Paper-10-Acta23.2010. 114KB Jun 04 2011 12:03:09 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 23/2010
pp. 107-115
SOME INCLUSION PROPERTIES FOR NEW SUBCLASSES OF
MEROMORPHIC P-VALENT STRONGLY STARLIKE AND
STRONGLY CONVEX FUNCTIONS ASSOCIATED WITH THE
EL-ASHWAH OPERATOR
S.P. Goyal and Rakesh Kumar
Abstract. The purpose of this paper is to derive some useful properties for
new subclasses of strongly starlike and strongly convex functions of order γ and type
(µ1 , µ2 ) in the open unit disk U using a multiplier transformation for meromorphic
p-valent functions introduced recently by R.M. El-Ashwah. Inclusion relationships
using these subclasses are established.
2010 Mathematics Subject Classification: Primery: 30C45; Secondary: 30C55.
Keywords and phrases: Analytic functions, Meromorphic p-valent functions,
Multiplier transformations, Strongly starlike functions, Strongly convex functions.
1. Introduction and definitions
Let Σp denote the class of functions of the form
∞
f (z) =
X
1
+
an z n
zp
(p ∈ N = {1, 2, ...}),
(1.1)
n=0
which are analytic in the punctured unit disk U∗ = {z : z ∈ C; 0 < |z| < 1} = U\{0},
where U = {z : z ∈ C; |z| < 1} is the open unit disk.
The classes of strongly starlike functions and strongly convex functions in the
open unit disk have earlier been introduced and studied by Takahashi and Nunokawa
[9], Shanmugam et al. [7] and others. A function f ∈ Σp is said to belong to the
107
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
class of meromorphically strongly starlike functions of order γ and type (µ1 , µ2 ) in
U, denoted by S∗p (µ1 , µ2 , γ), if it satisfies
′
π
π
z f (z)
− µ1 < arg
+ γ < µ2
(1.2)
2
f (z)
2
(z ∈ U, 0 < µ1 ≤ 1, 0 < µ2 ≤ 1, γ > p).
A function f ∈ Σp is said to belong to the class of meromorphically strongly
convex functions of order γ and type (µ1 , µ2 ) in U, denoted by Cp (µ1 , µ2 , γ), if it
satisfies
π
(z f ′ (z))′
π
+
γ
< µ2
(1.3)
− µ1 < arg
2
f ′ (z)
2
(z ∈ U, 0 < µ1 ≤ 1, 0 < µ2 ≤ 1, γ > p).
Recently El-Ashwah [4] defined the operator
∞
X
1
ℓ + λ(n + p) m
m
Ip (λ, ℓ)f (z) = p +
an z n
z
ℓ
(1.4)
n=0
(λ ≥ 0; ℓ > 0; m ∈ N0 = N ∪ {0}; z ∈ U∗ ).
It is easily verified from (1.4) that
λz(Ipm (λ, ℓ)f (z))′ = ℓIpm+1 (λ, ℓ)f (z) − (ℓ + λp) Ipm (λ, ℓ)f (z)
We note that
Ip0 (λ, ℓ)f (z) = f (z)
Ip1 (1, 1)f (z) =
(λ > 0).
(1.5)
and
(z p+1 f (z))′
= (p + 1)f (z) + z f ′ (z).
zp
Also by specializing the parameters λ, l and p, we obtain the following operators
studied earlier by various authors:
(i) I1m (1, ℓ)f (z) = I(m, ℓ)f (z) (see Cho et al. [2,3]);
(ii) Ipm (1, 1)f (z) = Dpm f (z) (see Aouf and Hossen [1], Liu and Owa [5] and Srivastava and Patel [8]);
(iii) I1m (1, 1)f (z) = I m f (z) (see Uralegaddi and Somanatha [10]).
Also we note that:
(i) Ipm (1, ℓ)f (z) = Ip (m, ℓ)f (z), where Ip (m, ℓ)f (z) is defined by
∞
X
1
Ip (m, ℓ)f (z) = p +
z
n=0
ℓ+n+p
ℓ
m
an z n (ℓ > 0; m ∈ N0 ; z ∈ U∗ );
108
(1.6)
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
m f (z), where D m f (z) is defined by
(ii) Ipm (λ, 1)f (z) = Dλ,p
λ,p
∞
m
Dλ,p
f (z) =
X
1
+
[λ(n + p) + 1]m an z n (λ ≥ 0; m ∈ N0 ; z ∈ U∗ ).
p
z
(1.7)
n=0
Now, we introduce the following new subclasses of meromorphically strongly
starlike and meromorphically strongly convex functions of order γ and type (µ1 , µ2 )
in the open unit disk U:
z (Ipm (λ, ℓ)f (z))′
∗
m
m
6= γ, z ∈ U
Rp (λ, ℓ; µ1 , µ2 , γ) = f ∈ Σp : Ip (λ, ℓ)f ∈ Sp (µ1 , µ2 , γ), − m
Ip (λ, ℓ)f (z)
(1.8)
and
[z((Ipm (λ, ℓ)f )′ (z))]′
m
m
Mp (λ, ℓ; µ1 , µ2 , γ) = f ∈ Σp : Ip (λ, ℓ)f ∈ Cp (µ1 , µ2 , γ), −
6= γ, z ∈ U .
