Argument Estimates of Certain Analytic Functions Associated with a Family of Multiplier Transformations

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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. Aouf

1∗

, A. Shamandy

2

, R. M. El-Ashwah

3

, and E. E. Ali

4

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

LetAdenote the class of the functions of the form:

f(z) =z+

∞

X

k=2

a_kzk, (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 functionw(z), which (by definition) is analytic inU 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 inU, 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 operatorIm(λ,ℓ)as follows:

∗_{Corresponding author.}

Email addresses:mkaouf127yahoo.om(M. Aouf),shamandy16hotmail.om(A. Shamandy),

r_elashwahyahoo.om(R. El-Ashway),ekram_008egyahoo.om(E. Ali)

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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+

∞

X

k=2

_ℓ₊₁₊_λ_(k−₁₎

ℓ+1

m

a_kzk (2)

(f ∈ A;λ≥0;ℓ≥0;m∈N0;z∈U).

We can write (2) as follows:

Im(λ,ℓ)f(z) = (Φ_λ,m_,_ℓ∗f)(z), (3)

where

Φ_λm_,_ℓ(z) =z+

∞

X

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)andI1(1, 0)f(z) =z f′(z).

Also by specializing the parametersλ,ℓandmwe 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) =D_λmf(z)(see AL-Oboudi[1]);

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

(iv) Im(1, 1) =Imf(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 classΩ_m_,λ,ℓ(q;A,B)if they satisfy the subordination condition:

z(Im(λ,ℓ)g_i(z))′

1 q

_Pq

j=1

Im(λ,ℓ)gj(z)

≺ 1+Az

1+Bz (z∈U;i=1, ...,q;−1≤B<A≤1), (5)

where

n

X

j=1 1

zI

m₍_λ,_ℓ₎_g

j(z)6=0 (z∈U). Forλ=1, m=ℓ=0 and

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Ω_m_,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 takeλ =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 inU considered by Lewandowski and Stankiewicz[9].

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

inequality

arg

     

z(Im(λ,ℓ)f(z))′

1 q

_Pq

j=1

Im(λ,ℓ)g_j(z)

     

< π

2α (6)

(z∈U,m∈N₀, 0< α≤1;g_j∈Ω_m_,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), C_m,λ,ℓ(q;A,B)

be-comes the familiar class of close-to-convex functions inU introduce by Kaplan[8]. Further, form= ℓ= 0, λ =1, q= 2, α=1, A=1 and B =−1, C_m_,_λ_,_ℓ(q;A,B) covers the class of close-to-convex functions inU 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 toA, which contain the basic inclusion relationships among the classesΩ_m_,_λ_,_ℓ(q;A,B)

andC_m_,λ,ℓ(q;A,B). The integral preserving properties in connection with the operatorIm(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 thatλ >0,ℓ≥ 0 andm∈N0.

In proving our main results, we need the following lemmas. Lemma 1. [7]Let h be convex univalent in U with h(0) =1and

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 andφbe analytic in U with R(φ(z))≥0 (z∈U).

implies that

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Lemma 3. [14]Let p be analytic in U with p(0) = 1and p(z) 6=0in U . If there exists two points z₁,z₂∈U such that

−π

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

2α2, (7)

for someα₁andα₂ (α₁,α₂>0)and for all z |z|<|z₁|=|z₂|

, then

z₁p′(z₁) p(z1)

=−i

_α

1+α2 2

m and z2p

′

(z₂) p(z2)

=i

_α

1+α2 2

m, (8)

where

m≥ 1− |a|

1+|a| and a=itan

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, ...,g_q∈Ω_m₊_1,_λ_,_ℓ(q;A,B), then g₁, ...,g_q∈Ω_m_,_λ_,_ℓ(q;A,B). Proof. Let

p_i(z) = z(I

m₍_λ,_ℓ₎_g i(z))

′

1 q

q P

j=1

Im₍_λ,_ℓ₎_g j(z)

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

By using the identity (4), we get 1

q

X

j=1

(Im(λ,ℓ)g_j(z))p_i(z) +1−λ+ℓ

λ (I

m₍_λ,_ℓ_)g i(z)) =

1+ℓ λ (I

m+1₍_λ,_ℓ_)g

i(z)). (11)

