Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 3(2011), Pages 115-121.
INCLUSION PROPERTIES OF CERTAIN CLASSES OF
MEROMORPHIC SPIRAL-LIKE FUNCTIONS OF COMPLEX
ORDER ASSOICATED WITH THE GENERALIZED
HYPERGEOMETRIC FUNCTION
(C O M M U N IC AT E D B Y S H IG E Y O S H I O WA )
ALI MUHAMMAD
Abstract. The purpose of the present paper is to introduce new classes of
meromorphic spiral-like functions de…ned by using a meromorphic analogue of
the Choi-Saigo-Srivastava operator for the generalized hypergeometric function
and investigate a number of inclusion relationships of these classes.
1. Introduction
Let M denote the class of functions of the form
f (z) =
1
1 X
+
ak z k ;
z
(1.1)
k=0
which are analytic in the punctured unit disk E = fz : 0 < jzj < 1g = Enf0g:
If f and g are analytic in E = E [ f0g; we say that f is subordinate to g;
written f
g or f (z) g(z); if there exists a Schwarz function w in E such that
f (z) = g(w(z)):
Let P be the class of all functions which are analytic and univalent in E and
for which (E) is convex with (0) = 1 and Re f (zg > 0 (z 2 E):
For a complex parameters 1 ; ::: q and 1 ; ::: s ( j 2 CnZ0 = f0; 1; 2; :::g;
j = 1; :::s); we now de…ne the generalized hypergeometric function [16; 17] as follows:
1
X
( 1 )k :::( q )k k
F
(
;
:::
;
;
:::
)
=
z ;
(1.2)
q s
q
s
1
1
( 1 )k :::( s )k k!
k=0
2000 Mathematics Subject Classi…cation. 30C45, 30C50.
Key words and phrases. Meromorphic functions; Spiral-like functions of complex order;
Hadamard product; Di¤erential subordination; Choi-Saigo-Srivastava operator.
c 2011 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted March 17, 2011. Accepted June 2, 2011.
Dedicated to Mr. and Mrs. Ibrahim Amodu Nigeria.
115
116
ALI M UHAM M AD
where (q s + 1; s 2 N [ f0g; N = f1; 2; :::g) and (v)k is the Pochhammer symbol
(or shifted factorial) de…ned in (terms of the Gamma function) by
(v)k =
(v + k)
=
(v)
1 if k = 0 and v 2 Cnf0g
v(v + 1):::(v + k 1) if k 2 N and v 2 C:
Corresponding to a function
F(
1
; :::
q;
1 ; ::: s ; z)
=z
1
q Fs (
1
; :::
q;
; :::
1
Liu and Srivastava [11] consider a linear operator H( 1 ; ::: q ;
de…ned by the following Hadamard product(or convolution):
H(
1
; ::
q;
1
; ::
s )f (z)
= h(
1 ; ::: q ;
1
; :::
s ; z)
s ; z):
(1.3)
1 ; ::: s )
:M !M
f (z):
We note that the linear operator H( 1 ; :: q ; 1 ; :: s ) was motivated essentially by