[Ipm (λ, ℓ)f (z)]′
(1.9)
m
The object of this paper is to derive some properties for the classes Rp (λ, ℓ; µ1 , µ2 , γ)
and Mpm (λ, ℓ; µ1 , µ2 , γ).
2. Main results
In order to prove our results, we need the following result of Nunokawa et al. [6]:
Lemma 2.1. Let q(z) be analytic in U with q(0) = 1 and q(z) 6= 0. If there exists
two points z1 , z2 ∈ U such that
π
π
− µ1 = arg q(z1 ) < arg q(z) < arg q(z2 ) = µ2 ,
2
2
(2.1)
for µ1 > 0, µ2 > 0, and for |z| < |z1 | = |z2 |, then we have
z1 q ′ (z1 )
µ1 + µ2
k,
= −i
q(z1 )
2
and
where k ≥ 1−|a|
1+|a|
Theorem 2.2.
z2 q ′ (z2 )
µ1 + µ2
=i
k,
q(z2 )
2
n
o
−µ1
and a = i tan π4 µµ22 +µ
.
1
109
(2.2)
(2.3)
(2.4)
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
Rpm+1 (λ, ℓ; µ1 , µ2 , γ) ⊂ Rpm (λ, ℓ; µ1 , µ2 , γ), for each m ∈ N0 .
Proof. Let f ∈ Rpm+1 (λ, ℓ; µ1 , µ2 , γ). Further suppose that
z (Ipm (λ, ℓ)f (z))′
= (γ − p)q(z) − γ ,
Ipm (λ, ℓ)f (z)
(2.5)
where q(z) = 1 + c1 z + c2 z 2 + ... is analytic in U, q(0) = 1, and q(z) 6= 0 ∀ z ∈ U.
Using (1.5), we get
ℓIpm+1 (λ, ℓ)f (z)
= λ(γ − p)q(z) + [ℓ − λ(γ − p)] .
Ipm (λ, ℓ)f (z)
(2.6)
By logarithmic differentiation, we easily get
z (Ipm+1 (λ, ℓ)f (z))′
Ipm+1 (λ, ℓ)f (z)
z (Ipm (λ, ℓ)f (z))′
λ(γ − p)z q ′ (z)
+γ =
+γ
+
Ipm (λ, ℓ)f (z)
λ(γ − p)q(z) + [ℓ − λ(γ − p)]
= (γ − p)q(z) +
λ(γ − p)z q ′ (z)
.
λ(γ − p)q(z) + [ℓ − λ(γ − p)]
(2.7)
Suppose that there exist two points z1 , z2 ∈ U such that,
− π2 µ1 = arg q(z1 ) < arg q(z) < arg q(z2 ) = π2 µ2 for |z| < |z1 | = |z2 | .
Then from the proof of the Nunokawa lemma [6], we have
and
where
and
ik(µ1 + µ2 )(1 + t21 )
z1 q ′ (z1 )
=−
q(z1 )
4t1
(2.8)
z2 q ′ (z2 )
ik(µ1 + µ2 )(1 + t22 )
=
,
q(z2 )
4t2
(2.9)
n π
o
q(z1 ) = (−it1 )(µ1 +µ2 )/2 exp i (µ2 − µ1 ) ,
4
o
n π
q(z2 ) = (it2 )(µ1 +µ2 )/2 exp i (µ2 − µ1 ) ,
4
k≥
t1 > 0
t2 > 0
1 − |a|
.
1 + |a|
Replacing z by z2 in (2.7) and using (2.9) and (2.11) therein, we find that
z2 (Ipm+1 (λ, ℓ)f (z2 ))′
Ipm+1 (λ, ℓ)f (z2 )
+γ
110
(2.10)
(2.11)
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
λz2 q ′ (z2 )/q(z2 )
λ(γ − p)q(z2 ) + [ℓ − (γ − p)]
#
"
π
2 )k
λi(µ
+
µ
)
(1
+
t
(µ +µ )/2
1
2
2
.
1+
= (γ−p) t2 1 2 exp i µ2
(µ1 +µ2 )/2
π
2
4t2 [λ(γ − p)t
exp i µ2 + {ℓ − λ(γ − p)}]
= (γ − p)q(z2 ) 1 +
2
2
This implies that
(
)
z2 (Ipm+1 (λ, ℓ)f (z2 ))′
arg
+γ
Ipm+1 (λ, ℓ)f (z2 )
)
(
+
t
)k
λi(µ1 + µ2 ) (t−1
π
2
2
= µ2 + arg 1 +
(µ +µ )/2
2
4[λ(γ − p)t2 1 2 exp i π2 µ2 + {ℓ − λ(γ − p)}]
(
)
(µ1 +µ2 )/2
k(µ1 + µ2 )(t−1
cos π2 µ2
π
2 + t2 )[ℓ − λ(γ − p)] + 4λ(γ − p) t2
−1
= µ2 +tan
2
4 ε(µ1 , µ2 , t2 )
≥ π2 µ2 ,
(2.12)
where
(µ +µ )/2
ε(µ1 , µ2 , t2 ) = [ℓ − λ(γ − p)]2 + 2λ [ℓ − λ(γ − p)] (γ − p) t2 1 2 cos π2 µ2
(µ1 +µ2 )/2
(µ +µ )
2
(t−1
sin π2 µ2 ,
+λ2 (γ −p)2 t2 1 2 +kλ2 µ1 +µ
2 +t2 )(γ −p)t2
4
and
1 − |a|
k≥
.
