Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 837913, 10 pages
doi:10.1155/2012/837913
Research Article
On Certain Sufficiency Criteria for p-Valent
Meromorphic Spiralike Functions
Muhammad Arif
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan
Correspondence should be addressed to Muhammad Arif, marifmaths@yahoo.com
Received 10 June 2012; Accepted 6 August 2012
Academic Editor: Allan Peterson
Copyright q 2012 Muhammad Arif. 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 consider some subclasses of meromorphic multivalent functions and obtain certain simple
sufficiency criteria for the functions belonging to these classes. We also study the mapping
properties of these classes under an integral operator.
1. Introduction
Let
p, n denote the class of functions fz of the form
fz z−p ∞
ak zk−p1
p∈N ,
1.1
kn
which are analytic and p-valent in the punctured unit disk U {z : 0 < |z| < 1}. Also let
∗
λ
p, n consisting of all functions fz
c p, n, α denote the subclasses of
λ p, n, α and
which are defined, respectively, by
z
> α cos λ z ∈ U,
Re −e
fz
zf z
iλ
> α cos λ z ∈ U.
Re −e
f z
of
iλ zf
1.2
We note that for λ 0 and n 1, the above classes reduce to the well-known subclasses
p consisting of meromorphic multivalent functions which are, respectively, starlike
2
Abstract and Applied Analysis
and convex of order α 0 ≤ α < p. For the detail on the subject of meromorphic spiral-like
functions and related topics, we refer the work of Liu and Srivastava 1, Goyal and Prajapat
2, Raina and Srivastava 3, Xu and Yang 4, and Spacek 5 and Robertson 6.
Analogous to the subclass of 1, 1 for meromorphic univalent functions studied by
Wang et al. 7 and Nehari and Netanyahu 8, we define a subclass λN p, n, α of p, n
consisting of functions fz satisfying
− Re e
zf z
f z
iλ
α > p, z ∈ U .
< α cos λ
1.3
For more details of the above classes see also 9, 10.
Motivated from the work of Frasin 11, we introduce the following integral operator
of multivalent meromorphic functions p
Hm,p z 1
z m
zp1
0 j1
up fj u
αj
du.
1.4
For p 1, 1.4 reduces to the integral operator introduced and studied by Mohammed
and Darus 12, 13. Similar integral operators for different classes of analytic, univalent, and
multivalent functons in the open unit disk are studied by various authors, see 14–19.
∗
In this paper, first, we find sufficient conditions for the classes
λ p, n, α and
λ
c p, n, α and then study some mapping properties of the integral operator given by
1.4.
We will assume throughout our discussion, unless otherwise stated, that λ is real with
|λ| < π/2, 0 ≤ α < p, p, n ∈ N, αj > 0 for j ∈ {1, . . . m}.
To obtain our main results, we need the following Lemmas.
Lemma 1.1 see 20. If qz ∈
1, n with n ≥ 1 and satisfies the condition
n
2 z q z 1 < √
n2 1
z ∈ U,
1.5
then
qz ∈
∗
1, n, 0.
1.6
0
Lemma 1.2 see 21. Let Ω be a set in the complex plane C and suppose that Ψ is a mapping
from C2 × U to C which satisfies Ψix, y, z ∈
/ Ω for z ∈ U, and for all real x, y such that y ≤
−n/21 x2 . If hz 1 cn zn · · · is analytic in U and Ψhz, zh z, z ∈ Ω for all z ∈ U,
then Re hz > 0.
Abstract and Applied Analysis
3
2. Some Properties of the Classes
Theorem 2.1. If fz ∈
∗
λ p, n, α
and
λ
c p, n, α
p, n satisfies
p
iλ
z fz e /p−α cos λ eiλ zf z α cos λ ip sin λ p − α cos λ
fz
n
<√
n2 1
then fz ∈
p − α cos λ
2.1
z ∈ U,
∗
λ p, n, α.
Proof. Let us set a function hz by
hz eiλ /p−α cos λ 1
eiλ an
1 p
zn · · ·
z fz
z
z
p − α cos λ
2.2
for fz ∈ p, n. Then clearly 2.2 shows that hz ∈ 1, n.
