General Mathematics Vol. 14, No. 4 (2006), 113–126
Meromorphic Starlike Functions with
Alternating and Missing Coefficients 1
M. Darus, S. B. Joshi
and
N. D. Sangle
In memoriam of Associate Professor Ph. D. Luciana Lupaş
Abstract
In this paper we introduced a new class Ωp (n, α) of meromorphic
starlike univalent functions with alternating coefficients in the punctured unit disk U ∗ = {z : 0 < |z| < 1}. We obtain among other
results are the coefficient inequalities, distortion theorem and class
preserving integral operator.
2000 Mathematics Subject Classification: 30C45
Keywords: meromorphic function, Starlike functions, Distortion theorem
1
Received 24 September, 2006
Accepted for publication (in revised form) 9 November, 2006
113
114
1
M. Darus, S. B. Joshi and N. D. Sangle
Introduction
Let Ap denote the class of functions of the form:
∞
(1)
f (z) =
a−1 X
ap+k z p+k ,
+
z
k=0
(a−1 6= 0,
p ∈ N = {1, 2, 3, . . .})
which are regular in the punctured unit disk U ∗ = {z : 0 < |z| < 1}.
Define
D0 f (z) = f (z)
(2)
∞
a−1 X
(p + k + 2)ap+k z p+k
+
Df (z) = D f (z) =
z
k=0
1
=
(3)
(z 2 f (z))′
z
D2 f (z) = D(D1 f (z)),
(4)
and for n = 1, 2, 3, . . .
∞
n
D f (z) = D(D
(5)
=
n−1
a−1 X
+
(p + k + 2)ap+k z p+k ,
f (z)) =
z
k=0
(z 2 Dn−1 f (z))′
z
Let Bn (α), denote the class consisting functions in Ap satisfying
(6) ½
¾
Dn+1 f (z)
Re
− 2 < −α,
Dn f (z)
(z ∈ U ∗ , 0 ≤ α < 1, n ∈ No = N ∪ {0}).
Let Λp be the subclass of Ap which consisting of functions of the form
∞
(7) f (z) =
a−1 X
+
(−1)p+k−1 ap+k z p+k ,
z
k=0
(a−1 > 0; ap+k ≥ 0,
p ∈ N).
Meromorphic Starlike Functions...
115
Further let
(8)
Ωp (n, α) = Bn (α) ∩ Λp .
We note that, in [1] Uralegaddi and Somanatha define a class Bn (α), which
consists functions of the form
∞
a−1 X
f (z) =
ak z k
+
z
k=0
(a−1 6= 0)
which are analytic in U ∗ and obtained inclusion relation, the class preserving
integral. In the same year, Uralegaddi and Ganigi [2] considered meromorphic starlike functions with alternating coefficients. Further, recently in [4],
Aouf and Darwish also considered meromorphic starlike univalent functions
with alternating coefficients and obtained coefficient inequalities, distortion theorem and integral operators. The class of meromorphic functions
have been studied by various authors and among are Darwish [3], Aouf and
Hossen [5], and Mogra et.al [6], few to mention.
In the present paper, we consider functions of the forms (7) and obtain
basic properties, which include for example the coefficient inequalities, distortion theorem, closure theorem and integral operators. Finally, the class
preserving integral operator of the form
(9)
Fc+1 (z) = (c + 1)
Z
1
uc+1 f (uz)du
(0 ≤ u < 1, 0 < c < ∞)
0
is considered. Techniques used are similar to those of Aouf and Hossen [5].
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2
M. Darus, S. B. Joshi and N. D. Sangle
Coefficient Inequality
Theorem 2.1. Let the function f (z) be defined by (1). If
∞
X
(10)
(p + k + 2)n (p + k + α)|ap+k | ≤ (1 − α)|a−1 |,
k=0
then f (z) ∈ Bn (α).