1 + |a|
Similarly, replacing z = z1 in (2.8) and using the same procedure as above, we obtain
that
)
(
z1 (Ipm+1 (λ, ℓ)f (z1 ))′
π
(2.13)
arg
+ γ ≤ − µ1 .
m+1
2
Ip (λ, ℓ)f (z1 )
Thus we get the contradiction to the hypothesis that f ∈ Rpm+1 (λ, ℓ; µ1 , µ2 , γ).
Hence the function q(z) defined by (2.5) yields
π
π
− µ1 < arg q(z) < µ2 ,
2
2
which implies that
π
− µ1 < arg
2
z(Ipm (λ, ℓ)f (z))′
+γ
Ipm (λ, ℓ)f (z)
<
π
µ2 .
2
Thus f ∈ Rpm (λ, ℓ; µ1 , µ2 , γ), which completes the proof of the Theorem 2.2.
On taking λ = 1 in Theorem 2.2 and using (1.6), we get the following result:
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S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
Corollary 2.3.
Rpm+1 (ℓ; µ1 , µ2 , γ) ⊂ Rpm (ℓ; µ1 , µ2 , γ), for each m ∈ N0
where
z(Ip (m, ℓ)f (z))′
∗
m
6= γ, z ∈ U ,
Rp (ℓ; µ1 , µ2 , γ) = f ∈ Σp : Ip (m, ℓ)f ∈ Sp (µ1 , µ2 , γ), −
Ip (m, ℓ)f (z)
(2.14)
and Ip (m, ℓ)f (z) is given by (1.6).
If we put ℓ = 1 in Theorem 2.2 and use (1.7) therein, we get
Corollary 2.4.
Rpm+1 (λ; µ1 , µ2 , γ) ⊂ Rpm (λ; µ1 , µ2 , γ), for each m ∈ N0
where
)
(
m f (z))′
z(D
λ,p
m
6= γ, z ∈ U ,
Rpm (λ; µ1 , µ2 , γ) = f ∈ Σp : Dλ,p
f ∈ S∗p (µ1 , µ2 , γ), −
m f (z)
Dλ,p
(2.15)
m f (z) is given by (1.7).
and Dλ,p
Theorem 2.5.
Mpm+1 (λ, ℓ; µ1 , µ2 , γ) ⊂ Mpm (λ, ℓ; µ1 , µ2 , γ), for each m ∈ N0 .
Proof.
f ∈ Mpm+1 (λ, ℓ; µ1 , µ2 , γ) ⇐⇒ Ipm+1 (λ, ℓ )f (z) ∈ Cp (µ1 , µ2 , γ)
⇐⇒ − p1 z (Ipm+1 (λ, ℓ )f (z))′ ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ Ipm+1 (λ, ℓ ) − p1 z f ′ (z) ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ − p1 z f ′ (z) ∈ Rpm+1 (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ − p1 z f ′ (z) ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ Ipm (λ, ℓ) − p1 z f ′ (z) ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ − p1 z (Ipm (λ, ℓ)f (z))′ ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ Ipm (λ, ℓ)f (z) ∈ Cp (µ1 , µ2 , γ)
⇐⇒ f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ).
For λ = 1 and ℓ = 1 in Theorem 2.5, we get results similar to Corollary 2.3 and
2.4 respectively.
It is obvious from Theorem 2.5 that Alexander’s type relationship holds between
the classes Mpm (λ; µ1 , µ2 , γ) and Rpm (λ; µ1 , µ1 , γ), which we state formally as:
Theorem 2.6.
f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ) ⇐⇒ − p1 z f ′ ∈ Rpm (λ, ℓ; µ1 , µ2 , γ).
112
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
3. Integral operators
Let f ∈ Σp . Then for µ > 0, we consider the integral operator
Fµ,p (f )(z) : Σp → Σp defined by
Fµ,p (f )(z) =
µ
z µ+p
Zz
tµ+p−1 f (t) dt
0
"
∞
X
1
= p+
z
n=0
µ
n+µ+p
z
n
#
∗ f (z).
(3.1)
Theorem 3.1. If f ∈ Rpm (λ, ℓ; µ1 , µ2 , γ), and for µ > 0,
−
z [Ipm (λ, ℓ)(Fµ,p (f )(z))]′
6= γ ∀z ∈ U,
[Ipm (λ, ℓ)(Fµ,p (f )(z))]
then Fµ,p (f )(z) ∈ Rpm (λ, ℓ; µ1 , µ2 , γ), where Fµ,p (f )(z) is given by (3.1).
