1.1 are meant as principle values. The functions f in the
class Bα, β, h, g are shown to be analytic and univalent, see 1. Bα, β, h, g is the largest
known subclass of univalent functions defined by an explicit formula and contains many of
the heavily researched subclasses of S. We note the following:
i B1, 0, 1, g C,
ii B1, 0, zg /g, g S∗ ,
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iii B1, 0, h, g K, where K is the class of close-to-convex functions introduced by
Kaplan 2,
iv Bcos γ, sin γ, coszg /g i sin γ, g is the class of γ-spiral like functions which are
univalent for |γ| < π/2.
∞
n
n
For analytic functions fz ∞
n0 an z and gz n0 bn z , by f ∗ g we denote the
Hadamard product convolution of f and g, defined by
∞
an bn zn .
f ∗ g z 1.2
n0
For k ∈ 0, ∞, the conic domain Ωk is defined in 3 as follows:
Ωk u iv : u > k u − 12 v2 .
1.3
For fixed k, Ωk represents the conic region bounded successively by the imaginary axis k 0, the right branch of hyperbola 0 < k < 1, a parabola k 1 and an ellipse k > 1.
The following univalent functions, defined by pk z with pk 0 1 and pk 0 > 0, map
the unit disc E onto Ωk :
pk z ⎧
1z
⎪
⎪
,
⎪
⎪
1−z ⎪
⎪
√ 2
⎪
⎪
2
1 z
⎪
⎪
1
,
log
⎪
√
⎪
⎪
π2
1− z
⎨
√ 2
⎪
⎪
1
sin h2 Akarc tanh z ,
⎪
2
⎪
1
−
k
⎪
⎪
⎪
⎪
⎪
z
2
π
⎪
2
⎪
F
, t
,
sin
⎪1 2
⎪
⎩
2Kt
t
k −1
k 0,
k 1,
0 < k < 1,
1.4
k > 1,
where Ak 2/πarc cos k, Fw, t is the Jacobi elliptic integral of the first kind:
Fw, t w
0
√
dx
,
√
1 − x 2 1 − t2 x 2
1.5
and t ∈ 0, 1 is chosen such that k coshπK t/2Kt, where Kt is the complete elliptic
√
integral of the first kind, Kt F1, t, K t K 1 − t2 .
It is known that pk z are continuous as regards to k and have real coefficients for
k ∈ 0, ∞.
Let P pk be the subclass of the class P of Caratheodory functions pz, analytic in E
with p0 1 and such that pz is subordinate to pk z, written as pz ≺ pk z in E.
We define the following.
Journal of Applied Mathematics
3
Definition 1.1. Let hz be analytic in E with h0 1. Then, h ∈ Pm pk if and only if, for
m ≥ 2,k ∈ 0, ∞, h1 , h2 ∈ P pk we can write
hz m 1
m 1
−
h1 z −
h2 z,
4 2
4 2
z ∈ E.
1.6
We note that P2 pk P pk , and Pm p0 Pm , see 4.
Definition 1.2. Let f ∈ A. Then, fz is said to belong to the class k − ∪Rm if and only if
zf /f ∈ Pm pk for k ∈ 0, ∞, m ≥ 2, and z ∈ E.
For m 2, k 0, the class 0 − ∪R2 R2 coincides with the class S∗ of starlike functions,
and 0 − ∪Rm Rm consists of analytic functions with bounded radius rotation, see 5, 6. Also
k − ∪R2 is the class ∪ST studied by several authors, see 7, 8.
Definition 1.3. Let f ∈ A. Then, f ∈ k − ∪Bm α, β, h, g if and only if fz is as given by 1.1
for some g ∈ k − ∪R2 , h ∈ Pm pk in E with k ∈ 0, ∞, m ≥ 2, α > 0 and β real.
When m 2 and k 0, we obtain the class Bα, β, h, g of Bazilevic functions.
We shall assume throughout, unless otherwise stated, that k ∈ 0, ∞,m ≥ 2, α > 0, β
real and z ∈ E.
2. Preliminary Results
Lemma 2.1 see 3. Let 0 ≤ k < ∞, and let β0 , δ be any complex numbers with β0 /
0 and
Reβ0 k/k 1 δ > 0. If hz is analytic in E,h0 1 and satisfies
hz zh z
β0 hz δ
≺ pk z
2.1
and qk z is analytic solution of
qk z zqk z
β0 qk z δ
pk z,
2.2
then qk z is univalent,
hz ≺ qk z ≺ pk z,
2.3
and qk z is the best dominant of 2.1.
Lemma 2.2 see 9. Let qz be convex in E and j : E → C with Re jz > 0, z ∈ E. If pz is
analytic in E with p0 1 and satisfies {pz jz · zp z} ≺ qz, then pz ≺ qz.
