Some Classes of Meromorphic Multivalent Functions with Positive Coefficients Involving Certain Linear Operator

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SOME CLASSES OF MEROMORPHIC MULTIVALENT

FUNCTIONS WITH POSITIVE COEFFICIENTS

INVOLVING CERTAIN LINEAR OPERATOR

Abdul Rahman S. Juma1_{, Hazha Zirar Hussain} 2

Department of Mathematics, Alanbar University, Ramadi-Iraq1,

Department of Mathematics, Salahaddin University,Erbil-Iraq2,

Abstract

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses Da,c,λ[p, α, A, B] and

P∗

p,a,c,λ[α, β] of meromorphically multivalent functions. In this paper, we obtain

co-efficient estimates, distortion theorems, radii of starlikeness and convexity and clo-sure theorems for the classDa,c,λ[p, α, A, B]. Several interesting results involving the

Hadamard product of functions belonging to the classDa,c,λ[p, α, A, B],P∗p,a,c,λ[α, β]

and P∗

p[α, β] are also derived. Also integral transforms of functions in the classes

P∗

p,a,c,λ[α, β] and

P∗

p[α, β] are studied.

Key Words : Linear operator, Meromorphic, Positive coefficients, Hadamard product.

AMS Subject Classification : Secondary 30C45.

1. Introduction

Let P

p denote the class of functions of the form:

f(z) = 1 zp +

∞ X

n=0

ap+nzp+n, (ap+n≥0;p∈N={1,2, ...}), (1)

which are analytic and p-valent in the punctured unit disk U∗ ₌ _{_z _∈

C : 0 < |z| <

1} =U − {0}; where U = {z ∈ _C: |z| < 1}. For functions f(z) ∈ P

p given by (1) and

g(z)∈P

p given by

g(z) = 1 zp +

∞ X

n=0

bp+nzp+n,(bp+n≥0),

1_E-mail:

dr−[email protected] 2_E-mail:

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we define the Hadamard product (or convolution) of two functions, f(z) and g(z) given

by

(f ∗g)(z) = 1 zp +

∞ X

n=0

ap+nbp+nzp+n = (g∗f)(z).

In terms of the Pochhammer symbol (θ)n given by

(θ)n=

Γ(θ+n) Γ(θ) =

1 (n = 0),

θ(θ+ 1)(θ+ 2)...(θ+n−1) (n∈_N),

we define the function ϕ(a, c,;z) by

ϕ(a, c,;z) = 1 zp +

∞ X

n=0

(a)n+1

(c)n+1

zn+p (2)

(z ∈ U∗_;_a_∈

R;c∈R−Z−0;Z

−

0 ={0,−1,−2, ...}).

Corresponding to the function ϕ(a, c,;z), Liu [8] and Liu and Srivastava [9] have

intro-duced a linear operator `p(a, c) which is defined by means of the following Hadamard

product (or convolution):

`p(a, c) = ϕ(a, c,;z)∗f(z). (3)

Just as in [8] and [9], it is easily verified from the definitions (2) and (3) that

z(`p(a, c))f(z))0 =a`p(a+ 1, c)f(z)−(a+p)`p(a, c)f(z).

We also note, for any integer m >−p and for f(z)∈P

p, that

`p(n+p,1)f(z) =Dm+p−1f(z) =

1

zp₍₁₋_z₎m+p ∗f(z),

where Dm+p−1f(z) is the differential operator studied by (among others) Uralegaddi and Somanatha [17] and Aouf [3].

Further M. K. Aouf et. al. [4] considered the generalized operators as follows:

Let

Fp,a,c,λ(z) = (1−λ)`p(a, c)f(z) +

λ

pz(`p(a, c)f(z))

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(f ∈P

p;p∈N; 0≤λ <

1 2),

so that, obviously,

Fp,a,c,λ(z) =

1−2λ zp +

∞ X

n=0

[1 +λ(n p]

(a)n+1

(c)n+1

ap+nzp+n (4)

(p∈_N; 0≤λ < 1 2),

since f(z)∈P

p is given by (1). From (4), it is easily verified that

zF_p,a,c,λ0 (z) =aFp,a+1,c,λ(z)−(a+p)Fp,a,c,λ(z).

We say that a function f(z) ∈ P

p is in the class Da,c,λ[p, α, A, B] if it satisfies the

following inequality:

zF_p,a,c,λ0 (z)

Fp,a,c,λ(z) +p

BzF 0

p,a,c,λ(z)

Fp,a,c,λ(z) + [pB+ (A−B)(p−α)]

<1 (z ∈ U∗), (5)

where the parameters A, B, α, p and λ are constrained as follows:

−1≤A < B ≤1, A+B ≥0,0≤α < p, p∈_N and 0≤λ < 1 2.

The class Da,a,0[p, α, A, B] = Q∗[p, α, A, B] was studied by Aouf [2] and Srivastava et

al.[15].

We observe also that

1. Da,c,λ[p, α,−β, β] =

P∗

p,a,c,λ[α, β] = {f(z) ∈

P∗

p :

zF_p,a,c,λ0 (z) Fp,a,c,λ(z)+p zF_p,a,c,λ0 (z)

Fp,a,c,λ(z)+2α−p

< β, 0 ≤ α <

p, p ∈_N,0< β ≤1,0≤α < p, z∈ U∗_}

2. Da,a,0[p, α,−β, β] =

P∗

p[α, β], the class of meromorphic p-valent starlike functions

of order α and type β,

={f(z)∈

∗ X

p

:

zf0(z)

f(z) +p

f0₍_z₎

f(z) + 2α−p

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Meromorphically multivalent functions have been extensively studied by (for example)

Mogra [10, 11], Uralegaddi and Ganigi [16], Aouf [1, 2], Srivastava et al. [15], Owa et al.