Proof. Let
z [Ipm (λ, ℓ)(Fµ,p (f )(z))]′
= (γ − p) q(z) − γ,
[Ipm (λ, ℓ)(Fµ,p (f )(z))]
(3.2)
where q(z) is analytic in U, q(0) = 1, and q(z) 6= 0 (z ∈ U).
Using (3.1), we have
z [Ipm (λ, ℓ)Fµ,p (f )(z))′ = µ Ipm (λ, ℓ)f (z) − (µ + p)Ipm (λ, ℓ)Fµ,p (f )(z).
(3.3)
By (3.2) and (3.3), we get
µ
Ipm (λ, ℓ)f (z)
= (γ − p)q(z) + [µ − (γ − p)].
Ipm (λ, ℓ)Fµ,p (f )(z)
(3.4)
Differentiating (3.4) logarithmically, multiplying by z and using (3.2), it follows that
z (Ipm (λ, ℓ)f (z))′
(γ − p)z q ′ (z)
.
+
γ
=
(γ
−
p)
q(z)
+
Ipm (λ, ℓ)f (z)
(γ − p) q(z) + [µ − (γ − p)]
(3.5)
The remaining part of the proof is similar to that of Theorem 2.2 and so is omitted.
113
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
Theorem 3.2. If f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ), and for µ > 0,
[z [Ipm (λ, ℓ)Fµ,p (f )(z)]′ ]′
6= γ ∀z ∈ U,
[Ipm (λ, ℓ)Fµ,p (f )(z)]′
−
then Fµ,p (f )(z) ∈ Mpm (λ, ℓ; µ1 , µ2 , γ), where Fµ,p (f )(z) is given by (3.1).
Proof.
f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ) ⇐⇒ − p1 z f ′ ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ Fµ,p − p1 z f ′ (z) ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ − p1 z (Fµ,p (f )(z))′ ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ Fµ,p (f )(z) ∈ Mpm (λ, ℓ; µ1 , µ2 , γ).
Letting λ = 1 in Theorem 3.1, we get the following result:
Corollary 3.3. If f ∈ Rpm (ℓ; µ1 , µ2 , γ), and for µ > 0,
−
z [Ip (m, ℓ)(Fµ,p (f )(z))]′
6= γ ∀z ∈ U,
[Ip (m, ℓ)(Fµ,p (f )(z))]
then Fµ,p (f )(z) ∈ Rpm (ℓ; µ1 , µ2 , γ), where Ip (m, ℓ)f (z), Rpm (ℓ; µ1 , µ2 , γ) and Fµ,p (f )(z)
are given by (1.6), (2.14) and (3.1) respectively.
If we set ℓ = 1 in Theorem 3.1, we can easily get the following result:
Corollary 3.4. If f ∈ Rpm (λ; µ1 , µ2 , γ), and for µ > 0,
−
m (F
′
z [Dλ,p
µ,p (f )(z))]
m (F
[Dλ,p
µ,p (f )(z))]
6= γ ∀z ∈ U,
m f (z), Rm (λ; µ , µ , γ) and F
then Fµ,p (f )(z) ∈ Rpm (λ; µ1 , µ2 , γ), where Dλ,p
1 2
µ,p (f )(z)
p
are given by (1.7), (2.15) and (3.1) respectively.
For λ = 1 and ℓ = 1 in Theorem 3.2, we get results similar to Corollary 3.3 and
3.4 respectively.
Acknowledgement: The authors are thankful to worthy referee for his useful suggestions. The first author (S. P. G.) is thankful to CSIR, New Delhi, India for awarding Emeritus Scientistship, under scheme number 21(084)/10/EMRII. The second author is also thankful to CSIR, India, for providing JRF No.
09/149(0498)/2008-EMR-I.
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S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
References
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functions, Tsukuba J. Math., 17 (1993), 481-486.
[2] N.E. Cho, O.S. Kwon and H.M. Srivastava, Inclusion and argument properties
for certain subclasses of meromorphic functions associated with a family of multiplier
transformations, J. Math. Anal. Appl., 300 (2004), 505-520.
[3] N.E. Cho, O.S. Kwon and H.M. Srivastava, Inclusion relationships for certain
subclasses of meromorphic functions associated with a family of multiplier transformations , Integral Transforms Spec. Funct., 16 (18) (2005), 647-659.
[4] R.M. El-Ashwah, A note on certain meromorphic p-valent functions, Appl.
Math. Lett., 22 (2009), 1756-1759.
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[6] M. Nunokawa, S. Owa, H. Saitoh, N.E. Cho and N. Takahashi, Some properties of analytic functions of extremal points for arguments, preprint.
[7] T.N. Shanmugam, S. Sivasubramanian and B.A. Frasin, Argument estimates
of strongly starlike functions associated with Noor-integral operator, Tamkang Journal of Mathematics, 40 (1) (2009), 7-13.