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

p_i(z) + zp

′ i(z)

1 q

_Pq

i=1

pi(z) +1−_λλ+ℓ

= z(I

m+1₍_λ,_ℓ₎_g i(z))

′

1 q

_Pq

j=1

Im+1(λ,ℓ)gj(z)

≺ 1+Az

1+Bz ≡h(z),

(z∈U;i=1, ...,q), (12)

g₁, ...,g_q ∈Ω_m+1,λ,ℓ(q;A,B). Sincehis 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

X

i=1

(5)

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

φ(z) = 1

X(z) +1−_λλ+ℓ

to (12) again, we find thatp_i ≺hfor alli(i=1, ...,q). Next, we prove that q

X

j=1 1

zI

m₍_λ,_ℓ₎_g

j(z)6=0 (z∈U).

Since g1, ...,gq ∈Ωm+1,λ,ℓ(q;A,B) and his convex, we find that there exists a point ζ0 ∈U such that, for anyz₀∈U,

r(z₀) = z₀

q

P

j=1

Im+1(λ,ℓ)g_j(z₀)

!′

q

P

j=1

Im+1(λ,ℓ)g_j(z₀)

=h(ζ0),

and hence,r≺h. We note also that q

X

j=1

Im(λ,ℓ)gj(z) = 1−λ+ℓ

λ +1

z(1−λ+ℓ)/λ

z

Z

0

t1−λλ+ℓ−1 q

X

j=1

Im+1(λ,ℓ)g_j(t)d t.

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

X

j=1 1

zI

m₍_λ,_ℓ₎_g

j(z)6=0 (z∈U). This evidently completes the proof of Proposition 1.

Proposition 2. If g₁, ...,g_q ∈Ω_m_,_λ_,_ℓ(q;A,B), then F_c(g₁), ...,F_c(g_q)∈Ω_m_,_λ_,_ℓ(q;A,B), where F_c is the integral operator defined by

F_c(g_i) =F_c(g_i)(z) = c+1 zc

z

Z

0

tc−1g_i(t)d t (i=1, ...,q,c≥0). (13)

Proof. From (13), we have

z(Im(λ,ℓ)F_c(g_i)(z))′= (c+1)Im(λ,ℓ)g_i(z)−cIm(λ,ℓ)F_c(g_i)(z). (14) Let

pi(z) =

z(Im(λ,ℓ)F_c(g_i)(z))′

1 q

_Pq

j=1

Im₍_λ,_ℓ_)F

c(gj)(z)

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Then by using (14), we obtain

₁

q

q X

j=1

(Im(λ,ℓ)F_c(g_j)(z))p_i(z) +cIm(λ,ℓ)F_c(g_i)(z) = (c+1)Im+1(λ,ℓ)g_i(z). (15)

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

p_i(z) + zp

′ i(z)

1 q

q P

j=1

p_i(z) +c

= z(I

m₍_λ,_ℓ_)g i(z))

′

1 q

q P

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 g_i(z) = w−1f(wi_z)(_f _∈ _A_;_i ₌ _{1, ...,}_q_;_w ₌ _e2πqi₎ _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 and0< δ1, δ2≤1. If

−π

2δ1<arg

     

z(Im+1₍_λ,_ℓ₎_f_(z))′

1 q

_Pq

j=1

Im+1(λ,ℓ)g_j(z)

     

<π 2δ2,

where g₁, ...,g_q∈Ω_m₊_1,_λ_,_ℓ(q;A,B), then

−π

2α1<arg

     

z(Im₍_λ,_ℓ₎_f_(z))′

1 q

q P

j=1

Im₍_λ,_ℓ₎_g j(z)

     

< π 2α2,

whereα1 andα2 (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

21₁+₊A

B+ 1−λ+ℓ

λ

(1+|a|)+(α1+α2)(1−|a|)sin _π

2

t1

(B6=−1),

α1 (B=−1),

(7)

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

(B6=−1),

α2 (B=−1),

(17)

a being given by (9), and

t1= 2 πsin

−1 A−B

1−AB+1−λ+ℓ

λ (1−B

2₎

!