[12], Joshi and Srivastava [7], Liu [8], Liu and Srivastava [9], Aouf et al. [5], Raina and

Srivastava [13] and Yang [18].

In this paper we investigate various important properties and characteristics of the

class Da,c,λ[p, α, A, B], we obtain coefficient estimates, distortion theorems, radii of

star-likeness and convexity and closure theorems. Several interesting results involving the

Hadamard product of functions belonging to the classesDa,c,λ[p, α, A, B],

P∗

p,a,c,λ[α, β] and

P∗

p[α, β] are also derived. Also integral transforms for functions in the classes

P∗

p,a,c,λ[α, β]

and P∗_p[α, β] are studied.

2. Coefficient Estimates

Theorem 2.1 : A function f(z) defined by (1) is said to be in the class Da,c,λ[p, α, A, B]

if and only if

∞ X

n=0

[1+λ(n p)]

(a)n+1

(c)n+1

[(1+B)n+2p+2αB+(B+A)(p−α)]ap+n≤(1−2λ)(B−A)(p−α), (6)

where 1≤A < B ≤1, A+B ≥0,0≤α < p, p ∈_N and 0≤λ < 1₂.

proof : We assume that the inequality (6) holds true. Then, if we let z ∈ ∂U, we find

from (1) and (6) that

zF_p,a,c,λ0 (z)

Fp,a,c,λ(z) +p

BzF 0

p,a,c,λ(z)

≤

P∞

n=0[1 +λ(

n p)]

(a)n+1

(c)n+1(2p+n)ap+n

(1−2λ)(B−A)(p−α)−P∞_n₌₀[1 +λ(n_p)](a)n+1

(c)n+1[B(n+ 2α) + (B+A)(p−α)]ap+n

≤1(z ∈∂U ={z∈_C:|z|= 1}

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Conversely, let f(z)∈Da,c,λ[p, α, A, B] be given by (1). Then, from (1) and (6), we have

zF_p,a,c,λ0 (z)

Fp,a,c,λ(z) +p

BzF 0

p,a,c,λ(z)

=

P∞

n=0[1 +λ(

n p)]

(a)n+1

(c)n+1(2p+n)ap+nz 2p+n

(1−2λ)(B−A)(p−α)−P∞_n₌₀[1 +λ(n_p](a)n+1

(c)n+1[B(n+ 2α) + (B+A)(p−α)]ap+nz 2p+n

<1 forz ∈ U∗

Since |<(z)≤ |z|(z ∈_C), we have

<

( P∞

n=0[1 +λ(

n p)]

(a)n+1

(c)n+1(2p+n)ap+nz 2p+n

(1−2λ)(B−A)(p−α)−P∞_n₌₀[1 +λ(n_p)](a)n+1

(c)n+1[B(n+ 2α) + (B +A)(p−α)]ap+nz 2p+n

)

<1.

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Choose values ofzon the real axis so that zF 0

p,a,c,λ(z)

Fp,a,c,λ(z) is real. Upon clearing the denominator

in (7) and letting z →1− through real values, we obtain (7).

Hence the proof is complete.

Corollary 2.1Let the functionf(z) defined by (1) be in the classDa,c,λ[p, α, A, B]. Then

ap+n ≤

(1−2λ)(B −A)(p−α)

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]. (c)n+1

(a)n+1

, (n≥0)

The result is sharp for the function:

f(z) = 1 zp +

(1−2λ)(B−A)(p−α)

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]. (c)n+1

(a)n+1

zp+n (8)

Putting A=−β and B =β(0< β≤1) in Theorem 2.1, we obtain:

Corollary 2.2 A function f(z) defined by (1) is in the classP∗_p,a,c,λ[α, β] if and only if

∞ X

n=0

[1 +λ(n p]

(a)n+1

(c)n+1

[(1 +B)n+ 2(p+αβ)]ap+n≤(1−2λ)2β(p−α).

Putting λ= 0 anda =c in Corollary 2.2, we obtain:

Corollary 2.3 A function f(z) defined by (1) is in the classP∗_p[α, β] if and only if

∞ X

n=0

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Putting A=−1, B = 1, a=cand λ= 0 in Theorem 2.1, we obtain:

Corollary 2.4 A function f(z) defined by (1) is in the classP∗_p[α] if and only if

∞ X

n=0

(n+p+α)ap+n ≤(p−α).

3. Distortion Theorem

Theorem 3.1 : If the function f(z) defined by (1) is in the class Da,c,λ[p, α, A, B], then

for 0<|z|=r <1, we have

1 rp −

c(1−2λ)(B−A)(p−α) a[2p+ 2αB+ (B+A)(p−α)]r

p _{≤ |}_f₍_z_{)| ≤} 1

rp +

c(1−2λ)(B−A)(p−α) a[2p+ 2αB+ (B+A)(p−α)]r

p_.

The result is sharp.

proof : In view of Theorem 2.1, we have

a

c[2p+2αB+(B+A)(p−α)]

∞ X

n=0

ap+n≤

∞ X

n=0

[1+λ(n p)]

(a)n+1

(c)n+1

[(1+B)n+2p+2αB+(B+A)(p−α)]ap+n

≤(1−2λ)(B−A)(p−α),

that is, that

∞ X

n=0

ap+n≤

(1−2λ)(B−A)(p−α) [2p+ 2αB+ (B+A)(p−α)].

c a.