[8] H.M. Srivastava and J. Patel, Applications of differential subordination to
certain classes of meromorphically multivalent functions, J. Inequal. Pure. Appl.
Math., 6 (3) (2005), 1-5.
[9] N. Takahashi and M. Nunokawa, A certain connection between starlike and
convex functions, Appl. Math. Lett., 16 (2003), 653-655.
[10] B.A. Uralegaddi and C. Somanatha, New criteria for meromorphic starlike
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S.P. Goyal
Department of Mathematics
University of Rajasthan
Jaipur (INDIA)- 302055
email:[email protected]
Rakesh Kumar
Department of Mathematics
University Rajasthan
Jaipur (INDIA) - 302055
email:[email protected]
115
ISSN: 1582-5329
No. 23/2010
pp. 107-115
SOME INCLUSION PROPERTIES FOR NEW SUBCLASSES OF
MEROMORPHIC P-VALENT STRONGLY STARLIKE AND
STRONGLY CONVEX FUNCTIONS ASSOCIATED WITH THE
EL-ASHWAH OPERATOR
S.P. Goyal and Rakesh Kumar
Abstract. The purpose of this paper is to derive some useful properties for
new subclasses of strongly starlike and strongly convex functions of order γ and type
(µ1 , µ2 ) in the open unit disk U using a multiplier transformation for meromorphic
p-valent functions introduced recently by R.M. El-Ashwah. Inclusion relationships
using these subclasses are established.
2010 Mathematics Subject Classification: Primery: 30C45; Secondary: 30C55.
Keywords and phrases: Analytic functions, Meromorphic p-valent functions,
Multiplier transformations, Strongly starlike functions, Strongly convex functions.
1. Introduction and definitions
Let Σp denote the class of functions of the form
∞
f (z) =
X
1
+
an z n
zp
(p ∈ N = {1, 2, ...}),
(1.1)
n=0
which are analytic in the punctured unit disk U∗ = {z : z ∈ C; 0 < |z| < 1} = U\{0},
where U = {z : z ∈ C; |z| < 1} is the open unit disk.
The classes of strongly starlike functions and strongly convex functions in the
open unit disk have earlier been introduced and studied by Takahashi and Nunokawa
[9], Shanmugam et al. [7] and others. A function f ∈ Σp is said to belong to the
107
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
class of meromorphically strongly starlike functions of order γ and type (µ1 , µ2 ) in
U, denoted by S∗p (µ1 , µ2 , γ), if it satisfies
′
π
π
z f (z)
− µ1 < arg
+ γ < µ2
(1.2)
2
f (z)
2
(z ∈ U, 0 < µ1 ≤ 1, 0 < µ2 ≤ 1, γ > p).
A function f ∈ Σp is said to belong to the class of meromorphically strongly
convex functions of order γ and type (µ1 , µ2 ) in U, denoted by Cp (µ1 , µ2 , γ), if it
satisfies
π
(z f ′ (z))′
π
+
γ
< µ2
(1.3)
− µ1 < arg
2
f ′ (z)
2
(z ∈ U, 0 < µ1 ≤ 1, 0 < µ2 ≤ 1, γ > p).
Recently El-Ashwah [4] defined the operator
∞
X
1
ℓ + λ(n + p) m
m
Ip (λ, ℓ)f (z) = p +
an z n
z
ℓ
(1.4)
n=0
(λ ≥ 0; ℓ > 0; m ∈ N0 = N ∪ {0}; z ∈ U∗ ).
It is easily verified from (1.4) that
λz(Ipm (λ, ℓ)f (z))′ = ℓIpm+1 (λ, ℓ)f (z) − (ℓ + λp) Ipm (λ, ℓ)f (z)
We note that
Ip0 (λ, ℓ)f (z) = f (z)
Ip1 (1, 1)f (z) =
(λ > 0).
(1.5)
and
(z p+1 f (z))′
= (p + 1)f (z) + z f ′ (z).
zp
Also by specializing the parameters λ, l and p, we obtain the following operators
studied earlier by various authors:
(i) I1m (1, ℓ)f (z) = I(m, ℓ)f (z) (see Cho et al. [2,3]);
(ii) Ipm (1, 1)f (z) = Dpm f (z) (see Aouf and Hossen [1], Liu and Owa [5] and Srivastava and Patel [8]);
(iii) I1m (1, 1)f (z) = I m f (z) (see Uralegaddi and Somanatha [10]).