Lemma 2.3 see 9. Let u u1 i u2 , v v1 iv2 , and let ψu, v be a complex-valued function
satisfying the conditions
i ψu, v is continuous in a domain D ⊂ C2 ,
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Journal of Applied Mathematics
ii 0, 1 ∈ D and Re ψ1, 0 > 0,
iii Re ψi u2 , v1 ≤ 0, whenever i u2 , v1 ∈ D and v1 ≤ −1/21 u22 .
If hz 1 c1 z c2 z2 · · · is a function analytic in E such that hz, zh z ∈ D and
Re ψ{hz, zh z} > 0 for z ∈ E, then Re hz > 0 in E.
3. Main Results
Theorem 3.1. Let 1/1 − γ{zf z/fz − γ} ∈ Pm pk , for z ∈ E and γ ∈ 0, 1. Define
z
1/α
tc−1 f α tdt
,
gz c 1z−c
α > 0, c ∈ C, Re c ≥ 0.
3.1
0
Then, 1/1 − γ{zg z/gz − γ} ∈ Pm pk in E. In particular g ∈ k − ∪Rm in E.
Proof. Let
zg z 1 − γ pz γ,
gz
3.2
where pz is analytic in E with p0 1, and let
m 1
m 1
−
p1 z −
p2 z.
4 2
4 2
3.3
g α z α 1 − γ pz c αγ f α z.
3.4
pz From 3.1 and 3.2, we have
Logarithmic differentiation of 3.4 and some computation yield
zp z
1
pz 1−γ
α 1 − γ pz c αγ
zf z
−γ .
fz
3.5
That is
zp z
pz ∈ P m pk
α 1 − γ pz c αγ
Let φa,b z z ∞
n2
in E.
3.6
zn /n − 1a b. Then,
φa,b z
pz ∗
z
a zp z
pz .
pz b
3.7
Journal of Applied Mathematics
5
Using convolution technique 3.7 with a 1/α1 − γ, b c αγ/α1 − γ, we obtain, from
3.3 and 3.6,
zpi z
pi z α 1 − γ pi z c αγ
≺ pk z
in E, i 1, 2.
3.8
Since Re{α1 − γk/k 1 c αγ} ≥ 0, we apply Lemma 2.1 with β0 α1 − γ, δ c αγ
to obtain pi z ≺ qk z ≺ pk z, where qk z is the best dominant and is given as
⎡
qk z ⎣β0
1 tβ0 δ−1 exp
0
tz
z
pk u − 1
du
u
β0
⎤−1
dt⎦
−
δ
.
β0
3.9
Consequently, p ∈ Pm pk in E, and this completes the result.
As a special case, we prove the following.
Corollary 3.2. Let k 0 and let 1/1 − γ1 {zf z/fz − γ1 } ∈ Pm in E. Then, for g defined by
3.1, 1/1 − γ{zg z/gz − γ} ∈ Pm in E where
γ
2
.
2
2c − 2αγ1 1 2c − 2αγ1 1 8α
3.10
Proof. We can write
zf z 1 − γ1 hz γ1 ,
fz
3.11
where h ∈ Pm in E.
Now proceeding as before, we have, with
m 1 m 1 1 − γ p1 z γ −
−
1 − γ p2 z γ
4 2
4 2
3.12
1 − γ z p z
zf z
1 − γ pz γ .
3.13
fz
α 1 − γ pz c αγ
zg z 1 − γ pz γ gz
Using convolution technique together with 3.11, we obtain
Re
for i 1, 2.
1 − γ z pi z
1 − γ pi z γ − γ1 > 0,
α 1 − γ pi z c αγ
3.14
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Journal of Applied Mathematics
We construct the functional ψu, v by taking u pi z, v zpi z as
ψu, v 1 − γ u γ − γ1
1−γ v
.
α 1 − γ u c αγ
3.15
The first two conditions of Lemma 2.3 are clearly satisfied. We verify condition iii as follows.
Re ψi u2 , v1 γ
γ
≤ γ
1 − γ v1
− γ1 Re
,
iα 1 − γ u2 c αγ
c αγ 1 − γ v1
− γ1 2
2 ,
c αγ α2 1 − γ u22
c αγ 1 − γ 1 u22
− γ1 ,
2
2
2 c αγ α2 1 − γ u22
1 u22
v1 ≤ −
2
3.16
,
A Bu22
,
2C
where
A 2γ − γ1 c αγ2 − 1 − γc αγ, B 2α2 γ − γ1 1 − γ2 − 1 − γc αγ, C c αγ2 α2 1 − γ2 u22 > 0.
Re ψiu2 , v1 ≤ 0 if and only if A ≤ 0, B ≤ 0. From A ≤ 0, we obtain γ as given by 3.10
and B ≤ 0 ensures that γ ∈ 0, 1.
Now proceeding as before, it follows from 3.12 that p ∈ Pm , and this proves our
result.
By assigning certain permissible values to different parameters, we obtain several new
and some known result.
Corollary 3.3. Let f ∈ k − ∪R2 k − ∪ST . Then, it is known that f ∈ S∗ γ1 ,γ1 k/k 1 and,
form Corollary 3.2, it follows that g ∈ S∗ γ where γ is given by 3.10. Also a starlike function is
k-uniformly convex for |z| < rk ,
rk 1
, see 8.