Then, for o <|z|= 1<1,

|f(z)|=| 1 zp +

∞ X

n=0

ap+nzp+n|,

|f(z)| ≤ 1 rp +

∞ X

n=0

ap+nrp+n,

≤ 1 rp +r

p

∞ X

n=0

ap+n

≤ 1 rp +

(1−2λ)(B−A)(p−α) [2p+ 2αB+ (B+A)(p−α)].

c ar

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and

|f(z)|=| 1 zp +

∞ X

n=0

ap+nzp+n|,

|f(z)| ≥ 1 rp −

∞ X

n=0

ap+nrp+n,

≥ 1 rp −r

p

∞ X

n=0

ap+n

≥ 1 rp −

(1−2λ)(B−A)(p−α) [2p+ 2αB+ (B+A)(p−α)].

c ar

p

The bounds for |f(z)| are sharp and are attained for the function

f(z) = 1 zp +

(1−2λ)(B−A)(p−α) [2p+ 2αB+ (B+A)(p−α)].

c az

p

at z =r, z =re2piπ_.

Next we proof the following growth and distortion properties for the classDa,c,λ[p, α, A, B].

Theorem 3.2 : If the functionf(z) defined by (1) is in the classDa,c,λ[p, α, A, B]. Then

(p+m−1)! (p−1)! r

−(p+m)₋ c.p!(1−2λ)(B−A)(p−α)

a(p−m)![2p+ 2αB+ (B+A)(p−α)].r

p+n−m _{≤ |}

f(m)(z)|

≤ (p+m−1)! (p−1)! r

−(p+m)₊ c.p!(1−2λ)(B −A)(p−α)

a(p−m)![2p+ 2αB+ (B +A)(p−α)].r

p+n−m_.

(0<|z|=r <1;a > c >0;m ∈_N0 =N∪ {0};p∈N;p > m).

The result is sharp for the function f(z) given by

f(z) = 1 zp +

c(1−2λ)(B −A)(p−α) a[2p+ 2αB+ (B+A)(p−α)]z

p

at z =r, z =re2piπ.

proof : In view of Theorem 2.1, we have

a

p!c[2p+ 2αB+ (B+A)(p−α)]

∞ X

n=0

(8)

≤

∞ X

n=0

[1 +λ(n p)]

(a)n+1

(c)n+1

[(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]ap+n

≤(1−2λ)(B−A)(p−α),

which yields

∞ X

n=0

(p+n)!ap+n ≤

c.p!(1−2λ)(B−A)(p−α)

a[2p+ 2αB+ (B+A)(p−α)](p∈N). (9)

Now, by differentiating both sides of (1) m times with respect toz, we have

f(m)(z) = (−1)m(p+m−1)!

(p−1)! z

−(p+m)₊

∞ X

n=0

(p+n)!

(p+n−m)!ap+nz

p+n−m₍_p_∈

N, m∈N0;p > m),

(10)

and Theorem 3.2 follows easily from (9) and (10).

Next we determine the radii of meromorphically p-valent starlikeness of order δ(0 ≤

δ < p) and meromorphically p-valent convexity of order δ(0≤δ < p) for functions in the

class Da,c,λ[p, α, A, B].

4. Radii of Sarlikness and Convexity

Theorem 4.1 : Let the function f(z) defined by (1) be in the class Da,c,λ[p, α, A, B].

Then

1. f(z) is meromorphically p-valent starlike of order δ(0≤δ < p) in the disk |z|< r1,

that is,

<{−zf_f₍0_z(z₎)}> δ (|z|< r1; 0≤δ < p, n ∈N),

where

r1(p, α, A, B, a, c) =

h₍_p₋_δ_)[1+_λ₍n

p)][(1+B)n+2p+2αB+(B+A)(p−α)] (n+p+δ)(1−2λ)(B−A)(p−α)

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

i_2p+n1

(n≥0).

2. f(z) is meromorphically p-valent convex of order δ(0≤δ < p) in the disk |z|< r1,

that is,

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where

r2(p, α, A, B, a, c) =

h_p₍_p₋_δ_)[1+_λ₍n

p)][(1+B)n+2p+2αB+(B+A)(p−α)] (p+n)(p+n+δ)(1−2λ)(B−A)(p−α) .

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

i_2p+n1

(n≥0).

Each of these results is sharp for the function f(z) given by (8).

proof : 1. It is sufficient to show that

zf0₍_z₎

f(z) +p

zf0₍_z₎

f(z) −p+ 2δ

≤1.

Note that

zf0(z)

f(z) +p

zf0₍_z₎

f(z) −p+ 2δ

≤ P∞

n=0(2p+n)ap+n|z|p+n

2(p−δ)−P∞_n₌₀(n+ 2δ)ap+n|z|2p+n

.

Thus, we have the desired inequality

zf0₍_z₎

f(z) +p

zf0₍_z₎

f(z) −p+ 2δ

≤1 (0≤δ < p, n∈_N), (11)

if

∞ X

n=0

((n+p+δ)

(p−δ) ap+n|z|

2p+n_≤₁_.

But Theorem 2.1 ensures that

∞ X

n=0

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)] (1−2λ)(B−A)(p−α) .