Also we note that:
(i) Ipm (1, ℓ)f (z) = Ip (m, ℓ)f (z), where Ip (m, ℓ)f (z) is defined by
∞
X
1
Ip (m, ℓ)f (z) = p +
z
n=0
ℓ+n+p
ℓ
m
an z n (ℓ > 0; m ∈ N0 ; z ∈ U∗ );
108
(1.6)
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
m f (z), where D m f (z) is defined by
(ii) Ipm (λ, 1)f (z) = Dλ,p
λ,p
∞
m
Dλ,p
f (z) =
X
1
+
[λ(n + p) + 1]m an z n (λ ≥ 0; m ∈ N0 ; z ∈ U∗ ).
p
z
(1.7)
n=0
Now, we introduce the following new subclasses of meromorphically strongly
starlike and meromorphically strongly convex functions of order γ and type (µ1 , µ2 )
in the open unit disk U:
z (Ipm (λ, ℓ)f (z))′
∗
m
m
6= γ, z ∈ U
Rp (λ, ℓ; µ1 , µ2 , γ) = f ∈ Σp : Ip (λ, ℓ)f ∈ Sp (µ1 , µ2 , γ), − m
Ip (λ, ℓ)f (z)
(1.8)
and
[z((Ipm (λ, ℓ)f )′ (z))]′
m
m
Mp (λ, ℓ; µ1 , µ2 , γ) = f ∈ Σp : Ip (λ, ℓ)f ∈ Cp (µ1 , µ2 , γ), −
6= γ, z ∈ U .
[Ipm (λ, ℓ)f (z)]′
(1.9)
m
The object of this paper is to derive some properties for the classes Rp (λ, ℓ; µ1 , µ2 , γ)
and Mpm (λ, ℓ; µ1 , µ2 , γ).
2. Main results
In order to prove our results, we need the following result of Nunokawa et al. [6]:
Lemma 2.1. Let q(z) be analytic in U with q(0) = 1 and q(z) 6= 0. If there exists
two points z1 , z2 ∈ U such that
π
π
− µ1 = arg q(z1 ) < arg q(z) < arg q(z2 ) = µ2 ,
2
2
(2.1)
for µ1 > 0, µ2 > 0, and for |z| < |z1 | = |z2 |, then we have
z1 q ′ (z1 )
µ1 + µ2
k,
= −i
q(z1 )
2
and
where k ≥ 1−|a|
1+|a|
Theorem 2.2.
z2 q ′ (z2 )
µ1 + µ2
=i
k,
q(z2 )
2
n
o
−µ1
and a = i tan π4 µµ22 +µ
.
1
109
(2.2)
(2.3)
(2.4)
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
Rpm+1 (λ, ℓ; µ1 , µ2 , γ) ⊂ Rpm (λ, ℓ; µ1 , µ2 , γ), for each m ∈ N0 .
Proof. Let f ∈ Rpm+1 (λ, ℓ; µ1 , µ2 , γ). Further suppose that
z (Ipm (λ, ℓ)f (z))′
= (γ − p)q(z) − γ ,
Ipm (λ, ℓ)f (z)
(2.5)
where q(z) = 1 + c1 z + c2 z 2 + ... is analytic in U, q(0) = 1, and q(z) 6= 0 ∀ z ∈ U.
Using (1.5), we get
ℓIpm+1 (λ, ℓ)f (z)
= λ(γ − p)q(z) + [ℓ − λ(γ − p)] .
Ipm (λ, ℓ)f (z)
(2.6)
By logarithmic differentiation, we easily get
z (Ipm+1 (λ, ℓ)f (z))′
Ipm+1 (λ, ℓ)f (z)
z (Ipm (λ, ℓ)f (z))′
λ(γ − p)z q ′ (z)
+γ =
+γ
+
Ipm (λ, ℓ)f (z)
λ(γ − p)q(z) + [ℓ − λ(γ − p)]
= (γ − p)q(z) +
λ(γ − p)z q ′ (z)
.
λ(γ − p)q(z) + [ℓ − λ(γ − p)]
(2.7)
Suppose that there exist two points z1 , z2 ∈ U such that,
− π2 µ1 = arg q(z1 ) < arg q(z) < arg q(z2 ) = π2 µ2 for |z| < |z1 | = |z2 | .
Then from the proof of the Nunokawa lemma [6], we have
and
where
and
ik(µ1 + µ2 )(1 + t21 )
z1 q ′ (z1 )
=−
q(z1 )
4t1
(2.8)
z2 q ′ (z2 )
ik(µ1 + µ2 )(1 + t22 )
=
,
q(z2 )
4t2
(2.9)
n π
o
q(z1 ) = (−it1 )(µ1 +µ2 )/2 exp i (µ2 − µ1 ) ,
4
o
n π
q(z2 ) = (it2 )(µ1 +µ2 )/2 exp i (µ2 − µ1 ) ,
4
k≥
t1 > 0
t2 > 0
1 − |a|
.
1 + |a|
Replacing z by z2 in (2.7) and using (2.9) and (2.11) therein, we find that
z2 (Ipm+1 (λ, ℓ)f (z2 ))′
Ipm+1 (λ, ℓ)f (z2 )
+γ
110
(2.10)
(2.11)
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
λz2 q ′ (z2 )/q(z2 )
λ(γ − p)q(z2 ) + [ℓ − (γ − p)]
#
"
π
2 )k
λi(µ
+
µ
)
(1
+
t
(µ +µ )/2
1
2
2
.