. (18)

Proof. Let

p(z) = z(I

m₍_λ,_ℓ₎_f_(z))′

1 q

_Pq

j=1

Im₍_λ,_ℓ₎_g j(z)

and Q(z) = 1 q

q

X

i=1

Qi(z),

where

Q_i(z) = z(I

m₍_λ,_ℓ₎_g i(z))

′

1 q

_Pq

j=1

Im₍_λ,_ℓ₎_g j(z)

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

using (10) withg_ireplaced by f, we have

₁

q

q X

j=1

(Im(λ,ℓ)g_j(z))p(z) +1−λ+ℓ

λ I

m₍_λ,_ℓ₎_f_{(z) =} 1+ℓ λ I

m+1₍

λ,ℓ)f(z). (19)

Differentiating (19) with respect toz, and simplifying, we obtain

z(Im+1₍_λ,_ℓ₎_f_(z))′

1 q

q P

j=1

Im+1₍_λ,_ℓ_)g j(z)

=p(z) + zp

′

(z)

Q(z) +1−λ+ℓ

λ

.

Since g₁, ...,g_n ∈Ω_m₊_1,_λ_,_ℓ(q;A,B), by Proposition 1, we know that g₁, ...,g_q ∈Ω_m_,_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;B6=−1), (20)

and

Re(Q(z))> 1−A

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

Q(z) + 1−λ+ℓ

λ =ρe

φπi/2_, where

1−A

1−B+

1−λ+ℓ λ < ρ <

1+A

1+B +

1−λ+ℓ

λ ,

−t1< φ <t1 (B6=−1),

t1 being given by (18), and

1−A

2 +

1−λ+ℓ

λ < ρ <∞

−1< φ <1 (B=−1).

We note thatp is analytic inU with p(0) =1. Let hbe the function which maps U onto the angular domain

§

φ:−π

2δ1<arg(φ)< π

2δ2

ª

, withh(0) =1. Applying Lemma 1 for thishwith

φ(z) = 1

Q(z) +1−λ_λ+ℓ ,

we see that

R(p(z))>0 (z∈U), and hence,p(z)6=0 inU.

If there exist two pointsz1andz2inU such that condition (7) is satisfied, then (By Lemma 3) we obtain (8) under restriction (9). For the caseB6=−1, we first obtain

arg p(z₁) + z1p

′

(z₁)

Q(z₁) + 1−λ+ℓ

λ

!

= −π

2α1+arg

1−iα1+α2

2 m

ρeφπi/2−1

≤ −π

2α1−tan

−1 (α1+α2)msin

_π

2

(1−φ)

2ρ+ (α1+α2)mcos

_π

2

(1−φ)

!

≤ −π

2α1−tan −1

  

(α1+α2)(1− |a|)cos

_π

2

t1

2₁1+₊A B+

1−λ+ℓ λ

(1+|a|) + (α1+α2)(1− |a|)sin

_π

2

t₁

  

= −π

(9)

and

arg p(z₂) + z2p

′

(z₂)

Q(z₂) +1−λ+ℓ

λ

!

≥ π

2α2+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

2δ2 ,

where we have used inequality (9), δ1, δ2 and t1 being given by (16), (17), and (18), re-spectively. Similarly, for the caseB=−1, we have

arg p(z1) +

z₁p′(z₁)

Q(z₁) +1−λ_λ+ℓ

!

≤ −π

2α1 and

arg p(z₂) + z2p

′

(z₂)

Q(z₂) +1−λ_λ+ℓ

!

≥ π

2α2 .

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

Puttingδ1=δ2=δin Theorem 1, we obtain the following corollary. Corollary 1. Let f ∈A and0< δ≤1. If

arg      

z(Im+1(λ,ℓ)f(z))′

1 q

_Pq

j=1

Im+1(λ,ℓ)gj(z)

      < π 2δ,

where g1, ...,gq∈Ωm,λ,ℓ(q;A,B), then

arg      

z(Im₍_λ,_ℓ₎_f_(z))′

1 q

q P

j=1

Im₍_λ,_ℓ_)g j(z)       <π 2α,

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

δ=

  

α+ 2

πtan

−1

αcosπ₂t1

1+A 1+B+

1−λ+ℓ λ

+αsinπ

2

t1

(B6=−1)

