15_488_aouf.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 2, 2010, 317-330

ISSN 1307-5543 – www.ejpam.com

Argument Estimates of Certain Analytic Functions Associated

with a Family of Multiplier Transformations

M. K. Aouf1∗, A. Shamandy2, R. M. El-Ashwah3, and E. E. Ali4

1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Abstract. The purpose of the present paper is to derive some inclusion properties and argument es-

timates of certain normalized analytic functions in the open unit disk, which are defined by means of

a class of multiplier transformations. Furthermore, the integral preserving properties in a sector are

investigated for these multiplier transformations.

2000 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Analytic functions, multiplier transformation, differential subordination,

close- to-convex functions, argument estimates.

1. Introduction

Let A denote the class of the functions of the form:

f (z) = z +

∞
∑

k=2

akzk, (1)

which are analytic in the open unit disc U = {z : |z| < 1}. If f (z) and g(z) are analytic in U ,

we say that f (z) is subordinate to g(z) written symbolically as follows:

f ≺ g (z ∈ U) or f (z)≺ g(z) (z ∈ U),

if there exists a Schwarz function w(z), which (by definition) is analytic in U with w(0) = 0

and |w(z)| < 1 (z ∈ U), such that f (z) = g(w(z)) (z ∈ U). In particular, if the function g(z)

is univalent in U , then we have the following equivalent (cf., e.g., [2]; see also [10], [11, p.

4])

f (z) ≺ g(z)(z ∈ U)⇔ f (0) = g(0) and f (U)⊂ g(U).

Many essentially equivalent definitions of multiplier transformation have been given in litera-

ture (see [4], [5], and [20]). In [3] Catas defined the operator Im(λ,ℓ) as follows:

∗Corresponding author.

Email addresses: mkaouf127�yahoo.
om (M. Aouf), shamandy16�hotmail.
om (A. Shamandy),r_elashwah�yahoo.
om (R. El-Ashway), ekram_008eg�yahoo.
om (E. Ali)

http://www.ejpam.com 317 c© 2010 EJPAM All rights reserved.



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 318

Definition 1. [3] Let the function f (z) ∈ A. for m ∈ N0 = N ∪ {0}, where N = {1,2, . . . }, λ≥
0, ℓ≥ 0. The extended multiplier transformation Im(λ,ℓ) on A is defined by the following infinite

series:

Im(λ,ℓ) f (z) = z +

∞
∑

k=2

�

ℓ+ 1+λ(k− 1)

ℓ+ 1

�m

akzk (2)

( f ∈ A;λ≥ 0;ℓ≥ 0; m ∈ N0; z ∈ U).

We can write (2) as follows:

Im(λ,ℓ) f (z) = (Φ
,m

λ,ℓ
∗ f )(z), (3)

where

Φm
λ,ℓ(z) = z +

∞
∑

k=2

�

ℓ+ 1+λ(k− 1)

ℓ+ 1

�m

zk.

It is easily verified from (2), that

λz(Im(λ,ℓ) f (z))
′
= (1+ ℓ)Im+1(λ,ℓ) f (z)− [1−λ+ ℓ]Im(λ,ℓ) f (z) (λ > 0). (4)

We note that:

I0(λ,ℓ) f (z) = f (z) and I1(1,0) f (z) = z f
′
(z).

Also by specializing the parameters λ,ℓ and m we obtain the following operators studied by

various authors:

(i) Im(1,ℓ) = Im(ℓ) f (z) (see Cho and Srivastava [4] and Cho and Kim [5]);

(ii) Im(λ, 0) f (z) = Dm
λ

f (z) (see AL-Oboudi [1]);

(iii) Im(1,0) = Dm f (z) (see Salagean [18]);

(iv) Im(1,1) = Im f (z) (see Uralegaddi and Somanatha [20]).

Let the functions g1, ..., gq be in the class A. Then we say that the functions g1, ..., gq are in

the class Ωm,λ,ℓ(q; A, B) if they satisfy the subordination condition:

z(Im(λ,ℓ)gi(z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)

≺
1+ Az

1+ Bz
(z ∈ U; i = 1, ...,q;−1≤ B < A≤ 1) , (5)

where
n
∑

j=1

1

z
Im(λ,ℓ)g j(z) 6= 0 (z ∈ U).