1+
= (γ−p) t2 1 2 exp i µ2
(µ1 +µ2 )/2
π
2
4t2 [λ(γ − p)t
exp i µ2 + {ℓ − λ(γ − p)}]
= (γ − p)q(z2 ) 1 +
2
2
This implies that
(
)
z2 (Ipm+1 (λ, ℓ)f (z2 ))′
arg
+γ
Ipm+1 (λ, ℓ)f (z2 )
)
(
+
t
)k
λi(µ1 + µ2 ) (t−1
π
2
2
= µ2 + arg 1 +
(µ +µ )/2
2
4[λ(γ − p)t2 1 2 exp i π2 µ2 + {ℓ − λ(γ − p)}]
(
)
(µ1 +µ2 )/2
k(µ1 + µ2 )(t−1
cos π2 µ2
π
2 + t2 )[ℓ − λ(γ − p)] + 4λ(γ − p) t2
−1
= µ2 +tan
2
4 ε(µ1 , µ2 , t2 )
≥ π2 µ2 ,
(2.12)
where
(µ +µ )/2
ε(µ1 , µ2 , t2 ) = [ℓ − λ(γ − p)]2 + 2λ [ℓ − λ(γ − p)] (γ − p) t2 1 2 cos π2 µ2
(µ1 +µ2 )/2
(µ +µ )
2
(t−1
sin π2 µ2 ,
+λ2 (γ −p)2 t2 1 2 +kλ2 µ1 +µ
2 +t2 )(γ −p)t2
4
and
1 − |a|
k≥
.
1 + |a|
Similarly, replacing z = z1 in (2.8) and using the same procedure as above, we obtain
that
)
(
z1 (Ipm+1 (λ, ℓ)f (z1 ))′
π
(2.13)
arg
+ γ ≤ − µ1 .
m+1
2
Ip (λ, ℓ)f (z1 )
Thus we get the contradiction to the hypothesis that f ∈ Rpm+1 (λ, ℓ; µ1 , µ2 , γ).
Hence the function q(z) defined by (2.5) yields
π
π
− µ1 < arg q(z) < µ2 ,
2
2
which implies that
π
− µ1 < arg
2
z(Ipm (λ, ℓ)f (z))′
+γ
Ipm (λ, ℓ)f (z)
<
π
µ2 .
2
Thus f ∈ Rpm (λ, ℓ; µ1 , µ2 , γ), which completes the proof of the Theorem 2.2.
On taking λ = 1 in Theorem 2.2 and using (1.6), we get the following result:
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S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
Corollary 2.3.
Rpm+1 (ℓ; µ1 , µ2 , γ) ⊂ Rpm (ℓ; µ1 , µ2 , γ), for each m ∈ N0
where
z(Ip (m, ℓ)f (z))′
∗
m
6= γ, z ∈ U ,
Rp (ℓ; µ1 , µ2 , γ) = f ∈ Σp : Ip (m, ℓ)f ∈ Sp (µ1 , µ2 , γ), −
Ip (m, ℓ)f (z)
(2.14)
and Ip (m, ℓ)f (z) is given by (1.6).
If we put ℓ = 1 in Theorem 2.2 and use (1.7) therein, we get
Corollary 2.4.
Rpm+1 (λ; µ1 , µ2 , γ) ⊂ Rpm (λ; µ1 , µ2 , γ), for each m ∈ N0
where
)
(
m f (z))′
z(D
λ,p
m
6= γ, z ∈ U ,
Rpm (λ; µ1 , µ2 , γ) = f ∈ Σp : Dλ,p
f ∈ S∗p (µ1 , µ2 , γ), −
m f (z)
Dλ,p
(2.15)
m f (z) is given by (1.7).
and Dλ,p
Theorem 2.5.
Mpm+1 (λ, ℓ; µ1 , µ2 , γ) ⊂ Mpm (λ, ℓ; µ1 , µ2 , γ), for each m ∈ N0 .
Proof.
f ∈ Mpm+1 (λ, ℓ; µ1 , µ2 , γ) ⇐⇒ Ipm+1 (λ, ℓ )f (z) ∈ Cp (µ1 , µ2 , γ)
⇐⇒ − p1 z (Ipm+1 (λ, ℓ )f (z))′ ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ Ipm+1 (λ, ℓ ) − p1 z f ′ (z) ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ − p1 z f ′ (z) ∈ Rpm+1 (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ − p1 z f ′ (z) ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ Ipm (λ, ℓ) − p1 z f ′ (z) ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ − p1 z (Ipm (λ, ℓ)f (z))′ ∈ S∗p (µ1 , µ2 , γ)
⇐⇒ Ipm (λ, ℓ)f (z) ∈ Cp (µ1 , µ2 , γ)
⇐⇒ f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ).
For λ = 1 and ℓ = 1 in Theorem 2.5, we get results similar to Corollary 2.3 and
2.4 respectively.
It is obvious from Theorem 2.5 that Alexander’s type relationship holds between
the classes Mpm (λ; µ1 , µ2 , γ) and Rpm (λ; µ1 , µ1 , γ), which we state formally as:
Theorem 2.6.
f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ) ⇐⇒ − p1 z f ′ ∈ Rpm (λ, ℓ; µ1 , µ2 , γ).