(a)n+1

(c)n+1

ap+n≤1. (12)

In view of (12), it follows that (11) will be true if

(n+p+δ p−δ )|r|

2p+n _≤ [1 +λ( n p)]

(a)n+1

(c)n+1[(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]

(1−2λ)(B−A)(p−α) ,

or if

r ≤

"

(p−δ)[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B +A)(p−α)] (n+p+δ)(1−2λ)(B −A)(p−α) .

(a)n+1

(c)n+1

#_2p+n1

(n ≥0).

(13)

(10)

2. In order to prove the second assertion of Theorem 4.1, it sufficient to show that

1 + zf_f000₍(_zz₎) +p 1 + 1+_ff0₍00_z(₎z)−p+ 2δ

≤1 (0≤δ < p, n∈_N).

Note that

1 + zf_f000₍(_zz₎) +p 1 + f_f000₍(_zz₎) −p+ 2δ

≤

P∞

n=0(n+p)(n+ 2p)ap+n|z|2p+n

2p(p−δ)−P∞_n₌₀(p+n)(n+ 2δ)ap+n|z|2p+n

.

Thus we have the desired inequality

1 + zf_f000₍(_zz₎) +p 1 + f_f000₍(_zz₎) −p+ 2δ

≤1 (0 ≤δ < p, n ∈_N),

∞ X

n=0

(p+n)(p+n+δ) p(p−δ) ap+nr

2p+n_≤₁_. ₍₁₄₎

By Theorem 2.1, (14) will be true if

(p+n)(p+n+δ) p(p−δ) r

2p+n _≤ [1 +λ( n

p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]

(1−2λ)(B−A)(p−α) . (a)n+1

(c)n+1

.

or if

r≤

"

p(p−δ)[1 +λ(n_p)](a)n+1

(c)n+1[(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]

(p+n)(p+n+δ)(1−2λ)(B−A)(p−α)

# 1 2p+n

(n≥0).

(15)

Setting r=r2(p, α, A, B, λ, a, c) in (15), the result follows, and the proof of Theorem 4.1

completed by merely verifying that each assertion is sharp for the function f(z) given by

(8).

5. Closure Theorems

Theorem 5.1 : Let

fp−1(z) =

1 zp,

and

fp+n(z) =

1 zp +

(1−2λ)(B−A)(p−α) [1 +λ(n

p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]

(c)n+1

(a)n+1

(11)

Then f(z) is in the class Da,c,λ[p, α, A, B] if and only if it can be expressed in the form

f(z) =

∞ X

n=−1

µp+nfp+n(z),

where µp+n≥0 and

P∞

n=−1µp+n = 1.

proof : First suppose that f(z) can be expressed of the form

f(z) =

∞ X

n=−1

µp+nfp+n(z)

= 1 zp +

∞ X

n=0

µp+n

(1−2λ)(B−A)(p−α)

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)] (c)n+1

(a)n+1

zp+n

Then

∞ X

n=0

µp+n

(1−2λ)(B−A)(p−α)

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)] (c)n+1

(a)n+1

.[1 +λ(

n

p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]

(1−2λ)(B−A)(p−α) . (a)n+1

(c)n+1

=

∞ X

n=0

µp+n = 1−µp−1 ≤1,

which shows that f(z)∈Da,c,λ[p, α, A, B].

Conversely, suppose that f ∈Da,c,λ[p, α, A, B]. Then

ap+n ≤

(1−2λ)(B−A)(p−α)

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)] (c)n+1

(a)n+1

(n≥0),

and setting

µp+n=

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)] (1−2λ)(B −A)(p−α) .

(a)n+1

(c)n+1

ap+n (n≥0),

and

µp−1 = 1−

∞ X

n=0

λp+n,

it follows that f(z) =P∞_n₌₋₁µp+nfp+n(z).

(12)

Theorem 5.2: The classf ∈Da,c,λ[p, α, A, B] is closed under convex linear combinations.

proof : Let each of the functions

fj(z) =

1 zp +

∞ X

n=0

ap+n,jzp+n, (j = 1,2) (16)

be in the class Da,c,λ[p, α, A, B]. It is sufficient to show that the functionh(z) defined by

h(z) = (1−t)f1(z) +tf2(z), (0≤t≤1)

is also in the class Da,c,λ[p, α, A, B]. Since

h(z) = 1 zp +

∞ X

n=0

[(1−t)ap+n,1+tap+n,2]zp+n (0≤t ≤1),

with the aid of Theorem 2.1, we have

∞ X

n=0

[1 +λ(n p)]

(a)n+1

(c)n+1

[(1 +B)n+ 2p+ 2αB+ (B +A)(p−α)][(1−t)ap+n,1+tap+n,2]

= (1−t)

∞ X

n=0

[1 +λ(n p)]

(a)n+1

(c)n+1

[(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]ap+n,1

+t

∞ X

n=0

[1 +λ(n p)]

(a)n+1

(c)n+1

[(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]ap+n,2

≤(1−t)(1−2λ)(B −A)(p−α) +t(1−2λ)(B −A)(p−α)

= (1−2λ)(B −A)(p−α),

which shows that h(z)∈Da,c,λ[p, α, A, B].

This completes the proof of the theorem.

