HADAMARD PRODUCT OF CERTAIN MEROMORPHIC p−VALENT STARLIKE AND p−VALENT CONVEX FUNCTIONS

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HADAMARD PRODUCT OF CERTAIN MEROMORPHICp−VALENT STARLIKE

ANDp−VALENT CONVEX FUNCTIONS

M. K. AOUF

DEPARTMENT OFMATHEMATICS,

FACULTY OFSCIENCE, MANSOURAUNIVERSITY, MANSOURA35516, EGYPT.

mkaouf127@yahoo.com

Received 12 April, 2008; accepted 30 April, 2009 Communicated by G. Kohr

ABSTRACT. In this paper, we establish some results concerning the Hadamard product of certain meromorphicp-valent starlike and meromorphicp-valent convex functions analogous to those obtained by Vinod Kumar (J. Math. Anal. Appl. 113(1986), 230-234) and M. L. Mogra (Tamkang J. Math. 25(1994), no. 2, 157-162).

Key words and phrases: Meromorphic,p-valent, Hadamard product.

2000 Mathematics Subject Classification. 30C45.

1. INTRODUCTION

Throughout this paper, letp∈N={1,2, . . .}and let the functions of the form:

ϕ(z) =c_pz^p−

∞

X

n=1

c_p+nz^p+n (c_p >0;c_p+n≥0),

Ψ(z) =d_pz^p−

∞

X

n=1

d_p+nz^p+n (d_p >0;d_p+n≥0)

be regular andp−valent in the unit discU ={z :|z|<1}. Also, let (1.1) f(z) = ap−1

z^p +

∞

X

n=1

ap+n−1z^p+n−1 (ap−1 >0;ap+n−1 ≥0),

f_i(z) = ap−1,i

z^p +

∞

X

n=1

ap+n−1,iz^p+n−1 (ap−1,i >0;ap+n−1,i ≥0),

g(z) = bp−1

z^p +

∞

X

n=1

bp+n−1z^p+n−1 (bp−1 >0;bp+n−1 ≥0)

The author is thankful to the referee for his comments and suggestions.

117-08

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and

g_j(z) = bp−1,j

z^p +

∞

X

n=1

bp+n−1,jz^p+n−1 (bp−1,j >0;bp+n−1,j ≥0).

be regular andp−valent in the punctured discD={z : 0<|z|<1}.

LetS₀^∗(p, α, β)denote the class of functionsϕ(z)which satisfy the condition

zϕ⁰(z) ϕ(z) −p

zϕ⁰(z)

ϕ(z) +p−2α

< β

for some α, β(0 ≤ α < p, 0 < β ≤ 1, p ∈ N)and for allz ∈ U; and let C₀(p, α, β)be the class of functions ϕ(z) for which ^zf

0(z)

p ∈ S₀^∗(p, α, β). It is well known that the functions in S₀^∗(p, α, β)andC₀^∗(p, α, β)are, respectively,p−valent starlike andp−valent convex functions of orderαand typeβ with negative coefficients inU (see Aouf [1]).

Denote byΣS₀^∗(p, α, β), the class of functionsf(z)which satisfy the condition (1.2)

zf⁰(z) f(z) +p

zf⁰(z)

f(z) + 2α−p

< β

for some α, β(0 ≤ α < p, 0 < β ≤ 1, p ∈ N) and for all z ∈ D, and ΣC₀^∗(p, α, β) be the class of functions f(z) for which ^−zf

0(z)

p ∈ ΣS₀^∗(p, α, β). The functions in ΣS₀^∗(p, α, β) and ΣC₀^∗(p, α, β) are, respectively, called meromorphic p−valent starlike and meromorphic p−valent convex functions of order α and type β with positive coefficients in D. The class ΣS₀^∗(p, α, β)withap−1 = 1has been studied by Aouf [2] and Mogra [9].

Using similar arguments as given in ([2] and [9]), we can prove the following result for functions inΣS₀^∗(p, α, β).

A functionf(z)∈ΣS₀^∗(p, α, β)if and only if (1.3)

∞

X

n=1

{[(n+ 2p−1) +β(n+ 2α−1)]an+p−1} ≤2β(p−α)ap−1.

The result is sharp for the functionf(z)given by f(z) = 1

z^p + 2β(p−α)ap−1

(n+ 2p−1) +β(n+ 2α−1)z^p+n−1 (p, n∈N).

Proof Outline. Letf(z)∈ΣS₀^∗(p, α, β)be given by (1.1). Then, from (1.2) and (1.1), we have (1.4)

P∞

n=1(n+ 2p−1)ap+n−1z^2p+n−1 2(p−α)ap−1 −P∞

n=1(n+ 2α−1)ap+n−1z^2p+n−1

< β (z ∈U).

Since |Re(z)| ≤ |z| (z ∈ C), choosing z to be real and letting z → 1⁻ through real values, (1.4) yields

∞

X

n=1

(n+ 2p−1)ap+n−1 ≤2β(p−α)ap−1−

∞

X

n=1

β(n+ 2α−1)ap+n−1, which leads us to (1.3).

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In order to prove the converse, we assume that the inequality (1.3) holds true. Then, if we let z ∈∂U, we find from (1.1) and (1.3) that

zf⁰(z) f(z) +p

zf⁰(z)

f(z) + 2α−p

≤

P∞

n=1(n+ 2p−1)ap+n−1

(p−α)ap−1−P∞

n=1(n+ 2α−1)ap+n−1

< β (z ∈∂U ={z :z ∈C and |z|= 1}).