For λ= 1, m = ℓ= 0 and

g j(z) = w− j f (w jz) ( f ∈ A; j = 1, ...,q; w = e2πi/n) ,



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 319

Ωm,λ,ℓ(q; A, B) reduces to the class of starlike functions in U with respect to q symmetric

points [12] (see also [17]). If we take λ = 1, ℓ = 0, m = 0, q = 2, A = 1 and B = −1 in

(5), then we obtain the class of mutually adjoint close-to-convex functions in U considered by

Lewandowski and Stankiewicz [9].

Let Cm,λ,ℓ(q; A, B) be the class of functions of functions f ∈ A satisfying the argument

inequality
�

�

�

�

�

�

�

�

�

arg













z(Im(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)













�

�

�

�

�

�

�

�

�

<
π

2
α (6)

(z ∈ U , m ∈ N0, 0< α≤ 1; g j ∈ Ωm,λ,ℓ(q; A, B); j = 1, ...,q) .

If we take m = ℓ = 0, λ = 1, q = 1, α = 1, A = 1 and B = −1 in (6), Cm,λ,ℓ(q; A, B) be-

comes the familiar class of close-to-convex functions in U introduce by Kaplan [8]. Further,

for m = ℓ = 0, λ = 1, q = 2, α = 1, A = 1 and B = −1, Cm,λ,ℓ(q; A, B) covers the class of

close-to-convex functions in U with respect to symmetric points studied by Das and Singh [6].

In this present paper, we give some argument properties and estimates of analytic func-

tions belonging to A, which contain the basic inclusion relationships among the classesΩm,λ,ℓ(q; A, B)

and Cm,λ,ℓ(q; A, B). The integral preserving properties in connection with the operator Im(λ,ℓ)

defined by (2) are also considered.

2. The Main Results And Their Consequences

Unless otherwise mentioned,we shall assume in the reminder of this paper that λ > 0,ℓ ≥
0 and m ∈ N0.

In proving our main results, we need the following lemmas.

Lemma 1. [7] Let h be convex univalent in U with h(0) = 1 and

R(βh(z) + γ)> 0 (z ∈ U;β ,γ ∈ C) .

If p is analytic in U with p(0) = 1, then

p(z) +
zp
′
(z)

βp(z) + γ
≺ h(z) (z ∈ U) ,

implies that

p(z) ≺ h(z) (z ∈ U) .

Lemma 2. [10] Let h be convex univalent in U and φ be analytic in U with

R(φ(z))≥ 0 (z ∈ U) .

implies that

p(z) ≺ h(z) (z ∈ U) .



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 320

Lemma 3. [14] Let p be analytic in U with p(0) = 1 and p(z) 6= 0 in U . If there exists two

points z1, z2 ∈ U such that

−π

2
α1 = arg(p(z1))< arg(p(z))< arg(p(z2)) =

π

2
α2 , (7)

for some α1 and α2 (α1,α2 > 0) and for all z
�

|z| < |z1| = |z2|
�

, then

z1p
′
(z1)

p(z1)
= −i

�

α1 +α2

2

�

m and
z2p

′
(z2)

p(z2)
= i

�

α1+α2

2

�

m , (8)

where

m≥
1− |a|

1+ |a|
and a = i tan

π

4

�

α2 −α1

α1 +α2

�

(9)

First of all, with the help of Lemma 1 and 2, we obtain the following.

Proposition 1. If g1, ..., gq ∈ Ωm+1,λ,ℓ(q; A, B), then g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B).

Proof. Let

pi(z) =
z(Im(λ,ℓ)gi(z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)

(i = 1, ...,q) . (10)

By using the identity (4), we get

1

q

q
∑

j=1

(Im(λ,ℓ)g j(z))pi(z) +
1−λ+ ℓ

λ
(Im(λ,ℓ)gi(z)) =

1+ ℓ

λ
(Im+1(λ,ℓ)gi(z)) . (11)

Differentiating both sides of (11) with respect to z, and simplifying, we obtain

pi(z) +
zp
′

i
(z)

�

1

q

� q
∑

i=1

pi(z) +
1−λ+ℓ
λ

=
z(Im+1(λ,ℓ)gi(z))

′

�

1

q

� q
∑

j=1

Im+1(λ,ℓ)g j(z)

≺
1+ Az

1+ Bz
≡ h(z),

(z ∈ U; i = 1, ...,q) , (12)

g1, ..., gq ∈ Ωm+1,λ,ℓ(q; A, B). Since h is convex, for any z0 ∈ U , there exists a point ζ0 ∈ U

such that

X (z0) +
z0X

′
(z0)

X (z0) +
1−λ+ℓ
λ

= h(ζ0) ,

where

X (z) =
1

q

q
∑

i=1

pi(z) .