Dzoik and Srivastava [4]: Some interesting developments with the generalized hypergeometric function were considered recently by Dzoik and Srivastava [5; 6] and Liu
and Srivastava [9; 10]: Corresponding to the function h( 1 ; ::: q ; 1 ; ::: s ; z) de…ned
by (1:3); we introduce a function h ( 1 ; ::: q ; 1 ; ::: s ; z) given by
h(
1 ; ::: q ;
1 ; ::: s ; z)
h (
1 ; ::: q ;
1 ; ::: s ; z)
=
1
z(1
( > 0):
z)
(1.5)
Analogous to H( 1 ; :: q ; 1 ; :: s ) de…ned by (1:4); we now de…ne the linear operator
H ( 1 ; :: q ; 1 ; :: s ) on M as follows:
H (
1 ; :: q ;
1 ; :: s )f (z)
=h (
where i ; j 2 CnZ0 ; i = 1; ::q; j = 1; ::s;
For convenience, we write
H
;q;s ( 1 )
=H (
1 ; ::: q ;
1 ; ::: s ; z)
f (z);
(1.6)
+ 1)f (z);
(1.7)
> 0; z 2 E ; f 2 M:
1 ; :: q ;
1 ; :: s ):
It is easily veri…ed from the de…nition (1:5) and (1:6) that
z(H
;q;s ( 1
0
+ 1)f (z)) =
1 H ;q;s ( 1 )f (z)
(
1
+ 1)H
;q;s ( 1
and
z(H
0
;q;s ( 1 )f (z))
= H
+1;q;s ( 1 )f (z)
( + 1)H
;q;s ( 1 )f (z):
(1.8)
We note that the operator H ;q;s ( 1 ) is closely related to the Choi-Saigo-Srivastava
operator [3] for analytic functions, which includes the integral operator studied by
Liu [8] and Noor et al [13; 15]: The interested readers are refered to the work done
by the authors [1; 2; 14]:
De…nition 1.1. Using the subordination principle between two analytic functions, we introduce the subclasses M Sb ( (z)); M Cb ( (z)) and M Kb;c; ( (z); (z))
of the class M as follows:
ei
zf 0 (z)
M Sb ( (z)) =
f (z) 2 M : 1 +
1
(z) in E
b cos
f (z)
ei
(zf 0 (z))0
M Cb ( (z)) =
f (z) 2 M : 1 +
1
(z) in E
b cos
f 0 (z)
M Kb;c; ( (z); (z))
=
f (z) 2 M : 1 +
with g(z) 2 M Sb ( (z)) in E and ;
and (z); (z) 2 P; z 2 E:
zf 0 (z)
g(z)
ei
b cos
2R:j j<
2;
j j<
1
2 ; b; c
(z),
;
6= 0 with b:c 2 C
INCLUSION PROPERTIES OF CERTAIN CLASSES
Now by using the operator (H
morphic functions.
M Sb;
M Cb;
;
M Kb;c;
;q;s;;
;q;s;
1
( (z))
;q;s ( 1 );we
introduce a new subclasses of mero-
= ff (z) 2 M : H
= ff (z) 2 M : H
n
(
(z);
(z))
=
f (z) 2 M : H
1
;q;s;
1
( (z))
where ; 2 R : j j < 2 ; j j <
(1:9) and (1:10); it is clear that
f (z) 2 M Cb;
;q;s;
1
2;
117
;q;s ( 1 )f (z)
;q;s ( 1 )f (z)
;q;s ( 1 )f (z)
2 M Sb ( (z)) g
2 M Cb ( (z)) g
2 M Kb;c;
(1.10)
o
( (z); (z)) ;
(1.11)
b; c 6= 0 with b:c 2 C and (z);
( (z)) ()
(1.9)
zf 0 (z) 2 M Sb;
;q;s;
1
(z) 2 P: From
( (z)):
(1.12)
2. Preliminary Results
To establish our main results we need the following Lemmas.