6. Convolution Properties

For functions fj(z)(j = 1,2) defined by (16) belonging to the class

P∗

p, we denote by

(f1∗f2)(z) the convolution (or Hadamard product) of the functionsf1(z) and f2(z); that

is

(f1 ∗f2)(z) =

1 zp +

∞ X

n=0

(13)

Theorem 6.1 : Let the functions f1(z) defined by (16) be in the class Da,c,λ[p, α, A, B],

and the function f2(z) defined by (16) be in the classDa,c,[p, γ, A, B]. Then (f1∗f2)(z)∈

Da,c,λ[p, τ, A, B], where

τ ≤pn1− _a_[2_p₊₂_αB₊₍_B₊_A₎₍_p₋2_αc(1_)][2−_p2₊₂λ)(1+_γB₊₍B)(_BB₊−_AA₎₍)(_p₋p−_γ_)]+(1α)(p−₋γ₂)_λ₎₍_B₋_A₎2₍_p₋_α₎₍_p₋_γ₎ o

.

The result is sharp for the functions fj(z)(j = 1,2) given by

f1(z) =

1 zp +

c(1−2λ)(B −A)(p−α) a[2p+ 2αB+ (B +A)(p−α)]z

p

, (p∈_N)

f2(z) =

1 zp +

c(1−2λ)(B−A)(p−γ) a[2p+ 2γB+ (B+A)(p−γ)]z

p_, ₍_p_∈

N).

proof : Employing the technique used earlier by Schild and Silverman [13], we need to

find the largest τ such that

∞ X

n=0

[1 +λ(n_p)][(1 +B)n+ 2p+ 2τ B+ (B +A)(p−τ)] (1−2λ)(B−A)(p−τ) .

(a)n+1

(c)n+1

ap+n,1ap+n,2 ≤1,

for f1(z)∈Da,c,λ[p, α, A, B], and f2(z)∈Da,c,λ[p, γ, A, B].

Since f1(z)∈Da,c,λ[p, α, A, B], and f2(z)∈Da,c,λ[p, γ, A, B], we readily see that

∞ X

n=0

(a)n+1

(c)n+1

ap+n,1 ≤1,

∞ X

n=0

[1 +λ(n_p)][(1 +B)n+ 2p+ 2γB + (B+A)(p−γ)] (1−2λ)(B−A)(p−γ) .

(a)n+1

(c)n+1

ap+n,2 ≤1.

Therefore, by the Cauchy- Schwarz inequality, we obtain

P∞

n=0

[1+λ(n_p)] (1−2λ)(B−A)).

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

q

[(1+B)n+2p+2αB+(B+A)(p−α)][(1+B)n+2p+2γB+(B+A)(p−γ)] (p−α)(p−γ)

√

ap+n,1ap+n,2 ≤

1.(17)

This implies that we need only to show that

[(1 +B)n+ 2p+ 2τ B+ (B+A)(p−τ)]

(p−τ) ap+n,1ap+n,2 ≤

q

[(1+B)n+2p+2αB+(B+A)(p−α)][(1+B)n+2p+2γB+(B+A)(p−γ)]

(p−α)(p−γ) .

√

ap+n,1ap+n,2 (n≥0),

(14)

√

ap+n,1ap+n,2 ≤

(p−τ)√[(1+B)n+2p+2αB+(B+A)(p−α)][(1+B)n+2p+2γB+(B+A)(p−γ)]

√

(p−α)(p−γ)[(1+B)n+2p+τ B+(B+A)(p−τ)] (n≥0).

Hence, by the inequality (17), it is sufficient to prove that

(1−2λ)(B−A)√(p−α)(p−γ)

[1+λ(n_p)]√[(1+B)n+2p+2αB+(B+A)(p−α)][(1+B)n+2p+2γB+(B+A)(p−γ)]. (c)n+1 (a)n+1

≤ (p−τ)

p

[(1 +B)n+ 2p+αB+ (B +A)(p−α)][(1 +B)n+ 2p+ 2γB+ (B+A)(p−γ)]

p

(p−α)(p−γ)[(1 +B)n+ 2p+ 2τ B+ (B +A)(p−τ)] (n ≥0) (18)

It follows from (18) that

τ ≤p−_[1+_λ₍n (n+2p)(1−2λ)(1+B)(B−A)(p−α)(p−γ)

p)][(1+B)n+2p+2αB+(B+A)(p−α)][(1+B)n+2p+2γB+(B+A)(p−γ)]+(1−2λ)(B−A)2(p−α)(p−γ) (c)n+1 (a)n+1 (n≥

0).

Now, defining the function ϕ(n) by

ϕ(n) =p−_[1+_λ₍n (n+2p)(1−2λ)(1+B)(B−A)(p−α)(p−γ)

p)][(1+B)n+2p+2αB+(B+A)(p−α)][(1+B)n+2p+2γB+(B+A)(p−γ)]+(1−2λ)(B−A)

2₍_p₋_α₎₍_p₋_γ₎. (c)n+1 (a)n+1.

We see that ϕ(n) is an increasing function ofn(n ≥0).

Therefore, we conclude that

τ ≤ϕ(0) =pn1− _a_[2_p₊₂_αB₊₍_B₊_A₎₍_p₋2_αc_)][2(1−_p2₊₂λ)(1+_γB₊₍B_B)(B₊_A−A₎₍_p)(₋p_γ−_)]+(1α)(p−₋γ₂)_λ₎₍_B₋_A₎2₍_p₋_α₎₍_p₋_γ_)] o

.

