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volume 7, issue 4, article 138, 2006.

Received 06 October, 2006;

accepted 15 November, 2006.

Communicated by:N.E. Cho

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Journal of Inequalities in Pure and Applied Mathematics

ON CERTAIN CLASSES OF MEROMORPHIC FUNCTIONS INVOLVING INTEGRAL OPERATORS

KHALIDA INAYAT NOOR

Mathematics Department

COMSATS Institute of Information Technology Islamabad, Pakistan.

EMail:khalidanoor@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 252-06

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On Certain Classes of Meromorphic Functions Involving Integral Operators

Khalida Inayat Noor

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J. Ineq. Pure and Appl. Math. 7(4) Art. 138, 2006

http://jipam.vu.edu.au

Abstract

We introduce and study some classes of meromorphic functions defined by using a meromorphic analogue of Noor [also Choi-Saigo-Srivastava] operator for analytic functions. Several inclusion results and some other interesting properties of these classes are investigated.

2000 Mathematics Subject Classification:30C45, 30C50.

Key words: Meromorphic functions, Functions with positive real part, Convolution, Integral operator, Functions with bounded boundary and bounded radius rotation, Quasi-convex and close-to-convex functions.

This research is supported by the Higher Education Commission, Pakistan, through research grant No: 1-28/HEC/HRD/2005/90.

1. Introduction

LetMdenote the class of functions of the form f(z) = 1

z +

∞

X

n=0

a_nzⁿ,

which are analytic inD={z : 0<|z|<1}.

LetP_k(β) be the class of analytic functions p(z) defined in unit discE = D∪ {0},satisfying the propertiesp(0) = 1and

(1.1)

Z 2π

0

Rep(z)−β 1−β

dθ≤kπ,

wherez = re^iθ, k ≥ 2and0 ≤ β < 1.Whenβ = 0,we obtain the classP_k defined in [14] and for β = 0, k = 2, we have the classP of functions with positive real part.

Also, we can write (1.1) as

(1.2) p(z) = 1

2 Z 2π

0

1 + (1−2β)ze^−it 1−ze^−it dµ(t),

whereµ(t)is a function with bounded variation on[0,2π]such that (1.3)

Z 2π

0

dµ(t) = 2, and

Z 2π

0

|dµ(t)| ≤k.

From (1.1), we can write, forp∈P_k(β),

(1.4) p(z) =

k 4 + 1

2

p₁(z)− k

4 − 1 2

p₂(z),

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where p₁, p₂ ∈P₂(β) =P(β), z∈E.

We define the functionλ(a, b, z)by λ(a, b, z) = 1

z +

∞

X

n=0

(a)_n+1 (c)n+1

zⁿ, z ∈D,

c 6= 0,−1,−2, . . . , a >0,where(a)nis the Pochhamer symbol (or the shifted factorial) defined by

(a)₀ = 1, (a)_n =a(a+ 1)· · ·(a+n−1), n >1.

We note that

λ(a, c, z) = 1

z²F₁(1, a;c, z),

2F₁(1, a;c, z)is Gauss hypergeometric function.

Letf ∈ M.Denote byL(a, c);˜ M −→ M,the operator defined by L(a, c)f(z) =˜ λ(a, c, z)? f(z), z∈D,

where the symbol ? stands for the Hadamard product (or convolution). The operator L(a, c)˜ was introduced and studied in [5]. This operator is closely related to the Carlson-Shaeffer operator [1] defined for the space of analytic and univalent functions inE,see [11,13].

We now introduce a function(λ(a, c, z))⁽⁻¹⁾ given by λ(a, c, z)?(λ(a, c, z))⁽⁻¹⁾ = 1

z(1−z)^µ, (µ >0), z∈D.

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Analogous toL(a, c),˜ a linear operatorI_µ(a, c)onMis defined as follows, see [2].

Iµ(a, c)f(z) = (λ(a, c, z))⁽⁻¹⁾? f(z), (1.5)

(µ >0, a >0, c6= 0,−1,−2, . . . , z ∈D).