Lemma 2.1 [7]: Let be convex univalent in E with (0) = 1 and Ref
t) > 0 ( ,t 2 C): If p is analytic in E with p(0) = 1; then
p(z) +
z p0 (z)
p(z) + t
(z) (z 2 E) , ) p(z)
(z) +
(z):
Lemma 2.2 [12]: Let (z) 2 P be convex univalent in E and !(z) be analytic
in E with Ref!(z)g 0: If p is analytic in E with p(0) = (0); then
p(z) + !(z)zp0 (z)
(z) (z 2 E) =) p(z)
(z):
3. Main Results
Theorem 3.1. Let
(z) 2 P for z 2 E ( ;
M Sb;
+1;q;s;
1
2 R; where j j <
1 > 0): Then
( (z))
M Sb;
;q;s;
1
2
and let b = b1 + ib2 6= 0; tan
( (z))
M Sb;
;q;s;
1 +1
=
b2
b1 ;
( (z));
for Im (z) < (Re (z) 1) cot(
v):
Proof. To prove the …rst part of Theorem 3.1, let f 2 M Sb; +1;q;s; 1 ( (z)) and
set
1
z(H ;q;s ( 1 )f (z))0
p(z) =
ei
(1 b) cos
i sin
:
(3.1)
b cos
H ;q;s ( 1 )f (z)
Then p(z) is analytic in E with p(0) = 1: Applying (1:8) in (3:1) and with a simple
computations, we have for > 0
( 1 )f (z))0
( 1 )f (z)
+1 ;q;s
zp0 (z)
e i b cos (p(z) 1) + + 1
(3.2)
Since Ref e i b cos ( (z) 1) + + 1) > 0 for Im (z) < (Re (z) 1) cot(
v)
and where tan = bb12 ; so by Lemma 2.1 and (3:2); we have p(z)
(z): This proves
that
M Sb; +1;q;s; 1 ( (z)) M Sb; ;q;s; 1 ( (z)):
(3.3)
1+
ei
b cos
z(H
H
+1 ;q;s
1
= p(z)+
To prove the second part of Theorem 3.1, we consider
p(z) =
1
b cos
ei
z(H
H
+ 1)f (z))0
;q;s ( 1 + 1)f (z)
;q;s ( 1
(1
b) cos
i sin
:
(3.4)
(z):
118
ALI M UHAM M AD
Then p(z) is analytic in E with p(0) = 1: Applying (1:7) in (3:1) and with a simple
computation, we have for 1 > 0
1+
ei
b cos
0
;q;s ( 1 )f (z))
z(H
H
= p(z)+
+1;q;s;
1
( (z))
M Cb;
;q;s;
( (z))
1
e
i
zp0 (z)
b cos (p(z) 1) +
+1
(3.5)
Since Ref e i b cos ( (z) 1) + 1 + 1) > 0 for Im (z) < (Re (z) 1) cot(
v)
and where tan = bb12 ; so by Lemma 2.1 and (3:5); we have p(z)
(z): This
complete the proof of second inclusion.
Theorem 3.2. Let 2 R; where j j < 2 and let b = b1 + ib2 6= 0; tan = bb21 ;
(z) 2 P for z 2 E ( ; 1 > 0): Then
M Cb;
;q;s ( 1 )f (z)
1
M Cb;
;q;s;
(z):
1
( (z)):
1 +1
for Im (z) < (Re (z) 1) cot(
v); z 2 E:
Proof. The proof follows from Theorem 3.1 and (1:12):
Taking
(z) =
1 + Az
( 1<B<A
1 + Bz
1; z 2 E)
Corollary 3.3. Let 2 R; where j j < 2 and let b = b1 + ib2 =
6 0; tan =
1+A
i
i
b cos ; 1 + 1=e b cos g; 1 < B < A 1: Then
1+B < minf + 1=e
M Sb;
+1;q;s;
1
(A; B))
M Sb;
;q;s;
1
M Cb;
+1;q;s;
1
(A; B)
M Sb;
;q;s;
1 +1
(A; B);
(A; B)
M Cb;
;q;s;
1
(A; B)
M Cb;
;q;s;
1 +1
(A; B):
b2
b1 ;
and
Next, by using Lemma 2.2, we obtain the following Inclusion relation for the class
of meromorphically close to convex functions.