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 321

Then we find from Lemma 1 that X ≺ h. Applying Lemma 2 with

φ(z) =
1

X (z) + 1−λ+ℓ
λ

to (12) again, we find that pi ≺ h for all i(i = 1, ...,q). Next, we prove that

q
∑

j=1

1

z
Im(λ,ℓ)g j(z) 6= 0 (z ∈ U) .

Since g1, ..., gq ∈ Ωm+1,λ,ℓ(q; A, B) and h is convex, we find that there exists a point ζ0 ∈ U

such that, for any z0 ∈ U ,

r(z0) =

z0

 

q
∑

j=1

Im+1(λ,ℓ)g j(z0)

!′

q
∑

j=1

Im+1(λ,ℓ)g j(z0)

= h(ζ0) ,

and hence, r ≺ h. We note also that

q
∑

j=1

Im(λ,ℓ)g j(z) =

1−λ+ℓ
λ
+ 1

z(1−λ+ℓ)/λ

z
∫

0

t
1−λ+ℓ
λ
−1

q
∑

j=1

Im+1(λ,ℓ)g j(t)d t .

Thus, by applying Lemma A of [12], we conclude that

q
∑

j=1

1

z
Im(λ,ℓ)g j(z) 6= 0 (z ∈ U) .

This evidently completes the proof of Proposition 1.

Proposition 2. If g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B), then Fc(g1), ..., Fc(gq) ∈ Ωm,λ,ℓ(q; A, B), where Fc

is the integral operator defined by

Fc(gi) = Fc(gi)(z) =
c + 1

zc

z
∫

0

t c−1 gi(t)d t (i = 1, ...,q, c ≥ 0) . (13)

Proof. From (13), we have

z(Im(λ,ℓ)Fc(gi)(z))
′
= (c + 1)Im(λ,ℓ)gi(z)− cIm(λ,ℓ)Fc(gi)(z). (14)

Let

pi(z) =
z(Im(λ,ℓ)Fc(gi)(z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)Fc(g j)(z)

(i = 1, ...,q) .



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 322

Then by using (14), we obtain

�

1

q

� q
∑

j=1

(Im(λ,ℓ)Fc(g j)(z))pi(z) + cIm(λ,ℓ)Fc(gi)(z) = (c + 1)Im+1(λ,ℓ)gi(z) . (15)

Differentiating both sides of (15) with respect to z, and simplifying, we obtain

pi(z) +
zp
′

i(z)

�

1

q

� q
∑

j=1

pi(z) + c

=
z(Im(λ,ℓ)gi(z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)

.

Then, by the same arguments as in the proof of Proposition 1, it follows that Proposition 2

holds true as stated.

Remark 1.

(i) Putting m = ℓ = 0,λ = 1 and gi(z) = w−1 f (w iz)( f ∈ A; i = 1, ...,q; w = e
2πi

q ) in

Proposition 2, we obtain the result obtained by Mocanu [12];

(ii) Putting m = ℓ = 0,λ = 1,q = 2, g1(z) = f (z), and g2(z) = − f (−z) in Proposition 2,

we obtain the result obtained by Padmanabhan and Thangamani [16], which (in turn)

includes the result given by Das and Singh [6] as a special case.

Next, we prove the following theorem.