We note that

I₂(2,1)f(z) =f(z), and I₂(1,1)f(z) = zf⁰(z) + 2f(z).

It can easily be verified that

z(I_µ(a+ 1, c)f(z))⁰ =aI_µ(a, c)f(z)−(a+ 1)I_µ(a+ 1, c)f(z), (1.6)

z(I_µ(a, c)f(z))⁰ =µI_µ+1(a, c)f(z)−(µ+ 1)I_µ(a, c)f(z).

(1.7)

We note that the operator Iµ(a, c)is motivated essentially by the operators defined and studied in [2,11].

Now, using the operatorI_µ(a, c),we define the following classes of meromorphic functions forµ >0, 0≤η, β <1, α≥0, z∈D.

We shall assume, unless stated otherwise, that a 6= 0,−1,−2, . . . , c 6=

0,−1,−2, . . .

Definition 1.1. A function f ∈ Mis said to belong to the class M Rk(η)for z ∈D,0≤η <1, k ≥2,if and only if

−zf⁰(z)

f(z) ∈P_k(η)

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andf ∈M V_k(η),for z ∈D, 0≤η <1, k ≥2,if and only if

−(zf⁰(z))⁰

f⁰(z) ∈P_k(η).

We callf ∈M R_k(η),a meromorphic function with bounded radius rotation of orderηandf ∈M V_ka meromorphic function with bounded boundary rotation.

Definition 1.2. Letf ∈ M, 0≤η <1, k ≥2, z∈D.Then

f ∈M R_k(µ, η, a, c) if and only if I_µ(a, c)f ∈M R_k(η).

Also

f ∈M V_k(µ, η, a, c) if and only if I_µ(a, c)f ∈M V_k(η), z ∈D.

We note that, forz ∈D,

f ∈M Vk(µ, η, a, c) ⇐⇒ −zf⁰ ∈M Rk(µ, η, a, c).

Definition 1.3. Let α ≥ 0, f ∈ M, 0 ≤ η, β < 1, µ > 0andz ∈ D. Then f ∈ B_k^α(µ, β, η, a, c), if and only if there exists a functiong ∈ M C(µ, η, a, c), such that

(1−α)(I_µ(a, c)f(z))⁰ (I_µ(a, c)g(z))⁰ +α

−(z(I_µ(a, c)f(z))⁰)⁰ (I_µ(a, c)g(z))⁰

∈Pk(β).

In particular, for α = 0, k = a = µ = 2, andc = 1, we obtain the class of meromorphic close-to-convex functions, see [4]. For α = 1, k = µ = a = 2, c = 1,we have the class of meromorphic quasi-convex functions defined for z ∈ D.We note that the classC^? of quasi-convex univalent functions, analytic inE,were first introduced and studied in [7]. See also [9,12].

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The following lemma will be required in our investigation.

Lemma 1.1 ([6]). Let u = u1 +iu2 and v = v1 +iv2 and let Φ(u, v) be a complex-valued function satisfying the conditions:

(i) Φ(u, v)is continuous in a domainD ⊂ C², (ii) (1,0)∈ DandΦ(1,0)>0.

(iii) Re Φ(iu2, v1)≤0 whenever (iu2, v1)∈ Dandv1 ≤ −¹₂(1 +u²₂).

Ifh(z) = 1+P∞

m=1cmz^mis a function, analytic inE,such that(h(z), zh⁰(z))∈ Dand Re(h(z), zh⁰(z))>0forz ∈E,thenReh(z)>0inE.

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2. Main Results

Theorem 2.1.

M Rk(µ+ 1, η, a, c)⊂M Rk(µ, β, a, c)⊂M Rk(µ, γ, a+ 1, c).

Proof. We prove the first part of the result and the second part follows by using similar arguments. Let

f ∈M R_k(µ+ 1, η, a, c), z ∈D and set

H(z) = k

4 +1 2

h₁(z)− k

4 − 1 2

h₂(z)

=−

z(I_µ(a, c)f(z))⁰ I_µ(a, c)f(z)

(2.1) ,

whereH(z)is analytic inEwithH(0) = 1.