α (B=−1),

(10)

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= 1and B = −1, the class C_m_,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≤1and c≥0. If

−π

2δ1<arg

     

z(Im(λ,ℓ)f(z))′

1 q

_Pq

j=1

Im₍_λ,_ℓ₎_g j(z)

     

< π 2δ2,

−π

2α1<arg

     

z(Im(λ,ℓ)F_c(f)(z))′

1 q

q P

j=1

Im₍_λ,_ℓ_)F

c(gj)(z)

     

<π 2α2,

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

solutions of the following equations:

δ1=

  

α1+_π2tan−1

(α1+α2)(1−|a|)cos _π

2

t2

21+A

1+B+c

(1+|a|)+(α1+α2)(1−|a|)sin _π

2

t2

(B6=−1)

α1 (B=−1),

and

δ2=

  

α2+_π2tan−1

(α1+α2)(1−|a|)cos _π

2

t2

21+A

1+B+c

(1+|a|)+(α1+α2)(1−|a|)sin _π

2

t2

(B6=−1),

α1 (B=−1),

a being given by (9) and t₂ being the same as t₁given by (18) with c= 1−λ+ℓ

(11)

Proof. Let

p(z) = z(I

m₍_λ,_ℓ_)F

c(f)(z)) ′

1 q

_Pq

j=1

Im₍_λ,_ℓ_)F

c(gj)(z)

and Q(z) = 1 q

q

X

k=1

Q_k(z),

Q_k(z) = z(I

m₍_λ,_ℓ_)F

c(gk)(z)) ′

1 q

_Pq

j=1

Im₍_λ,_ℓ_)F

c(gj)(z) .

Using the relationship (14), we obtain

₁

q

q X

j=1

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

Differentiating (22) with respect toz, and simplifying, we get

z(Im(λ,ℓ)f(z))′

1 q

_Pq

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 haveFc(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.

Puttingδ1=δ2in Theorem 2 we obtain the following corollary. Corollary 3. Let f ∈A, 0< δ≤1and c≥0. If

arg      

z(Im(λ,ℓ)f(z))′

1 q

_Pq

j=1

Im₍_λ,_ℓ_)g j(z)       <π 2δ,

arg      

z(Im(λ,ℓ)F_c(f)(z))′

1 q

_Pq

j=1

Im₍_λ,_ℓ_)F

c(gj)(z)

(12)

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

δ=

  

α+ 2

πtan

−1

αcosπ₂t2

1+A 1+B+c

+αsinπ

2

t2

(B6=−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 ∈ C_m_,_λ_,_ℓ(q;A,B). Then F_c(f) ∈ C_m_,_λ_,_ℓ(q;A,B), where F_c 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=−1in 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≤1andγ≥0. If

−π

2δ1<arg

      γ z(I

m+1₍_λ,_ℓ₎_f_(z))′

1 q

_Pq

j=1

Im+1(λ,ℓ)g_j(z)

+ (1−γ) z(I

m₍_λ,_ℓ₎_f_(z))′

1 q

_Pq

j=1

Im(λ,ℓ)g_j(z)

      < π 2δ2,

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

−π

2α1<arg

     

z(Im(λ,ℓ)f(z))′

1 q

q

P

j=1

Im(λ,ℓ)g_j(z)

      < π 2α2 ,

whereα1 andα2 are the solutions of the following equation:

δ1=

  

α1+_π2tan−1

(α1+α2)(1−|a|)γcos _π

2

t1

21+A

1+B+ 1−λ+ℓ

λ

(1+|a|)+(α1+α2)(1−|a|)sin _π

2

t1

(B6=−1),

α1 (B=−1),

and

δ2=

  

α2+_π2tan−1

(α1+α2)(1−|a|)γcos _π 2 t1 2 1+A 1+B+

1−λ+ℓ λ

(1+|a|)+(α1+α2)(1−|a|)γsin _π

2

t1

(B6=−1),

α2 (B=−1),

(13)

Remark 4. For m=ℓ=0, λ=1, q=2, A=1, B=−1andδ=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. Puttingℓ=0in the above results, we obtain the corresponding results for the opera-tor D_λm.

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