Differentiating 2.2 logarithmically, we have
f z p
eiλ
1
h z
−
hz
fz
z
z
p − α cos λ
2.3
which gives
2 z h z 1
eiλ /p−α cos λ
1
p
iλ zf z
e
z fz
α cos λ ip sin λ 1.
fz
p − α cos λ
2.4
Thus using 2.1, we have
n
2 z h z 1 ≤ √
n2 1
Hence, using Lemma 1.1, we have hz
From 2.3, we can write
1
zh z
hz
p − α cos λ
Since hz ∈
∗
0 1, n, 0,
eiλ
z ∈ U.
∗
0 1, n, 0.
zf z α cos λ ip sin λ .
fz
2.6
it implies that Re−zh z/hz > 0. Therefore, we get
1
zh z
iλ zf z
− α cos λ Re −
>0
Re −e
fz
hz
p − α cos λ
2.5
2.7
4
Abstract and Applied Analysis
or
eiλ
Re −
zf z
fz
2.8
> α cos λ,
and this implies that fz ∈ ∗λ p, n, α.
If we take λ 0, we obtain the following result.
Corollary 2.2. If fz ∈
p, n satisfies
p
z fz 1/p−α eiλ zf z α p − α < √1 p − α
fz
2
then fz ∈
∗
z ∈ U,
p, n, α.
Theorem 2.3. If fz ∈
p, n satisfies
eiλ /p−α cos λ zp1 f z
zf
z
iλ
1 α cos λ ip sin λ p − α cos λ
e
−p
f z
n 1 p − α cos λ
<
n 12 1
then fz ∈
2.9
2.10
z ∈ U,
λ
c p, n, α.
Proof. Let us set
hz −
z
0
1
t2
−tp1 f t
p
eiλ /p−α cos λ
dt eiλ an
1 n−p1
zn · · · .
z
np
p − α cos λ
2.11
Also let
1
gz −zh z z
−zp1 f z
p
eiλ /p−α cos λ
Then clearly hz and gz ∈
1
gz z
eiλ an
1 p−n−1
zn · · · . 2.12
z
p
p − α cos λ
1, n. Now
−zp1 f z
p
eiλ /p−α cos λ
.
2.13
Abstract and Applied Analysis
5
Differentiating logarithmically and then simple computation gives us
2 z g z 1
ei λ/p−α cos λ
zp1 f z
zf
z
1
1
α
cos
λ
ip
sin
λ
1
eiλ
−p
f z
p − α cos λ
<√
n
n2 1
.
2.14
Therefore, by using Lemma 1.1, we have
gz −zh z ∈
∗
1, n, 0
2.15
0
which implies that hz ∈
1
0
c 1, n, 0.
Since
zf z zh z
eiλ
p
1
− 1,
h z
p − α cos λ f z
2.16
therefore
zh z
1
Re 1 Re eiλ 1 h z
p − α cos λ
1
iλ
Re e 1 p − α cos λ
Since hz ∈
0
c 1, n, 0,
zf z
f z
zf z
f z
peiλ − p − α cos λ
α cos λ .
2.17
so
zf z
−1
α cos λ > 0,
Re eiλ 1 f z
p − α cos λ
2.18
or
zf z
− Re eiλ 1 > α cos λ.
f z
It follows that fz ∈
Theorem 2.4. If fz ∈
2.19
λ
c p, n, α.
p, n satisfies
zf z
M2
iλ zf z
α N
Re e
−1 >
fz
f z
4L
z ∈ U,
2.20
6
Abstract and Applied Analysis
then fz ∈
∗
λ p, n, β,
where 0 ≤ α ≤ 1, 0 ≤ β < p and
n β − p cos 2λ cos λ,
Lα p−β
2
M α β − p 1 − β cos λ sin 2λ cos λ,
n
N α β2 cos2 λ sin2 λ −
β − p cos λ 1 αβ cos λ.
2
2.21
Proof. Let us set
eiλ
zf z β − p hz − β cos λ − i sin λ.
fz
2.22
Then hz is analytic in U with p0 1.