Simple computation together with (2.1) and (1.7) yields (2.2) −

z(I_µ+1(a, c)f(z))⁰ I_µ+1(a, c)f(z)

=

H(z) + zH⁰(z)

−H(z) +µ+ 1

∈P_k(η), z ∈E.

Let

Φ_µ(z) = 1 µ+ 1

"

1 z +

∞

X

k=0

z^k

#

+ µ

µ+ 1

"

1 z +

∞

X

k=0

kz^k

# ,

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then

(H(z)? zΦµ(z)) =H(z) + zH⁰(z)

−H(z) +µ+ 1

= k

4 + 1 2

(h₁(z)? zΦ_µ(z))− k

4 − 1 2

(h₂(z)? zΦ_µ(z))

= k

4 + 1

2 h₁(z) + zh⁰₁(z)

−h1(z) +µ+ 1

− k

4 − 1

2 h₂(z) + zh⁰₂(z)

−h2(z) +µ+ 1

. (2.3)

Sincef ∈M Rk(µ+ 1, η, a, c),it follows from (2.2) and (2.3) that

h_i(z) + zh⁰_i(z)

−h_i(z) +µ+ 1

∈P(η), i= 1,2, z ∈E.

Leth_i(z) = (1−β)p_i(z) +β.Then

(1−β)pi(z) +

(1−β)zp⁰_i(z)

−(1−β)p_i(z)−β+µ+ 1

+ (β−η)

∈P, z ∈E.

We shall show thatp_i ∈P, i= 1,2.

We form the functionalΦ(u, v)by takingu = pi(z), v = zp⁰_i(z)withu = u₁ +iu₂, v = v₁ +iv₂. The first two conditions of Lemma1.1 can easily be verified. We proceed to verify the condition (iii).

Φ(u, v) = (1−β)u+ (1−β)v

−(1−β)u−β+µ+ 1 + (β−η),

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implies that

Re Φ(iu₂, v₁) = (β−η) + (1−β)(1 +µ−β)v1

(1 +µ−β)²+ (1−β)²u²₂. By takingv1 ≤ −¹₂(1 +u²₂),we have

Re Φ(iu2, v1)≤ A+Bu²₂ 2C , where

A= 2(β−η)(1 +µ−β)²−(1−β)(1 +µ−β), B = 2(β−η)(1−β)²−(1−β)(1 +µ−β), C = (1 +µ−β)²+ (1−β)²u²₂ >0.

We note thatRe Φ(iu₂, v₁)≤ 0if and only ifA ≤0andB ≤ 0.FromA ≤ 0, we obtain

(2.4) β = 1

4 h

(3 + 2µ+ 2η)−p

(3 + 2µ+ 2η)²−8 i

, andB ≤0gives us0≤β <1.

Now using Lemma1.1, we see that p_i ∈ P forz ∈ E, i = 1,2and hence f ∈M R_k(µ, β, a, c)withβ given by (2.4).

In particular, we note that β = 1

4 h

(3 + 2µ)−p

4µ²+ 12µ+ 1i .

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Theorem 2.2.

M V_k(µ+ 1, η, a, c)⊂M V_k(µ, β, a, , c)⊂M V_k(µ, γ, a+ 1, c).

Proof.

f ∈M V_k(µ+ 1, η, a, c) ⇐⇒ −zf⁰ ∈M R_k(µ+ 1, η, a, c)

⇒ −zf⁰ ∈M Rk(µ, β, a, c)

⇐⇒f ∈M V_k(µ, β, a, c), whereβis given by (2.4).

The second part can be proved with similar arguments.

Theorem 2.3.

B_k^α(µ+ 1, β₁, η₁, a, c)⊂ B_k^α(µ, β₂, η₂, a, c)⊂ B_k^α(µ, β₃, η₃, a+ 1, c), whereη_i =η_i(β_i, µ), i = 1,2,3are given in the proof.