Corollary 6.1 : Let the functions f1(z) defined by (16) be in the class

P∗

p,a,c,λ[α, β],

and the function f2(z) defined by (16) be in the class

P∗

p,a,c,λ[γ, β]. Then (f1∗f2)(z) ∈

P∗

p,a,c,λ[τ, β], where

τ ≤p

1− 2c(1−2λ)β+ (1 +B)(p−α)(p−γ) a(p+αβ)(p−γβ) + (1−2λ)β2₍_p₋_α₎₍_p₋_γ₎

.

f1(z) =

1 zp +

c(1−2λ)β(p−α) a(p+αβ) z

p_, ₍_p_∈

N)

f2(z) =

1 zp +

c(1−2λ)β(p−γ) a(p+γβ) z

p_, ₍_p_∈

N).

(15)

Corollary 6.2 : Let the functions f1(z) defined by (16) be in the class

P∗

p[α, β], and

the function f2(z) defined by (16) be in the class

P∗

p[γ, β]. Then (f1∗f2)(z)∈

P∗

p[τ, β],

where

τ ≤p

1− cβ(1−β)(p−α)(p−γ)

a(p+αβ)(p+γβ) +β2₍_p₋_α₎₍_p₋_γ₎

.

f1(z) =

1 zp +

cβ(p−α) a(p+αβ)z

p_, ₍_p_∈

N)

f2(z) =

1 zp +

cβ(p−γ) a(p+γβ)z

p

, (p∈_N).

Using an argument similar to those in the proof of Theorem 6.1, we obtain the following

result:

Theorem 6.2: Let the functionsfj(z)(j = 1,2) defined by (16) be in the classDa,c,λ[p, α, A, B].

Then (f1∗f2)(z)∈Da,c,λ[p, γ, A, B], where

γ ≤p

1− 2c

2₍₁₋₂_λ_{)(1 +}_B₎₍_B₋_A₎₍_p₋_α₎2

a2_[2_p_{+ 2}_αB_{+ (}_B₊_A₎₍_p₋_α_)]2_{+ (1}₋₂_λ_)[(_B ₋_A₎₍_p₋_α_)]2

.

fj(z) =

1 zp +

c(1−2λ)(1 +B)(B−A)(p−α) a[2p+ 2αB+ (B +A)(p−α)] z

p_, ₍_j _{= 1}_,_2;_p_∈

N) (19)

Corollary 6.3 : Let the functions fj(z)(j = 1,2) defined by (16) be in the class

P∗

p,a,c,λ[α, β]. Then (f1∗f2)(z)∈

P∗

p,a,c,λ[γ, β], where

γ ≤p

1− c(1−2λ)β(1 +B)(p−α)

2

a(p+αβ)2_{+ (1}₋₂_λ₎_β2₍_p₋_α₎2

.

fj(z) =

1 zp +

c(1−2λ)β(p−α) a(p+αβ) z

p_, ₍_j _{= 1}_,_2;_p_∈

(16)

Putting λ= 0 in Corollary 6.3, we obtain:

Corollary 6.4: Let the functionsfj(z)(j = 1,2) defined by (16) be in the class

P∗

p[α, β].

Then (f1∗f2)(z)∈

P∗

p[γ, β], where

γ ≤p

1− cβ(1 +β)(p−α)

2

a(p+αβ)2₊_β2₍_p₋_α₎2

.

fj(z) =

1 zp +

cβ(p−α) a(p+αβ)z

p_, ₍_j _{= 1}_,_2;_p_∈

N)

Theorem 6.3 : If f1(z) = _z1p + P∞

n=0ap+n,1z

p+n _∈ _D

a,c,λ[p, α, A, B] and f2(z) = _z1p + P∞

n=0ap+n,2z

p+n _∈ _D

a,c,λ[p, α, A, B] with |ap+n,2| ≤ 1, n = 1,2, ...,, then (f1 ∗f2)(z) ∈

Da,c,λ[p, α, A, B].

proof : Since

∞ X

n=0

(a)n+1

(c)n+1

|ap+n,1ap+n,2|

=

∞ X

n=0

(a)n+1

(c)n+1

ap+n,1|ap+n,2|,

≤

∞ X

n=0

(a)n+1

(c)n+1

ap+n,1,

≤1.

By Theorem 2.1, it follows that (f1∗f2)(z)∈Da,c,λ[p, α, A, B].

Corollary 6.5 : If f1(z) = _z1p + P∞

n=0ap+n,1z

p+n _∈ _D

a,c,λ[p, α, A, B] and f2(z) = _z1p + P∞

n=0ap+n,2z

p+n_∈_D

a,c,λ[p, α, A, B] with (0 ≤ap+n,2 ≤1, n= 1,2, ...,), then (f1∗f2)(z)∈

Da,c,λ[p, α, A, B].

Theorem 6.4: Let the functionsfj(z)(j = 1,2) defined by (16) be in the classDa,c,λ[p, α, A, B]

and

2(p+αβ)a

(17)

then the function h(z) defined by

h(z) = 1 zp +

∞ X

n=0

(a2_p₊_n,₁+a_p2₊_n,₂)zp+n, (20)

belongs to the class Da,c,λ[p, α, A, B].

proof : Since f1(z)∈Da,c,λ[p, α, A, B], we get:

∞ X

n=0

(a)n+1

(c)n+1

ap+n,1 ≤1.

and so

∞ X

n=0

"

(a)n+1

(c)n+1

#2

a2_p₊_n,₁ ≤1.

Similarly, since f2(z)∈Da,c,λ[p, α, A, B], we have

∞ X

n=0

"

(a)n+1

(c)n+1

#2

a2_p₊_n,₂ ≤1.