Theorem 1. Let f ∈ A and 0< δ1, δ2 ≤ 1. If

−
π

2
δ1 < arg













z(Im+1(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im+1(λ,ℓ)g j(z)













<
π

2
δ2 ,

where g1, ..., gq ∈ Ωm+1,λ,ℓ(q; A, B), then

−
π

2
α1 < arg













z(Im(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)













<
π

2
α2 ,

where α1 and α2 (0< α1, α2 ≤ 1) are the solutions of the following equations:

δ1 =







α1 +
�

2

π

�

tan−1

�

(α1+α2)(1−|a|) cos
�

π
2

�

t1

2
�

1+A

1+B
+ 1−λ+ℓ

λ

�

(1+|a|)+(α1+α2)(1−|a|) sin
�

π
2

�

t1

�

(B 6= −1) ,

α1 (B = −1) ,

(16)



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 323

and

δ2 =







α2 +
�

2

π

�

tan−1

�

(α1+α2)(1−|a|) cos
�

π
2

�

t1

2
�

1+A

1+B
+ 1−λ+ℓ

λ

�

(1+|a|)+(α1+α2)(1−|a|) sin
�

π
2

�

t1

�

(B 6= −1) ,

α2 (B = −1) ,

(17)

a being given by (9), and

t1 =
2

π
sin−1

 

A− B

1− AB+ 1−λ+ℓ
λ
(1− B2)

!

. (18)

Proof. Let

p(z) =
z(Im(λ,ℓ) f (z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)

and Q(z) =
1

q

q
∑

i=1

Q i(z) ,

where

Q i(z) =
z(Im(λ,ℓ)gi(z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)

(i = 1, ...,q) .

using (10) with gi replaced by f , we have

�

1

q

� q
∑

j=1

(Im(λ,ℓ)g j(z))p(z) +
1−λ+ ℓ

λ
Im(λ,ℓ) f (z) =

1+ ℓ

λ
Im+1(λ,ℓ) f (z) . (19)

Differentiating (19) with respect to z, and simplifying, we obtain

z(Im+1(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im+1(λ,ℓ)g j(z)

= p(z) +
zp
′
(z)

Q(z) + 1−λ+ℓ
λ

.

Since g1, ..., gn ∈ Ωm+1,λ,ℓ(q; A, B), by Proposition 1, we know that g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B),

and so

Q(z)≺
1+ Az

1+ Bz
(z ∈ U;−1≤ B < A≤ 1) .

Hence, we observe from [19] that

�

�

�

�

Q(z)−
1− AB

1− B2

�

�

�

�

<
A− B

1− B2
(z ∈ U; B 6= −1) , (20)

and

Re(Q(z))>
1− A

2
(z ∈ U; B = −1) . (21)



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 324

Then, by using (20) and (21), we have

Q(z) +
1−λ+ ℓ

λ
= ρeφπi/2,

where
1− A

1− B
+

1−λ+ ℓ

λ
< ρ <

1+ A

1+ B
+

1−λ+ ℓ

λ
,

−t1 < φ < t1 (B 6= −1) ,

t1 being given by (18), and
1− A

2
+

1−λ+ ℓ

λ
< ρ <∞

−1< φ < 1 (B = −1) .

We note that p is analytic in U with p(0) = 1. Let h be the function which maps U onto the

angular domain
§

φ :−
π

2
δ1 < arg(φ)<

π

2
δ2

ª

, with h(0) = 1.

Applying Lemma 1 for this h with

φ(z) =
1

Q(z) + 1−λ+ℓ
λ

,

we see that

R(p(z))> 0 (z ∈ U) ,

and hence, p(z) 6= 0 in U .

If there exist two points z1 and z2 in U such that condition (7) is satisfied, then (By Lemma

3) we obtain (8) under restriction (9). For the case B 6= −1, we first obtain

arg

 

p(z1) +
z1p

′
(z1)

Q(z1) +
1−λ+ℓ
λ

!

= −
π

2
α1 + arg

�

1− i
α1 +α2

2
m
�

ρeφπi/2
�−1
�

≤ −
π

2
α1 − tan−1

 

(α1 +α2)m sin
�

π

2

�

(1−φ)

2ρ+ (α1+α2)m cos
�

π

2

�

(1−φ)

!

≤ −
π

2
α1 − tan−1







(α1 +α2)(1− |a|) cos
�

π

2

�

t1

2
�

1+A

1+B
+ 1−λ+ℓ

λ

�

(1+ |a|)+ (α1 +α2)(1− |a|) sin
�

π

2

�

t1







= −
π

2
δ1



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 325

and

arg

 

p(z2) +
z2p

′
(z2)

Q(z2) +
1−λ+ℓ
λ

!