Proof. We prove the first inclusion of this result and other part follows along similar lines. Let f ∈ B_k^α(µ+ 1, β₁, η₁, a, c). Then, by Definition 1.3, there exists a functiong ∈M V2(µ+ 1, η1, a, c)such that

(2.5) (1−α)

(I_µ+1(a, c)f(z))⁰ (I_µ+1(a, c)g(z))⁰

+α

−(z(I_µ+1(a, c)f(z))⁰)⁰ (I_µ+1(a, c)g(z))⁰

∈P_k(β₁).

Set

(2.6) p(z) = (1−α)

(Iµ(a, c)f(z))⁰ (I_µ(a, c)g(z))⁰

+α

−(z(Iµ(a, c)f(z))⁰)⁰ (I_µ(a, c)g(z))⁰

,

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wherepis an analytic function inEwithp(0) = 1.

Now,g ∈ M V₂(µ+ 1, η₁, a, c)⊂M V₂(µ, η₂, a, c),whereη₂ is given by the equation

(2.7) 2η₂²+ (3 + 2µ−2η₁)η₂−[2η₁(1 +µ) + 1] = 0.

Therefore,

q(z) =

−(z(I_µ(a, c)g(z))⁰)⁰ (I_µ(a, c)g(z))⁰

∈P(η₂), z ∈E.

By using (1.7), (2.5), (2.6) and (2.7), we have (2.8)

p(z) +α zp⁰(z)

−q(z) +µ+ 1

∈Pk(β1), q ∈P(η2), z ∈E.

With p(z) =

k 4 +1

2

[(1−β₂)p₁(z) +β₂]− k

4 − 1 2

[(1−β₂)p₂(z) +β₂], (2.8) can be written as

k 4 +1

2 (1−β2)p1(z) +α(1−β₂)zp⁰₁(z)

−q(z) +µ+ 1 +β2

− k

4 − 1

2 (1−β2)p2(z) +α(1−β2)zp⁰₂(z)

−q(z) +µ+ 1 +β2

,

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where

(1−β₂)p_i(z) +α(1−β₂)zp⁰_i(z)

−q(z) +µ+ 1 +β₂

∈P(β₁), z ∈E, i= 1,2.

That is

(1−β₂)p_i(z) +α(1−β₂)zp⁰_i(z)

−q(z) +µ+ 1 + (β₂−β₁)

∈P, z ∈E, i= 1,2.

We form the functionalΨ(u, v)by taking u= u₁+iu₂ =p_i, v =v₁+iv₂ = zp⁰_i,and

Ψ(u, v) = (1−β₂)u+α (1−β₂)v

(−q1+iq2) +µ+ 1 + (β₂ −β₁), (q=q₁+iq₂).

The first two conditions of Lemma1.1are clearly satisfied. We verify (iii), with v₁ ≤ −¹₂(1 +u²₂)as follows

Re Ψ(iu2, v1)

= (β₂−β₁) + Re

α(1−β₂)v₁{(−q₁+µ+ 1) +iq₂} (−q+µ+ 1)²+q²₂

≤ 2(β−2−β₁)| −q+µ+ 1|²−α(1−β₂)(−q₁+µ+ 1)(1 +u²₂) 2| −q+µ+ 1|²

= A+Bu²₂

2C , C =| −q+µ+ 1|² >0

≤0, if A≤0 and B ≤0,

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where

A= 2(β₂−β₁)| −q+µ+ 1|²−α(1−β₂)(−q₁+µ+ 1), B =−α(1−β₂)(−q₁+µ+ 1)≤0.

FromA≤0,we get

(2.9) β₂ = 2β₁| −q+µ+ 1|²+αRe(−q(z) +µ+ 1) 2| −q+µ+ 1|²+αRe(−q(z) +µ+ 1) .

Hence, using Lemma 1.1, it follows that p(z), defined by (2.6), belongs to P_k(β₂) and thus f ∈ B_k^α(µ, β₂, η₂, a, c), z ∈ D. This completes the proof of the first part. The second part of this result can be obtained by using similar arguments and the relation (1.6).

Theorem 2.4.