112
S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
3. Integral operators
Let f ∈ Σp . Then for µ > 0, we consider the integral operator
Fµ,p (f )(z) : Σp → Σp defined by
Fµ,p (f )(z) =
µ
z µ+p
Zz
tµ+p−1 f (t) dt
0
"
∞
X
1
= p+
z
n=0
µ
n+µ+p
z
n
#
∗ f (z).
(3.1)
Theorem 3.1. If f ∈ Rpm (λ, ℓ; µ1 , µ2 , γ), and for µ > 0,
−
z [Ipm (λ, ℓ)(Fµ,p (f )(z))]′
6= γ ∀z ∈ U,
[Ipm (λ, ℓ)(Fµ,p (f )(z))]
then Fµ,p (f )(z) ∈ Rpm (λ, ℓ; µ1 , µ2 , γ), where Fµ,p (f )(z) is given by (3.1).
Proof. Let
z [Ipm (λ, ℓ)(Fµ,p (f )(z))]′
= (γ − p) q(z) − γ,
[Ipm (λ, ℓ)(Fµ,p (f )(z))]
(3.2)
where q(z) is analytic in U, q(0) = 1, and q(z) 6= 0 (z ∈ U).
Using (3.1), we have
z [Ipm (λ, ℓ)Fµ,p (f )(z))′ = µ Ipm (λ, ℓ)f (z) − (µ + p)Ipm (λ, ℓ)Fµ,p (f )(z).
(3.3)
By (3.2) and (3.3), we get
µ
Ipm (λ, ℓ)f (z)
= (γ − p)q(z) + [µ − (γ − p)].
Ipm (λ, ℓ)Fµ,p (f )(z)
(3.4)
Differentiating (3.4) logarithmically, multiplying by z and using (3.2), it follows that
z (Ipm (λ, ℓ)f (z))′
(γ − p)z q ′ (z)
.
+
γ
=
(γ
−
p)
q(z)
+
Ipm (λ, ℓ)f (z)
(γ − p) q(z) + [µ − (γ − p)]
(3.5)
The remaining part of the proof is similar to that of Theorem 2.2 and so is omitted.
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S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
Theorem 3.2. If f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ), and for µ > 0,
[z [Ipm (λ, ℓ)Fµ,p (f )(z)]′ ]′
6= γ ∀z ∈ U,
[Ipm (λ, ℓ)Fµ,p (f )(z)]′
−
then Fµ,p (f )(z) ∈ Mpm (λ, ℓ; µ1 , µ2 , γ), where Fµ,p (f )(z) is given by (3.1).
Proof.
f ∈ Mpm (λ, ℓ; µ1 , µ2 , γ) ⇐⇒ − p1 z f ′ ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ Fµ,p − p1 z f ′ (z) ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ − p1 z (Fµ,p (f )(z))′ ∈ Rpm (λ, ℓ; µ1 , µ2 , γ)
⇐⇒ Fµ,p (f )(z) ∈ Mpm (λ, ℓ; µ1 , µ2 , γ).
Letting λ = 1 in Theorem 3.1, we get the following result:
Corollary 3.3. If f ∈ Rpm (ℓ; µ1 , µ2 , γ), and for µ > 0,
−
z [Ip (m, ℓ)(Fµ,p (f )(z))]′
6= γ ∀z ∈ U,
[Ip (m, ℓ)(Fµ,p (f )(z))]
then Fµ,p (f )(z) ∈ Rpm (ℓ; µ1 , µ2 , γ), where Ip (m, ℓ)f (z), Rpm (ℓ; µ1 , µ2 , γ) and Fµ,p (f )(z)
are given by (1.6), (2.14) and (3.1) respectively.
If we set ℓ = 1 in Theorem 3.1, we can easily get the following result:
Corollary 3.4. If f ∈ Rpm (λ; µ1 , µ2 , γ), and for µ > 0,
−
m (F
′
z [Dλ,p
µ,p (f )(z))]
m (F
[Dλ,p
µ,p (f )(z))]
6= γ ∀z ∈ U,
m f (z), Rm (λ; µ , µ , γ) and F
then Fµ,p (f )(z) ∈ Rpm (λ; µ1 , µ2 , γ), where Dλ,p
1 2
µ,p (f )(z)
p
are given by (1.7), (2.15) and (3.1) respectively.
For λ = 1 and ℓ = 1 in Theorem 3.2, we get results similar to Corollary 3.3 and
3.4 respectively.
Acknowledgement: The authors are thankful to worthy referee for his useful suggestions. The first author (S. P. G.) is thankful to CSIR, New Delhi, India for awarding Emeritus Scientistship, under scheme number 21(084)/10/EMRII. The second author is also thankful to CSIR, India, for providing JRF No.
09/149(0498)/2008-EMR-I.
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S.P. Goyal, R. Kumar - Some Inclusion Properties for New Subclasses of...
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S.P. Goyal
Department of Mathematics
University of Rajasthan
Jaipur (INDIA)- 302055
email:[email protected]
Rakesh Kumar
Department of Mathematics
University Rajasthan
Jaipur (INDIA) - 302055
email:[email protected]
115