≥
π

2
α2 + tan−1







(α1+α2)(1− |a|) cos
�

π

2

�

t1

2
�

1+A

1+B
+ 1−λ+ℓ

λ

�

(1+ |a|)+ (α1 +α2)(1− |a|) sin
�

π

2

�

t1







=
π

2
δ2 ,

where we have used inequality (9), δ1, δ2 and t1 being given by (16), (17), and (18), re-

spectively. Similarly, for the case B = −1, we have

arg

 

p(z1) +
z1p

′
(z1)

Q(z1) +
1−λ+ℓ
λ

!

≤ −
π

2
α1

and

arg

 

p(z2) +
z2p

′
(z2)

Q(z2) +
1−λ+ℓ
λ

!

≥
π

2
α2 .

These obviously contradict the assumption of Theorem 1. The proof of Theorem 1 is thus

completed.

Putting δ1 = δ2 = δ in Theorem 1, we obtain the following corollary.

Corollary 1. Let f ∈ A and 0< δ ≤ 1. If
�

�

�

�

�

�

�

�

�

arg













z(Im+1(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im+1(λ,ℓ)g j(z)













�

�

�

�

�

�

�

�

�

<
π

2
δ ,

where g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B), then

�

�

�

�

�

�

�

�

�

arg













z(Im(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)













�

�

�

�

�

�

�

�

�

<
π

2
α ,

where α (0< α ≤ 1) is the solution of the equation

δ =







α+ 2

π
tan−1

�

α cos
�

π
2

�

t1
�

1+A

1+B
+ 1−λ+ℓ

λ

�

+α sin
�

π
2

�

t1

�

(B 6= −1)

α (B = −1) ,

t1 being given by (18).



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 326

From Corollary 1, we immediately obtain the following corollary.

Corollary 2. The inclusion relation

Cm+1,λ,ℓ(q; A, B)⊂ Cm,λ,ℓ(q; A, B)

holds true for any integer m.

Remark 2. For m = ℓ = 0,λ = 1,q = 1,δ = 1,A = 1 and B = −1, the class Cm,λ,ℓ(q; A, B)

reduces to the class of quasiconvex functions in U introduced by Sakaguchi [17] (see also [13]).

Hence, we see from Corollary 2 that every quasiconvex function in U is close-to-convex in U.

Next, we prove the following theorem.

Theorem 2. Let f ∈ A, 0< δ1, δ2 ≤ 1 and c ≥ 0. If

−
π

2
δ1 < arg













z(Im(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)













<
π

2
δ2 ,

where g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B), then

−
π

2
α1 < arg













z(Im(λ,ℓ)Fc( f )(z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)Fc(g j)(z)













<
π

2
α2 ,

where Fc is the integral operator defined by (13), and α1 and α2 (0 < α1, α2 ≤ 1) are the

solutions of the following equations:

δ1 =







α1 +
2

π
tan−1

�

(α1+α2)(1−|a|) cos
�

π
2

�

t2

2
�

1+A

1+B
+c
�

(1+|a|)+(α1+α2)(1−|a|) sin
�

π
2

�

t2

�

(B 6=−1)

α1 (B =−1) ,

and

δ2 =







α2 +
2

π
tan−1

�

(α1+α2)(1−|a|) cos
�

π
2

�

t2

2
�

1+A

1+B
+c
�

(1+|a|)+(α1+α2)(1−|a|) sin
�

π
2

�

t2

�

(B 6=−1) ,

α1 (B =−1) ,

a being given by (9) and t2 being the same as t1 given by (18) with c =
1−λ+ ℓ

λ
.



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 327

Proof. Let

p(z) =
z(Im(λ,ℓ)Fc( f )(z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)Fc(g j)(z)

and Q(z) =
1

q

q
∑

k=1

Qk(z) ,

Qk(z) =
z(Im(λ,ℓ)Fc(gk)(z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)Fc(g j)(z)

.

Using the relationship (14), we obtain

�

1

q

� q
∑

j=1

(Im(λ,ℓ)Fc(g j)(z))p(z) + cIm(λ,ℓ)Fc( f )(z) = (c + 1)Im(λ,ℓ) f (z) . (22)

Differentiating (22) with respect to z, and simplifying, we get

z(Im(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)

= p(z) +
zp
′
(z)

Q(z) + c
.

Since g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B), by Proposition 2, we have Fc(g1), ..., Fc(gq) ∈ Ωm,λ,ℓ(q; A, B).