√
2k 1 4k2 6k 3
3.17
Therefore, for f ∈ k − ∪R2 , it follows that 1/1 − γ{zg z /g z − γ} ≺ pk for |z| < rk , where
γ is given by 3.10.
As special cases we note the following.
√
i For k 0, we have r0 1/2 3 and f ∈ S∗ 0 implies that g ∈ Cγ∗ , with
2
γ∗ .
2
2c 1 2c 1 8α
3.18
Journal of Applied Mathematics
7
ii When k 1, we have γ1 1/2, γ 2/2c − α 1 √
1/4 13.
2c − α 12 8α and r1 Theorem 3.4. Let F ∈ k − ∪Bm α, β, p, f, f ∈ k − ∪R2 , p ∈ Pm pk . Define, for Reαk/k 1 c iβ > 0,
z
1/αiβ
−c
c−1 αiβ
t F
.
Gz c 1z
tdt
3.19
0
Then, G ∈ k − ∪Bm α, β, h, g in E, where gz is given by 3.1, and hz is analytic in E with
h0 1.
Proof. Set
zG zGαiβ−1 z
hz ziβ g α z
m 1
m 1
−
h1 z −
h2 z.
4 2
4 2
3.20
We note that hz is analytic in E with h0 1. From 3.20, we have
zg z
c iβ
zF zF αiβ−1 z,
ziβ g α z zh z hz α
gz
3.21
using 3.1, we note that
fz
gz
α
α
zg z
c iβ.
gz
3.22
From 3.21 and 3.22, it follows that
zh z
hz αh0 z c iβ
∈ P m pk ,
3.23
where h0 z zg z/gz ∈ P pk since g ∈ k − ∪R2 by Theorem 3.1.
It can easily be seen that g ∈ S∗ k/k 1 and Re{αzg z/gz c iβ} > 0.
Now, using 3.8, we can easily derive
hi z jz zhi z ≺ pk z
in E,
i 1, 2,
3.24
where 1/jz {αzg z/gz c iβ} and Re jz > 0.
Applying Lemma 2.2, it follows from 3.24 hi z ≺ pk z in E and therefore h ∈ Pm pk in E. This completes the proof.
8
Journal of Applied Mathematics
Theorem 3.5. Let fz be given by 1.1 with hz 1,{α2 β2 1/2
α2 β2 eiγ α iβ, |γ| < π/2. Then, for z ∈ E
i eiγ zf z/fz cos γpz i sin γ,
1/2 iγ
e zg /g} ∈ Pm pk p ∈ Pm pk ,
ii For α iβ tα iβ, t ≥ 1,
k − ∪Bm α, β, 1, g ⊂ kt − ∪Bm α , β , 1, g .
3.25
Proof. i From 1.1, we have
1
zf z
zf z α − 1 iβ
f z
fz
zg z
α
iβ H2 z,
gz
H2 ∈ Pm pk in E.
3.26
Define a function pz analytic in E by
eiγ
zf z
cos γ pz i sin γ,
fz
β
γ tan−1 .
α
3.27
We can easily check that p0 1.
Now, from 3.26 and 3.27, we have
zp z
αpz iβ ∈ Pm pk
pz i tan γ
in E.
3.28
That is
αzp z
αpz iβ ∈ Pm pk ,
αpz iβ
3.29
and, with hz αpz iβ m/4 1/2h1 z − m/4 − 1/2h2 z, we apply convolution
technique used before to have
zhi z
hi z hi z
≺ pk z
3.30
in E.
Applying Lemma, it follows that
hi z ≺ qk z ≺ pk z,
z ∈ E,
3.31
where qk z is the best dominant and is given by
qk z ! 1
tz
exp
0
0
pk u − 1
du
u
"−1
.
From 3.31, we have hz αpz iβ ∈ Pm pk in E, and this proves part i.
3.32
Journal of Applied Mathematics
9
ii From part i, we have
#
α2 β2
$1/2
eiγ
zf z
H1 z,
fz
H1 ∈ Pm pk
in E.
3.33
Now,
zf z zf z
α
−
1
iβ
f z
fz
zf z
zf z α − 1 iβ
1 f z
fz
#
$1/2 zf z
t − 1 α2 β2
,
eiγ
fz
H2 z t − 1H1 z, Hi ∈ Pm pk ,
1
1
H1 z H2 z ,
t 1−
t
t
tH, t ≥ 1,
1
3.34
i 1, 2,
H ∈ Pm pk , since Pm pk is convex set, see 8.
Therefore, f ∈ kt − ∪Bm α , β , 1, g for z ∈ E. This completes the proof.
As a special case, with m 2,k 0, we obtain a result proved in 10.
By assigning certain permissible values to the parameters α, β and m, we have several
other new results.
Acknowledgment
The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information
Technology, Islamabad, Pakistan, for providing excellent research facilities and environment
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10
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