Hence

∞ X

n=0

1 2

"

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B +A)(p−α)] (1−2λ)(B −A)(p−α) .

(a)n+1

(c)n+1

#2

(a2_p₊_n,₂+a2_p₊_n,₂)≤1.

In view of Theorem 2.1, it is sufficient to show that

∞ X

n=0

"

(a)n+1

(c)n+1

#

(a2_p₊_n,₁+a2_p₊_n,₂)≤1.

(21)

Thus the inequality (21) will be satisfied if, for n= 0,1,2, ...

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B +A)(p−α)] (1−2λ)(B−A)(p−α) .

(a)n+1

(c)n+1

≤ 1 2

"

(a)n+1

(c)n+1

#2

,

or if

[1+λ(n p)]

(a)n+1

(c)n+1

(18)

for n = 0,1,2, .... Then left hand side of (22) is an increasing function of n, and hence

(22) is satisfied for all n if

2(p+αβ)a

c + (p−α)[3A−B+ 4λ(B−A)]≥0,

which is true by our assumption.

Putting λ= 0 in Theorem 6.4, we obtain:

Corollary 6.6 : Let the functions fj(z)(j = 1,2) defined by (16) be in the class

Da,c[p, α, A, B] and

2(p+αβ)c

a + (p−α)(3A−B)≥0,

then the function h(z) defined by (20) belongs to the class Da,c[p, α, A, B].

Theorem 6.5: Let the functionsfj(z)(j = 1,2) defined by (16) be in the classDa,c,λ[p, α, A, B].

Then the function h(z) defined by (20) belongs to the classDa,c,λ[p, γ, A, B], where

γ ≤p

1− 4c

2₍₁₋₂_λ_{)(1 +}_B₎₍_B₋_A₎₍_p₋_α₎2

a2_[2_p_{+ 2}_αB_{+ (}_B ₊_A₎₍_p₋_α_)]2_{) + 2(1}₋₂_λ_)[(_B ₋_A₎₍_p₋_α_)]2

.

The result is sharp for the functions fj(z)(j = 1,2) defined by (19).

proof : Noting that

∞ X

n=0

[[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B +A)(p−α)]]2 [(1−2λ)(B −A)(p−α)]2 .[

(a)n+1

(c)n+1

]2a2_p₊_n,j

≤

∞ X

n=0

(a)n+1

(c)n+1

ap+n,j

!2

≤1 (j = 1,2),

for fj(z)∈Da,c,λ[p, γ, A, B](j = 1,2), we have

∞ X

n=0

1 2

"

[1 +λ(n_p)][(1 +B)n+ 2p+ 2αB+ (B +A)(p−α)] (1−2λ)(B −A)(p−α) .

(a)n+1

(c)n+1

#2

(a2_p₊_n,₁+a2_p₊_n,₂)≤1.

Therefore, we have to find the largest γ such that

(19)

≤ [1 +λ(

n

p)][(1 +B)n+ 2p+ 2αB+ (B+A)(p−α)]

2

2(1−2λ)(B−A)(p−α)2 .

(a)n+1

(c)n+1

, (n≥0)

that is, that

γ ≤p−

2(2p+n)(1−2λ)(1 +B)(B −A)(p−α)2

[1 +λ(n_p)](a)n+1 (c)n+1]

2_{[(1 +}_B₎_n_{+ 2}_p_{+ 2}_αB_{+ (}_B ₊_A₎₍_p₋_α_)]2_{+ 2(1}₋₂_λ_)[(_B ₋_A₎₍_p₋_α_)]2. (n ≥0)

Now, defining the function ϕ(n) by

ϕ(n) = p−

2(2p+n)(1−2λ)(1 +B)(B −A)(p−α)2

[1 +λ(n_p)](a)n+1 (c)n+1]

2_{[(1 +}_B₎_n_{+ 2}_p_{+ 2}_αB_{+ (}_B ₊_A₎₍_p₋_α_)]2_{+ 2(1}₋₂_λ_)[(_B ₋_A₎₍_p₋_α_)]2. (n ≥0)

We see that ϕ(n) is an increasing function ofn.Thus, we conclude that

γ ≤ϕ(0) =p

1− 4c(1−2λ)(1 +B)(B −A)(p−α)

2

a[2p+ 2αB+ (B+A)(p−α)]2_{+ 2(1}₋₂_λ_)[(_B₋_A₎₍_p₋_α_)]2

.

Putting A=−β and B =β(0< β≤1) and λ= 0 in Theorem 6.5, we obtain:

Corollary 6.7: Let the functionsfj(z)(j = 1,2) defined by (16) be in the class

P∗

p[α, β].

Then the function h(z) defined by (20) belongs to the classP∗_p[γ, β], where

γ ≤p

1− 2β(1 +β)(p−α)

2

(p+αβ)2_{+ 2}_β2₍_p₋_α₎

.

The result is sharp for the functions fj(z)(j = 1,2) defined by (19).

7. Integral Transforms

In this section, we consider integral transforms of functions in the classesP∗_p,a,c,λ[α, β] and P∗_p[α, β].

Theorem 7.1 : If f(z) is in the class P∗_p,a,c,λ[α, β], then the integral transforms

Fd+p−1(z) =d

Z 1

0

(20)

are in the class P∗_p[δ],0≤δ < p, where

δ =δ(α, β, a, c, d, p, λ) = p

₍_d_{+ 2}_p₎₍_p₊_αβ₎a

c −β(p−α)d(1−2λ)

(d+ 2p)(p+αβ)a_c +β(p−α)d(1−2λ)

.