Hence, we find that

Q(z)≺
1+ Az

1+ Bz
(z ∈ U;−1≤ B < A≤ 1) .

The remaining part of the proof is similar to that in the proof of Theorem 1, and so we omit

the details involved.

Putting δ1 = δ2 in Theorem 2 we obtain the following corollary.

Corollary 3. Let f ∈ A, 0< δ ≤ 1 and c ≥ 0. If

�

�

�

�

�

�

�

�

�

arg













z(Im(λ,ℓ) f (z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)













�

�

�

�

�

�

�

�

�

<
π

2
δ ,

where g1, ..., gq ∈ Ωm,λ,ℓ(q; A, B), then

�

�

�

�

�

�

�

�

�

arg













z(Im(λ,ℓ)Fc( f )(z))
′

�

1

q

� q
∑

j=1

Im(λ,ℓ)Fc(g j)(z)













�

�

�

�

�

�

�

�

�

<
π

2
α ,



M. Aouf, A. Shamandy, R. El-Ashway, E. Ali / Eur. J. Pure Appl. Math, 3 (2010), 317-330 328

where α (0< α ≤ 1) is the solution of the following equation

δ =







α+ 2

π
tan−1

�

α cos
�

π
2

�

t2
�

1+A

1+B
+c
�

+α sin
�

π
2

�

t2

�

(B 6= −1) ,

α (B = −1) ,

t2 being the same as t1 given by (18) with c =
1−λ+ ℓ

λ
.

From Corollary 3, we readily obtain the following corollary.

Corollary 4. Let f ∈ Cm,λ,ℓ(q; A, B). Then Fc( f ) ∈ Cm,λ,ℓ(q; A, B), where Fc is the integral

operator defined by (13).

Remark 3. From Theorem 2 or Corollary 4, we see that every function in Cm,λ,ℓ(q; A, B) preserves

the angles under the integral operator defined by (13). If we put m = ℓ= 0, λ= 1, q = 2, A= 1

and B = −1 in Corollary 4, we are easily led to the result given earlier by Das and Singh [6].

Finally, we state Theorem 3 below. The proof is much akin to that of Theorem 1, and so

that details may be omitted.

Theorem 3. Let f ∈ A, 0< δ1, δ2 ≤ 1 and γ≥ 0. If

−
π

2
δ1 < arg













γ
z(Im+1(λ,ℓ) f (z))

′

�

1

q

� q
∑

j=1

Im+1(λ,ℓ)g j(z)

+ (1− γ)
z(Im(λ,ℓ) f (z))

′

�

1

q

� q
∑

j=1

Im(λ,ℓ)g j(z)













<
π

2
δ2 ,

where g1, ..., gq ∈ Ωm+1,λ,ℓ(q; A, B), then

−
π

2
α1 < arg













z(Im(λ,ℓ) f (z))
′

1

q

q
∑

j=1

Im(λ,ℓ)g j(z)













<
π

2
α2 ,

where α1 and α2 are the solutions of the following equation:

δ1 =







α1 +
2

π
tan−1

�

(α1+α2)(1−|a|)γ cos
�

π
2

�

t1

2
�

1+A

1+B
+ 1−λ+ℓ

λ

�

(1+|a|)+(α1+α2)(1−|a|) sin
�

π
2

�

t1

�

(B 6=−1) ,

α1 (B =−1) ,

and

δ2 =







α2 +
2

π
tan−1

�

(α1+α2)(1−|a|)γ cos
�

π
2

�

t1

2
�

1+A

1+B
+ 1−λ+ℓ

λ

�

(1+|a|)+(α1+α2)(1−|a|)γ sin
�

π
2

�

t1

�

(B 6= −1) ,

α2 (B = −1) ,

a and t1 being given by (9) and (18), respectively.



REFERENCES 329

Remark 4. For m= ℓ = 0, λ= 1, q = 2, A= 1, B = −1 and δ = 1, Theorem 3 reduces at once

to the result given earlier by Padmanabhan and Thangamani [15].

Remark 5. Putting λ = 1 in the above results, we obtain the results obtained by Cho and

Srivastava [5].

Remark 6. Putting ℓ= 0 in the above results, we obtain the corresponding results for the opera-

tor Dm
λ

.

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