Proof. It suffices to show that
¯
¯
¯
¯
Dn+1 f (z)
−
1
¯
¯
Dn f (z)
¯ < 1,
¯
(11)
¯
¯ Dn+1 f (z)
¯ Dn f (z) − (3 − 2α) ¯
we have
|z| < 1,
¯
¯
¯
¯
Dn+1 f (z)
−
1
¯
¯
¯ n+1 Dn f (z)
¯=
¯ D f (z)
¯
−
(3
−
2α)
¯ Dn f (z)
¯
¯
¯
∞
P
¯
¯
n
p+k
¯
¯
(p + k + 2) (p + k + 1)ap+k z
¯
¯
k=0
¯
¯≤
=¯
∞
¯
P
n (p + k + 2α − 1)a
p+k ¯
¯ (2 − 2α)a−1 −
(p
+
k
+
2)
z
p+k
¯
¯
k=0
∞
P
(p + k + 2)n (p + k + 1)|ap+k |
k=0
≤
(2 − 2α)a−1 −
∞
P
.
(p + k +
2)n (p
+ k + 2α − 1)|ap+k |
k=0
The last expression is bounded by 1 if
∞
X
k=0
∞
X
(p+k+2) (p+k+1)|ap+k | ≤ (2−2α)a−1 − (p+k+2)n (p+k+2α−1)|ap+k |,
n
k=0
which reduces to
(12)
∞
X
(p + k + 2)n (p + k + α)|ap+k | ≤ (1 − α)|a−1 |.
k=0
But (12) is true by hypothesis. Hence the result follows.
Meromorphic Starlike Functions...
117
Theorem 2.2. Let the function f (z) be defined by (7).
Then f (z) ∈
Ωp (n, α) if and only if
∞
X
(p + k + 2)n (p + k + α)|ap+k | ≤ (1 − α)a−1 ,
(13)
k=0
Proof. In view of Theorem 2.1, it suffices to prove the ”only if” part. Let
us assume that f (z) defined by (7) is in Ωp (n, α). Then
¾
¾
½ n+1
½ n+1
D f (z) − 2Dn f (z)
D f (z)
,
− 2 = Re
Re
Dn f (z)
Dn f (z)
(14)
= Re




a−1 −


 −a−1 −
∞
P
n
(p + k + 2) (p + k)ap+k z
k=0
∞
P
k=0




p+k
< −α,
|z| < 1.

(−1)p+k−1 (p + k + 2)n ap+k z p+k 

Choose value of z on the real axis so that
Dn+1 f (z)
Dn f (z)
− 2 is real. Upon clearing
the denominator in (14) and letting z → 1 through real values, we obtain
Ã
!
∞
∞
X
X
(p + k + 2)n ap+k α.
(p + k + 2)n (p + k)ap+k ≥ a−1 +
a−1 −
k=0
k=0
Thus
∞
X
(p + k + 2)n (p + k + α)ap+k ≤ (1 − α)a−1 .
k=0
Hence the result follows.
Corollary 1 Let the function f (z) be defined by (7) be in Ωp (n, α). Then
(15)
ap+k ≤
(1 − α)a−1
,
(p + k + 2)n (p + k + α)
The result is sharp for the function
(16)
f (z) =
a−1
(1 − α)a−1
+ (−1)p+k−1
z p+k ,
z
(p + k + 2)n (p + k + α)
(k ≥ 1).
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3
M. Darus, S. B. Joshi and N. D. Sangle
Distortion Theorem
Theorem 3.1. Let the function f (z) be defined by (7) be in Ωp (n, α). Then
for 0 < |z| < 1, we have
(17)
a−1
(1 − α)a−1
a−1
(1 − α)a−1
−
r ≤ |f (z)| ≤
+
r,
n
r
(p + 3) (p + α + 1)
r
(p + 3)n (p + α + 1)
where equality holds for the function
(18)
f (z) =
(1 − α)a−1
a−1
+
zp,
z
(p + 3)n (p + α + 1)
(z = ir, r),
and
(19)
(1 − α)a−1
a−1
(1 − α)a−1
a−1
−
≤ |f ′ (z)| ≤ 2 +
,
2
n
r
(p + 3) (p + α + 1)
r
(p + 3)n (p + α + 1)
where equality holds for the function f (z) given by (18) at z = ∓ir, ∓r.