Theorem 3.4. Let ; 2 R; where j j < 2 ; j j < 2 and let b = b1 + ib2 6= 0;
tan = bb12 ; (z); (z) 2 P for z 2 E: Then
;
M Kb;c;
+1;q;s;;
1
( (z); (z))
;
M Kb;c;
;q;s;;
1
( (z); (z))
;
M Kb;c;
;q;s;;
1 +1
( (z); (z));
for Im (z) < Re(Re (z) 1) cot(
v); Im(q(z) < (Re q(z) 1) cot(
v)(z 2 E);
( ; 1 > 0):
;
Proof. To prove the …rst inclusion of Theorem 3.4, let f 2 M Kb;c;
+1;q;s;; 1 ( (z); (z)):
;
Then from the de…nition of M Kb;c;
g 2 M Sb; +1;q;s; 1 ( (z))) such that
1
c cos
ei
z(H
H
0
+1;q;s ( 1 )f (z))
+1;q;s ( 1 )g(z)
+1;q;s;;
(1
1
( (z); (z)); there existsa function
c) cos
i sin
(z):(z 2 E):
(3.6)
Now let
p(z) =
1
c cos
ei
z(H
H
0
;q;s ( 1 )f (z))
;q;s ( 1 )g(z)
(1
c) cos
i sin
;
(3.7)
INCLUSION PROPERTIES OF CERTAIN CLASSES
119
where p(z) is analytic in E with p(0) = 1: Using (1:8); we obtain that
1
c cos
=
1
c cos
ei
0
@ei
Since g(z) 2 M Sb;
q(z) =
1
b cos
0
+1;q;s ( 1 )f (z))
z(H
H
(1
+1;q;s ( 1 )g(z)
c) cos
i sin
z(H ;q;s ( 1 )( zf (z))0
z(H
( 1 )( zf (z))
+ ( + 1) H;q;s
H ;q;s ( 1 )g(z)
;q;s ( 1 )g(z)
(H ;q;s ( 1 )g(z))0
H ;q;s ( 1 )g(z) + + 1
(1
c) cos
+1;q;s;
1
ei
( (z))
M Sb;
;q;s;
0
;q;s ( 1 )f (z))
z(H
H
;q;s ( 1 )g(z)
1
( (z)); by Theorem 3.1, we set
(1
b) cos
and > 0 in E with Re( e i (q(z) 1) +
1) cot(
v); Im(q(z) < (Re q(z) 1) cot(
!(z) =
i sin
;
(3.9)
1
e
i
(q(z)
1) +
+1
+ 1) > 0 for Im (z) <
v): Hence, by taking
;
in (3:10); and applying Lemma2.2, we can show that p(z)
(z) in E; so that
;
f (z) 2 M Kb;c;
(
(z);
(z)):
Moreover,
we
have
the
second
inclusion
by using
;q;s;; 1
the similar arguments to those detailed above with (1:7): Therefore we complete
the proof of the Theorem 3.4.
Inclusion properties involving the Integral operaton F
Consider the operator F ; de…ned by
F (f )(z) =
z
+1
Zz
t f (t)dt (f 2 M ;
> 0):
(3.11)
0
From the de…nition of F de…ned by (3:11); we observe that
z(H
;q;s ( 1 )F
i sin A :
(3.8)
where q
in E with assumption 2 P: Then , by virtue of (3:7); (3:8) and (3:9);
we obtain that
1
z(H +1;q;s ( 1 )f (z))0
ei
(1 c) cos
i sin
c cos
H +1;q;s ( 1 )g(z)
zp0 (z)
= p(z) +
(z) (z 2 E):
(3.10)
i
e (q(z) 1) + + 1
Since q
Re(Re (z)
1
f (z))0 = H
;q;s ( 1 )f (z)
( + 1)H
;q;s ( 1 )F
f (z):
(3.12)
Theorem 3.5. Let 2 R; where j j < 2 and let b = b1 + ib2 6= 0; (z) 2
P for z 2 E ( ; 1 > 0): Then for f (z) 2 M Sb; ;q;s; 1 ( (z)); then F (f )(z) 2
M Sb; ;q;s; 1 ( (z)); for Im (z) < (Re (z) 1) cot(
v); where tan v = bb21 ; z 2 E:
Proof. Consider
1
z(H ;q;s ( 1 )F (f )(z))0
p(z) =
ei
(1 b) cos
i sin
; (3.13)
b cos
H ;q;s ( 1 )F (f )(z)
where p(z) is analytic in E with p(0) = 1: Using (3:12) in (3:13) and after simple
computation we have
p(z) +
e
i
zp0 (z)
b cos (p(z)
1) +
(z):
120
ALI M UHAM M AD
For Im (z) < (Re (z)
1) cot(
Re( e
i
v); where tan v =
b cos (p(z)
b2
b1 ;
we have
1) + ) > 0:
Thus, by Lemma 2.1 yeilds p(z)
(z): Hence we have the desired proof.
Next, we derive an inclusion property involving F which is obtaind by applying
(1:8) and Theorem 3.5.