Taking logarithmic differentiation of 2.22 and then by simple computation, we obtain
eiλ
zf z
zf z
α − 1 Azh z Bh2 z Chz D
fz
f z
Ψ hz, zh z, z
2.23
with
A β − p α cos λ,
2
B αe−iλ β − p cos2 λ,
C p − β αe−iλ 2βcos2 λ i sin 2λ 1 αe−iλ cos λ ,
2.24
D αe−iλ β2 cos2 λ − sin2 λ iβ sin 2λ 1 α β cos λ − i sin λ .
Now for all real x and y satisfying y ≤ −n/21 x2 , we have
Ψ ix, y, z Ay − Bx2 Cix D.
2.25
Reputing the values of A, B, C, and D and then taking real part, we obtain
Re Ψ ix, y, z ≤ −Lx2 Mx N
−
<
where L, M, and N are given in 2.21.
√
M
Lx − √
2 L
M2
N,
4L
2
M2
N
4L
2.26
Abstract and Applied Analysis
7
/ Ω,
Let Ω {w : Re w > M2 /4L N}. Then Ψhz, zh z, z ∈ Ω and Ψix, y, z ∈
for all real x and y satisfying y ≤ −n/21 x2 , z ∈ U. By using Lemma 1.2, we have
Re hz > 0, that is fz ∈ ∗λ p, n, β.
If we put λ 0, we obtain the following result.
Corollary 2.5. If fz ∈
p, n satisfies
zf z
zf z
n
α − 1 > α β2 −
Re
β − p 1 αβ
fz
f z
2
then fz
∗
z ∈ U,
2.27
p, n, β, where 0 ≤ α ≤ 1, 0 ≤ β < p.
Theorem 2.6. For j ∈ {1, . . . m}, let fj z ∈
p, n and satisfy 2.9. If
m
αj <
j1
then Hm,p z ∈
N p, n, ζ,
p1
,
p−β
2.28
where ζ > p.
Proof. From 1.4, we obtain
m
α
p
zp1 Hm,p
z fj z j .
z p 1 zp Hm,p z 2.29
j1
Differentiating again logarithmically and then by simple computation, we get
⎤
⎡
m
Hm,p z
zfj z
H
z
m,p
⎣ αj
1 p2 − 1
2p 1 p 1
p − 1⎦,
fj z
Hm,p
zHm,p
zHm,p
z
z
z
j1
zHm,p
z
2.30
or, equivalently we can write
−
zHm,p
z
Hm,p
z
1
⎛
⎞
⎡
⎤
m
zfj z
2
⎣ p 1 ⎝ αj −
− p 1⎠ p − 1 ⎦
fj z
zHm,p
z
j1
Hm,p z
m
j1
αj
−
zfj z
fj z
−p
1 2p .
2.31
8
Abstract and Applied Analysis
Now taking real part on both sides, we obtain
− Re
zHm,p
z
Hm,p
z
1
⎞
⎛
⎡
⎤
m
zfj z
2
⎣ p 1 ⎝ αj −
− p 1⎠ p − 1 ⎦
Re
fj z
zHm,p
z
j1
Hm,p z
m
αj
− Re
zfj z
j1
fj z
−p
1 2p .
2.32
This further implies that
− Re
zHm,p
z
Hm,p
z
1
⎛
⎞
⎡
⎤
m
zfj z
Hm,p z 2
⎣ p 1 ⎝ αj −
≤ − p 1⎠ p − 1 ⎦
fj z
zHm,p z
j1
m
αj
− Re
j1
zfj z
fj z
−p
1 2p .
2.33
Let
⎛
⎞
⎡
⎤
m
zfj z
Hm,p z ⎣ p 1 ⎝ αj −
ζ − p 1⎠ p2 − 1 ⎦
fj z
zHm,p z
j1
⎡
⎤
m
zfj z
⎣ αj − Re
− p 1 2p ⎦.
fj z
j1
2.34
Clearly we have
⎡
⎤
m
zfj z
ζ > ⎣ αj − Re
− p 1 2p ⎦.
fj z
j1
2.35
Then by using 2.28 and Corollary 2.2, we obtain
ζ>
m
αj β − p 1 2p > p.
j1
Therefore Hm,p z ∈ N p, n, ζ with ζ > p.
Making use of 2.27 and Corollary 2.5, one can prove the following result.
2.36
Abstract and Applied Analysis
9
Theorem 2.7. For j ∈ {1, . . . m}, let fj z ∈
m
j1
then Hm,p z ∈
N p, n, ζ,
p, n and satisfy 2.27. If
αj <
p1
,
p−β
2.37
where ζ > p.