Proof. In view of Theorem 2.2, we have
(20)
∞
X
ap+k ≤
k=0
(1 − α)a−1
.
(p + k + 2)n (p + k + α)
Thus, for 0 < |z| < 1,
|f (z)| ≤
(21)
≤
∞
X
a−1
ap+k
+r
r
k=0
a−1
(1 − α)a−1
+
r
r
(p + 3)n (p + α + 1)
and
|f (z)| ≥
(22)
≥
∞
X
a−1
−r
ap+k
r
k=0
a−1
(1 − α)a−1
−
r.
r
(p + 3)n (p + α + 1)
Meromorphic Starlike Functions...
119
Thus (17) follows.
Since
n
(p+3) (p+α+1)
∞
X
(p+k)|ap+k | ≤
(p+k+2)n (p+k+α)|ap+k | ≤ (1−α)|a−1 |
k=0
k=0
where
∞
X
(p+k+2)n (p+k+α)
p+k
is an increasing function of k, from Theorem 2.2, it
follows that
(23)
∞
X
(p + k) ≤
k=0
a−1
(1 − α)a−1
+
.
r
(p + 3)n (p + α + 1)
Hence
∞
a−1 X
(p + k)ap+k rp+k−1
+
|f (z)| ≤
2
r
k=0
′
∞
a−1 X
≤
+
(p + k)ap+k
r
k=0
(24)
≤
(1 − α)a−1
a−1
+
r
(p + 3)n (p + α + 1)
and
∞
a−1 X
(p + k)ap+k rp+k−1
|f (z)| ≥
−
r2
k=0
′
∞
a−1 X
≥
−
(p + k)ap+k
r
k=0
(25)
≥
(1 − α)a−1
a−1
−
r
(p + 3)n (p + α + 1)
Thus (19) follows. It can be easily seen that the function f (z) defined by
(18) is extremal for the theorem.
120
4
M. Darus, S. B. Joshi and N. D. Sangle
Closure Theorem
Let the function fj (z) be defined for j ∈ {1, 2, 3, . . . , m}, by
∞
(26)
a−1,j X
(−1)p+k−1 ap+k,j z p+k ,
+
fj (z) =
z
k=0
(a−1,j > 0; ap+k,j ≥ 0)
for z ∈ U ∗ .
Now, we shall prove the following result for the closure of function in the
class Ωp (n, α).
Theorem 4.1. Let the functions fj (z) be defined by (26) be in the class
Ωp (n, α) for every j ∈ {1, 2, 3, . . . , m}. Then the function F (z) defined by
∞
b−1 X
(27) F (z) =
(−1)p+k−1 bp+k z p+k ,
+
z
k=0
(b−1 > 0; bp+k ≥ 0,
p ∈ N)
is a member of the class Ωp (n, α), where
∞
(28)
b−1
1 X
=
a−1,j
m j=1
∞
and
bp+k
1 X
=
ap+k,j
m j=1
(k = 1, 2, . . .).
Proof. Since fj (z) ∈ Ωp (n, α), it follows from Theorem 2.2, that
(29)
∞
X
(p + k + 2)n (p + k + α)|ap+k,j | ≤ (1 − α)|a−1,j |,
k=0
for every j ∈ {1, 2, 3, . . . , m}. Hence,
∞
X
(p + k + 2)n (p + k + α)bp+k =
k=0
=
∞
X
k=0
(p + k + 2)n (p + k + α)
Ã
∞
1 X
ap+k,j
m j=1
!
=
Meromorphic Starlike Functions...
121
!