B_k^α(µ, β, η, a, c)⊂ B_k⁰(µ, γ, η, a, c) (i)

B^α_k¹(µ, β, η, a, c)⊂ B_k^α²(µ, β, η, a, c), for 0≤α₂ < α₁. (ii)

Proof. (i). Let

h(z) = (I_µ(a, c)f(z))⁰ (Iµ(a, c)g(z))⁰, h(z)is analytic inEandh(0) = 1.Then

(2.10) (1−α)

(I_µ(a, c)f(z))⁰ (I_µ(a, c)g(z))⁰

+α

−(z(I_µ(a, c)f(z))⁰)⁰ (I_µ(a, c)g(z))⁰

=h(z) +α zh⁰(z)

−h₀(z),

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where

h₀(z) = −(z(I_µ(a, c)g(z))⁰)⁰

(I_µ(a, c)g(z))⁰ ∈P(η).

Since f ∈ B_k^α(µ, β, η, a, c),it follows that

h(z) +α zh⁰(z)

−h₀(z)

∈P_k(β), h₀ ∈P(η), for z ∈E.

Let

h(z) = k

4 +1 2

h₁(z)− k

4 − 1 2

h₂(z).

Then (2.10) implies that

hi(z) +α zh⁰_i(z)

−h₀(z)

∈P(β), z ∈E, i= 1,2,

and from use of similar arguments, together with Lemma 1.1, it follows that h_i ∈P(γ), i= 1,2,where

γ = 2β|h₀|²+αReh₀ 2|h₀|²+αReh₀ .

Therefore h ∈ Pk(γ), and f ∈ B_k⁰(µ, γ, η, a, c), z ∈ D. In particular, it can be shown that h_i ∈ P(β), i = 1,2. Consequently h ∈ P_k(β) and f ∈ B_k⁰(µ, β, η, a, c)inD.

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Forα₂ = 0,we have (i). Therefore, we letα₂ >0and f ∈ B_k^α¹(µ, β, η, a, c).

There exist two functionsH1, H2 ∈Pk(β)such that H₁(z) = (1−α₁)

(I_µ(a, c)f(z))⁰ (I_µ(a, c)g(z))⁰

+α₁

−(z(I_µ(a, c)f(z))⁰)⁰ (I_µ(a, c)g(z))⁰

H₂(z) = (Iµ(a, c)f(z))⁰

(I_µ(a, c)g(z))⁰, g ∈M V₂(µ, η, a, c).

Now

(2.11) (1−α₂)

(I_µ(a, c)f(z))⁰ (I_µ(a, c)g(z))⁰

+α₂

−(z(I_µ(a, c)f(z))⁰)⁰ (I_µ(a, c)g(z))⁰

= α₂ α1

H₁(z) +

1− α₂ α1

H₂(z).

Since the class P_k(β)is a convex set [10], it follows that the right hand side of (2.11) belongs toP_k(β)and this shows that f ∈ B_k^α²(µ, β, η, a, c)forz ∈ D.

This completes the proof.

Letf ∈ M, b > 0and let the integral operatorF_bbe defined by (2.12) F_b(f) = F_b(f)(z) = b

z^b+1 Z z

0

t^bf(t)dt.

From (2.12), we note that

(2.13) z(I_µ(a, c)F_b(f)(z))⁰ =bI_µ(a, c)f(z)−(b+ 1)I_µ(a, c)F_b(f)(z).

Using (2.12), (2.13) with similar techniques used earlier, we can prove the following:

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Theorem 2.5. Letf ∈M R_k(µ, β, a, c),orM V_k(µ, β, a, c),orB_k^α(µ, β, η, a, c), forz ∈D.ThenFb(f)defined by (2.12) is also in the same class forz ∈D.

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References

[1] B.C. CARLSON ANDD.B. SCHAEFFER, Starlike and prestarlike hyper- geometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.

[2] N.E. CHO AND K. INAYAT NOOR, Inclusion properties for certain classes of meromorphic functions associated with Choi-Saigo-Srivastava operator, J. Math. Anal. Appl., 320 (2006), 779–786

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Math. Sci., 10 (1987), 241–258.

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[10] K.I. NOOR, On subclasses of close-to-convex functions of higher order, Inter. J. Math. Math. Sci., 15 (1992), 279–290.

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