Acknowledgment
The author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.
References
1 J.-L. Liu and H. M. Srivastava, “A linear operator and associated families of meromorphically multivalent functions,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 566–581, 2001.
2 S. P. Goyal and J. K. Prajapat, “A new class of meromorphic multivalent functions involving certain
linear operator,” Tamsui Oxford Journal of Mathematical Sciences, vol. 25, no. 2, pp. 167–176, 2009.
3 R. K. Raina and H. M. Srivastava, “A new class of meromorphically multivalent functions with applications to generalized hypergeometric functions,” Mathematical and Computer Modelling, vol. 43, no.
3-4, pp. 350–356, 2006.
4 N. Xu and D. Yang, “On starlikeness and close-to-convexity of certain meromorphic functions,” Journal of the Korea Society of Mathematical Education B, vol. 10, no. 1, pp. 566–581, 2003.
5 L. Spacek, “Prispevek k teorii funkei prostych,” Časopis Pro Pestovánı́ Matematiky A Fysiky, vol. 62, pp.
12–19, 1933.
6 M. S. Robertson, “Univalent functions fz for which zf z is spirallike,” The Michigan Mathematical
Journal, vol. 16, pp. 97–101, 1969.
7 Z.-G. Wang, Y. Sun, and Z.-H. Zhang, “Certain classes of meromorphic multivalent functions,”
Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1408–1417, 2009.
8 Z. Nehari and E. Netanyahu, “On the coefficients of meromorphic schlicht functions,” Proceedings of
the American Mathematical Society, vol. 8, pp. 15–23, 1957.
9 Z.-G. Wang, Z.-H. Liu, and R.-G. Xiang, “Some criteria for meromorphic multivalent starlike
functions,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1107–1111, 2011.
10 Z.-G. Wang, Z.-H. Liu, and A. Catas, “On neighborhoods and partial sums of certain meromorphic
multivalent functions,” Applied Mathematics Letters, vol. 24, no. 6, pp. 864–868, 2011.
11 B. A. Frasin, “New general integral operators of p-valent functions,” Journal of Inequalities in Pure and
Applied Mathematics, vol. 10, no. 4, article 109, 2009.
12 A. Mohammed and M. Darus, “A new integral operator for meromorphic functions,” Acta Universitatis Apulensis, no. 24, pp. 231–238, 2010.
13 A. Mohammed and M. Darus, “Starlikeness properties of a new integral operator for meromorphic
functions,” Journal of Applied Mathematics, Article ID 804150, 8 pages, 2011.
14 M. Arif, W. Haq, and M. Ismail, “Mapping properties of generalized Robertson functions under
certain integral operators,” Applied Mathematics, vol. 3, no. 1, pp. 52–55, 2012.
15 K. I. Noor, M. Arif, and A. Muhammad, “Mapping properties of some classes of analytic functions
under an integral operator,” Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 593–600, 2010.
16 N. Breaz, V. Pescar, and D. Breaz, “Univalence criteria for a new integral operator,” Mathematical and
Computer Modelling, vol. 52, no. 1-2, pp. 241–246, 2010.
17 B. A. Frasin, “Convexity of integral operators of p-valent functions,” Mathematical and Computer
Modelling, vol. 51, no. 5-6, pp. 601–605, 2010.
18 B. A. Frasin, “Some sufficient conditions for certain integral operators,” Journal of Mathematical Inequalities, vol. 2, no. 4, pp. 527–535, 2008.
19 G. Saltik, E. Deniz, and E. Kadioğlu, “Two new general p-valent integral operators,” Mathematical and
Computer Modelling, vol. 52, no. 9-10, pp. 1605–1609, 2010.
10
Abstract and Applied Analysis
20 H. Al-Amiri and P. T. Mocanu, “Some simple criteria of starlikeness and convexity for meromorphic
functions,” Mathematica, vol. 3760, no. 1-2, pp. 11–20, 1995.
21 S. S. Miller and P. T. Mocanu, “Differential subordinations and inequalities in the complex plane,”
Journal of Differential Equations, vol. 67, no. 2, pp. 199–211, 1987.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014