̰
∞
1 X X
(p + k + 2)n (p + k + α)ap+k,j ≤
=
m j=1 k=0
≤ (1 − α)
Ã
∞
1 X
a−1,j
m j=1
!
= (1 − α)b−1 ,
which (in view of Theorem 2.2) implies that F (z) ∈ Ωp (n, α).
Theorem 4.2. The class Ωp (n, α) is closed under convex linear combination.
Proof. Let the fj (z)(j = 1, 2) defined by (26) be in the class Ωp (n, α), it is
sufficient to prove that the function
(30)
H(z) = λf1 (z) + (1 − λ)f2 (z)
(0 ≤ λ ≤ 1)
is also in the class Ωp (n, α). Since, for (0 ≤ λ ≤ 1),
∞
(31)
λa−1,1 + (1 − λ)a−1,2 X
{λap+k,1 + (1 − λ)ap+k,2 } z p+k ,
−
H(z) =
z
k=0
we observe that
∞
X
(p + k + 2)n (p + k + α) {λap+k,1 + (1 − λ)ap+k,2 } =
k=0
=λ
∞
X
(p+k+2)n (p+k+α)λap+k,1 +(1−λ)
k=0
∞
X
(p+k+2)n (p+k+α)λap+k,2 ≤
k=0
≤ (1 − α) {λa−1,1 + (1 − λ)a−1,2 }
with the aid of Theorem 2.2. Hence H(z) ∈ Ωp (n, α).
This completes the proof of Theorem 4.2.
122
M. Darus, S. B. Joshi and N. D. Sangle
Theorem 4.3. Let
a−1
z
(1 − α)a−1
a−1
(33)fp+k (z) =
+ (−1)p+k−1
z p+k ,
z
(p + k + 2)n (p + k + α)
(32) fo (z) =
(k ≥ 1).
Then f (z) ∈ Ωp (n, α) if and only if it can be expressed in the form
(34)
f (z) =
∞
X
λp+k fp+k (z),
k=0
where
∞
X
λp+k (k ≥ 0) and
λk = 1.
k=0
Proof. Let
f (z) =
∞
X
λp+k fp+k (z),
∞
X
where λp+k (k ≥ 0) and
k=0
k=0
Then
f (z) =
∞
X
λp+k fp+k (z) = λo fo (z) +
+
∞
X
λp+k fp+k (z) =
k=1
k=0
∞
X
λp+k
k=0
=
λk = 1.
½
Ã
1−
∞
X
λp+k
k=0
(1 − α)a−1
a−1
+ (−1)p+k−1
z p+k
n
z
(p + k + 2) (p + k + α)
¾
(1 − α)a−1
a−1
+ (−1)p+k−1
z p+k
z
(p + k + 2)n (p + k + α)
Since
∞
X
(p + k + 2)n (p + k + α) ·
k=0
= (1 − α)a−1 +
∞
X
k=0
(35)
≤ (1 − α)a−1 ,
(1 − α)a−1 λp+k
(p + k + 2)n (p + k + α)
λp+k = (1 − α)a−1 (1 − λo )
=
!
+
Meromorphic Starlike Functions...
123
by Theorem 2.2, f (z) ∈ Ωp (n, α). Conversely, we suppose that f (z) defined
by (7) is in the class Ωp (n, α). Then by using (15), we get
(36)
ap+k ≤
(1 − α)a−1
,
(p + k + 2)n (p + k + α)
(k ≥ 1).
Setting
(37)
λp+k =
(p + k + 2)n (p + k + α)
ap+k ,
(1 − α)a−1
(k ≥ 1).
and
(38)
λo = 1 −
∞
X
λp+k ,
k=0
we have (34). This completes the proof of the Theorem 4.3.
5
Integral Operator
In his section we consider integral transforms of functions in the class
Ωp (n, α).