The result is the best possible for the function f(z) given by

f(z) = 1 zp +

(1−2λ)β(p−α) p+αβ .

c az

p_, ₍_p_∈

N)

proof : Suppose f(z) = _z1p + P∞

n=0ap+nz

p+n_∈P∗

p,a,c,λ[α, β]. Then we have

Fd+p−1(z) = d

Z 1

0

ud+p−1f(uz)du= 1 zp +

∞ X

n=0

dap+n

n+d+ 2pz

p+n_.

In view of Corollary 2.2, it is sufficient to show that

∞ X

n=0

n+p+δ p−δ

dap+n

n+d+ 2p ≤1. (24)

Since f(z)∈P∗_p,a,c,λ[α, β], we have

∞ X

n=0

[1 +λ(n_p)][(1 +B)n+ 2p+αβ)] (1−2λ)2β(p−α) .

(a)n+1

(c)n+1

ap+n ≤1.

Thus (24) will be satisfied if

(n+p+δ)d (p−δ)(n+d+ 2p) ≤

[1 +λ(n_p)][(1 +B)n+ 2(p+αββ)] (1−2λ)2β(p−α) .

(a)n+1

(c)n+1

(n≥0),

or

δ≤ p(n+d+ 2p)[1 +λ(

n p)]

(a)n+1

(c)n+1[(1 +B)n+ 2(p+αβ)]−(n+p)d(1−2λ)2β(p−α)]

(n+d+ 2p)[1 +λ(n_p)]((a)n+1

(c)n+1)[(1 +B)n+ 2(p+αβ) +d(1−2λ)2β(p−α)

.

(25)

Since the right hand side of (25) is an increasing function of n, putting n= 0 in (25), we

get

δ≤p

(

(d+ 2p)(p+αβ)(a_c)−β(p−α)d(1−2λ) (d+ 2p)(p+αβ)(a)n+1

(c)n+1 +β(p−α)d(1−2λ) )

.

(21)

Putting λ= 0 in Theorem 7.1, we get:

Corollary 7.1 : If f(z) is in the class P∗_p[α, β], then the integral transforms (23)are in the class P∗_p[δ],0≤δ < p, where

δ =δ(α, β, a, c, d, p) =p

₍_d_{+ 2}_p₎₍_p₊_αβ₎a

c −β(p−α)d

(d+ 2p)(p+αβ)a_c +β(p−α)d

.

The result is the best possible for the function f(z) given by

f(z) = 1 zp +

β(p−α) p+αβ .

c az

p_,₍_p_∈

(22)

REFERENCES

[1] M. K. Aouf, A generalization of meromorphic multivalent functions with

positive coefficients. Math, Japonica 35(1990) 609-614.

[2] M. K. Aouf, On a class of meromorphic multivalent functions with positive

coefficients, Math. Japonica 25(1990)603- 608.

[3] M. K. Aouf, New criteria for multivalent meromorphic starlike functions of

order alpha, Proc. Japan. Acad. 69(1993) 66- 70.

[4] M. K. Aouf and B. A. Frasin, Properties of some families of meromorphic

multivalent functions involving certain linear operator, Filomat 24:3 (2010)

35-54.

[5] M. K. Aouf, H. M. Mossen, H. E.Elattar, A certain class of meromorphic

multivalent functions with positive and fixed second coefficients, Punjab

Univ. J. Math. 33(2000)115-124.

[6] S. B. Joshi, H. K. Aouf, Meromorphic multivalent functions with positive

and fixed second coefficients, Kyungpook Math. J. 35(1995) 163-169.

[7] S. B. Joshi, H. M. Srivastava, A certain family of meromorphically

multi-valent functions, Clmput. Math. Appl. 38(3-4)(1999)201-211.

[8] J. L. Liu, Properties of some families of meromorphic p-valent functions,

Math. Japonica 52(2000)425-434.

[9] J. L. Liu, H. M. Srivastava, A linear operator and associated families of

meromorphically multivalent functions, J. Math. Anal. Appl.

259(2001)566-581.

[10] M. L. Mogra, Meromorphic multivalent functions with positive

(23)

[11] M. L. Mogra, Meromorphic multivalent functions with positive

coeffi-cients. II, Math. Japonica 35(6)(1990)1089-1098.

[12] S. Owa, H. E. Darwish, M. K. Aouf, Meromorphic multivalent functions

with positive and fixed second coefficients, Math. Japonica

46(2)(1997)231-236.

[13] R. K. Rain, H. M. Srivastava, A new class of meromorphically multivalent

functions with applications to generalized hypergeometric functions, Math.

Comput. Modelling 43(2006)350-356.

[14] A. Schild, H. Silverman, Convolution of univalent functions with negative

coefficients, Ann. Univ. Mariae Curie-Sklodowska Sect A 29(1975) 99-107.

[15] H. M. Srivastava, H. M. Mossen, M. K. Aouf, A unified presentation of

some classes of meromorphically multivalent functions, Comput. Math.

Appl. 38(1999)63-70.

[16] B. A. Uralegaddi, M. D. Gannigi, Meromorphic multivalent functions with

positive coefficients, Nepali. Math. Sci. Rep. 11(1986)95-102.

[17] B. A. Uralegaddi and C. Somantha, Certain classes of meromorphic

mul-tivalent functions, Tamkang J. Math. 23(1992), 223- 231.

[18] D. G. Yang, On a new subclasses of meromorphic p-valent functions, J.