Theorem 3.6. Let 2 R; where j j < 2 and let b = b1 + ib2 6= 0; (z) 2
P for z 2 E ( ; 1 > 0): Then for f (z) 2 M Cb; ;q;s; 1 ( (z)); then F (f )(z) 2
M Cb; ;q;s; 1 ( (z)); for Im (z) < (Re (z) 1) cot(
v); where tan v = bb21 ; z 2 E:
Finally, we obtain Theorem 3.7 below by using the same lines of proof as we
used in the proof of Theorem3.4.
Theorem 3.7. Let ; 2 R;where j j < 2 ; j j < 2 and let b = b1 + ib2 6= 0;
;
tan = bb12 ; (z); (z) 2 P for z 2 E ( ; 1 > 0): If f 2 M Kb;c;
+1;q;s;; 1 ( (z); (z));
;
Then F (f )(z) 2 M Kb;c;
+1;q;s;;
Im (z) < (Re (z) 1) cot(
1
( (z); (z)) ( > 0) for
v); Im q(z) < (Re q(z) 1) cot(
v) and q(z)
(z); z 2 E:
Acknowledgments. The authors would like to thank S. Owa for his comments
that helped us improve this article.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Ali Muhammad, On certain class of meromorphic functions de…ned by means of a linear
operator, J. Acta. Univesitatis. Apulensis, no 23 (2010), 251-262.
N. E. Cho and In. Hwa. Kim, Inclusion properties of certain of certain classes of meromophic
functions associated with the gerneralized hepergeometric functions. Appl. Math. Comput.
187 (2007), 115-121.
J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion properties of a certain family of
integral operators, J. Math. Anal. Appl. 276 (2002), 432 -445.
J. Dziok, and H. M. Srivastava, Classes of analytic functions associated with the generalized
hypergeometric function, Appl. Math. Comput, 103 (1999), 1-13.
J. Dziok, and H. M. Srivastava, Some subclasses of analytic functions with …xed argument of
coe¢ cients associated with the generalized hypergeometric functions, Adv. Stud. Contemp.
Math. 5 (2002), 115 -125.
J. Dziok, and H. M. Srivastava, Certain subclasses of analytic functions associated with the
generalized hypergeometric function, Integral Trans. Spec. Funct. 14 (2003), 7-18.
P. Eenigenberg, S. S. Miller, P.T. Mocanu and M. O. Reade, On a Briot-Bouquet di¤erential
subordination, General Inequalities 3 (1983),339-348.
J. L. Liu, The Noor integral and strongly starlike functions, J. Math. Anal. Appl. 261 (2001),
441- 447.
J. L. Liu and H. M. Srivastava, A linear operator and associated families of meromorphically
multivalent functions, J. Math. Anal. Appl.259 (2001), 566-581.
J. L. Liu and H. M. Srivastava, Certain properties of the Dzoik Srivastava operator, Appl.
Math. Comput. 159 (2004), 485-493.
J. L. Liu and H. M. Srivastava, Classes of meromorphically multivalent functions associated
with the generalized hypergeormetric function, Math. Comut. Modell. 39 (2004), 21- 34.
S. S. Miller and P. T. Mocanu, Di¤erential subordinations and univalent functions, Michigan
Math. J. 28 (1981), 157-171.
K. I. Noor, On new classes of integral operators, J. Natur. Geom. 16 (1999), 71- 80.
K. I. Noor and Ali Muhammad, On certain subclasses of meromorphic univalent functions,
Bull. Institute. Maths. Academia. Sinica, Vol 5 (2010), 83-94.
K. I. Noor and M. A. Noor, On integral operators, J. Math. Anal. Appl. 238 (1999), 341- 352.
S. Owa and H. M. Srivastava, univalent and starlike generalized hypergeometric functions,
canad. J. Math. 39 (1987), 1057-1077.
INCLUSION PROPERTIES OF CERTAIN CLASSES
121
[17] H. M. Srivastava and S. Owa, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators,
and certain subclasses of analytic functions, Nagoya Math. J. 106 (1987), 1-28.
Ali Muhammad, Departement of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan
E-mail address : ali7887@gmail.com