Theorem 5.1. Let the function f (z) be defined by (7) be in the class Ωp (n, α),
then the integral transforms
Z 1
(39)
Fc+1 (z) = (c + 1)
uc+1 f (uz)du
(0 ≤ u < 1, 0 < c < ∞),
0
are in Ωp (n, α), where
(40)
δ(α, c, p) =
(p + k + 3)(p + α + 1) − (p + 1)(1 − α)(c + 1)
.
(p + k + 3)(p + α + 1) + (c + 1)(1 − α)
The result is sharp for the function
(41)
f (z) =
(1 − α)a−1
a−1
+ (−1)p+k−1
zp.
z
(p + 3)n (p + α + 1)
124
M. Darus, S. B. Joshi and N. D. Sangle
Proof. Let
(42) Fc+1 (z) = (c+1)
Z
1
u
c+1
0
∞
(c + 1)
a−1 X
−
ap+k z p+k .
f (uz)du =
z
p+k+c+2
k=0
In view of Theorem 2.2, it is sufficient to show that
µ
¶
∞
X
(p + k + 2)n (p + k + δ)
c+1
(43)
ap+k ≤ 1.
(1 − δ)a−1
p+k+c+2
k=0
Since f (z) ∈ Ωp (n, α), we have
∞
X
(p + k + 2)n (p + k + α)
(44)
k=0
(1 − α)a−1
ap+k ≤ 1.
Thus (42) will be satisfy if
(p + k + δ)(c + 1)
p+k+α
≤
(1 − δ)(p + k + c + 2)
1−α
for each k,
or
(45)
δ≤
(p + k + c + 2)(p + k + α) − (p + k)(1 − α)(c + 1)
.
(p + k + c + 2)(p + k + α) + (c + 1)(1 − α)
For each α, p, and c fixed, let
F (k) =
(p + k + c + 2)(p + k + α) − (p + k)(1 − α)(c + 1)
.
(p + k + c + 2)(p + k + α) + (c + 1)(1 − α)
Then
F (k + 1) − F (k) =
=
(c + 1)(1 − α)(p + k + 1)(p + k + 2)
>
[(p + k + c + 2)(p + k + α) + (c + 1)(1 − α)][(p + k + c + 3)(p + k + α + 1) + (c + 1)(1 − α)]
for each k. Hence, F (k) is an increasing function of k.
Since
F (1) =
the result follows.
(p + c + 3)(p + α + 1) − (p + 1)(1 − α)(c + 1)
,
(p + c + 3)(p + α + 1) + (c + 1)(1 − α)
Meromorphic Starlike Functions...
125
References
[1] B. A. Uralegaddi and C. Somanatha, New criteria for meromorphic
starlike univalent functions, Bull. Austral. Math. Soc. 43, (1991), 137140.
[2] B. A. Uralegaddi and M. D. Ganigi, Meromorphic starlike functions
with alternating coefficients, Rend. Math. 11 (7), Roma(1991), 441446.
[3] H. E. Darwish, Meromorphic p-valent starlike functions with negative
coefficients, Indian J. Pure Appl. Math. 33(7), (2002), 967-976.
[4] M. K. Aouf and H. E. Darwish, Meromorphic starlike univalent functions with alternating coefficients, Utilitas Math. 47, (1995),137-144.
[5] M. K. Aouf and H. M. Hossen, New criteria for meromorphic p-valent
starlike functions, Tsukuba J. Math. 17(2), (1993), 1-6.
[6] M. L. Mogra, T. R. Reddy and O. P. Juneja, Meromorphic univalent functions with positive coefficients, Bull. Austral. Math. Soc. 32,
(1985), 161-176.
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi, 43600, Selangor D.E., Malaysia
E-mail address: maslina@pkrisc.cc.ukm.my
126
M. Darus, S. B. Joshi and N. D. Sangle
Department of Mathematics
Walchand College of Engineering
Sangli (M. S), India 416415
E-mail address:joshisb@hotmail.com
Department of Mathematics
Annasaheb Dange College of Engineering
Ashta, Sangli (M. S), India 416301
E-mail address: navneet− sangle@rediffmail.com