Hence, by the maximum modulus theorem, we havef(z) ∈ ΣS₀^∗(p, α, β). This completes the

proof of (1.3).

Also we can prove thatf(z)∈ΣC₀^∗(p, α, β)if and only if (1.5)

∞

X

n=1

p+n−1 p

[(n+ 2p−1) +β(n+ 2α−1)]an+p−1

≤2β(p−α)ap−1.

The result is sharp for the functionf(z)given by f(z) = 1

z^p + 2β(p−α)ap−1

n+2p−1 p

[(n+ 2p−1) +β(n+ 2α−1)]

z^p+n−1 (p, n∈N).

The quasi-Hadamard product of two or more functions has recently been defined and used by Owa ([11], [12] and [13]), Kumar ([6], [7] and [8]), Aouf et al. [3], Hossen [5], Darwish [4]

and Sekine [14]. Accordingly, the quasi-Hadamard product of two functionsϕ(z)andΨ(z)is defined by

ϕ∗Ψ(z) =c_pd_pz^p−

∞

X

n=1

c_p+nd_p+nz^p+n .

Let us define the Hadamard product of two functionsf(z)andg(z)by f∗g(z) = ap−1bp−1

z^p +

∞

X

n=1

ap+n−1bp+n−1z^p+n−1.

Similarly, we can define the Hadamard product of more than two meromorphicp−valent functions.

We now introduce the following class of meromorphicp−valent functions inD.

A functionf(z)∈Σ^∗_k(p, α, β)if and only if (1.6)

∞

X

n=1

(

p+n−1 p

k

[(n+ 2p−1) +β(n+ 2α−1)]an+p−1

)

≤2β(p−α)ap−1. where0≤α < p, 0< β ≤1, p∈N, andk is any fixed nonnegative real number.

Evidently,Σ^∗₀(p, α, β)≡ΣS₀^∗(p, α, β)andΣ^∗₁(p, α, β)≡ΣC₀^∗(p, α, β). Further,Σ^∗_k(p, α, β)⊂ Σ^∗_h(p, α, β)ifk > h ≥0, the containment being proper. Moreover, for any positive integerk, we have the following inclusion relation

Σ^∗_k(p, α, β)⊂Σ^∗_k−1(p, α, β)⊂ · · · ⊂Σ^∗₂(p, α, β)⊂ΣC₀^∗(p, α, β)⊂ΣS₀^∗(p, α, β). We also note that for every nonnegative real numberk, the classΣ^∗_k(p, α, β)is nonempty as the functions

f(z) = ap−1

z^p +

∞

X

n=1

p+n−1 p

^−k

2β(p−α)

(n+ 2p−1) +β(n+ 2α−1)

ap−1λp+n−1z^p+n−1, whereap−1 >0,0≤α < p, 0< β ≤1, p∈N, ap−1 >0, λp+n−1 ≥0andP∞

n=1λp+n−1 ≤1, satisfy the inequality (1.1).

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In this paper we establish certain results concerning the Hadamard product of meromorphic p−valent starlike and meromorphicp−valent convex functions of orderαand typeβanalogous to Kumar [7] and Mogra [10].

2. THEMAINTHEOREMS

Theorem 2.1. Let the functionsf_i(z)belong to the classΣC₀^∗(p, α, β)for everyi= 1,2, . . . , m.

Then the Hadamard productf₁∗f₂∗ · · · ∗f_m(z)belongs to the classΣ^∗_2m−1(p, α, β).

Proof. It is sufficient to show that

∞

X

n=1

(

p+n−1 p

2m−1

[(n+ 2p−1) +β(n+ 2α−1)]

m

Y

i=1

ap+n−1,i

)

≤2β(p−α)

" _m Y

i=1

ap−1,i

# .

Sincef_i(z)∈ΣC₀^∗(p, α, β), we have (2.1)

∞

X

n=1

p+n−1 p

[(n+ 2p−1) +β(n+ 2α−1)]ap+n−1,i

≤2β(p−α)ap−1,i,

fori= 1,2, . . . , m. Therefore, p+n−1

p

[(n+ 2p−1) +β(n+ 2α−1)]ap+n−1,i ≤2β(p−α)ap−1,i

or

ap+n−1,i≤





2β(p−α) p+n−1

p

[(n+ 2p−1) +β(n+ 2α−1)]



ap−1,i,

for everyi = 1,2, . . . , m. The right-hand expression of the last inequality is not greater than p+n−1

p

⁻²

ap−1,i. Hence

(2.2) ap+n−1,i ≤

p+n−1 p

⁻² ap−1,i, for everyi= 1,2, . . . , m.

Using (2.2) fori= 1,2, . . . , m−1, and (2.1) fori=m, we obtain

∞

X

n=1

(

p+n−1 p

2m−1

[(n+ 2p−1) +β(n+ 2α−1)]

m

Y

i=1

ap+n−1,i

)

≤

∞

X

n=1

(

p+n−1 p

^2m−1

[(n+ 2p−1) +β(n+ 2α−1)]

×

p+n−1 p

^−2(m−1)

·

m−1

Y

i=1

ap−1,i

!

ap+n−1,m

)

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=

"_m−1 Y

i=1

ap−1,i

# _∞ X

n=1

p+n−1 p

[(n+ 2p−1) +β(n+ 2α−1)]an+p−1,m

≤2β(p−α)

" _m Y

i=1

ap−1,i

# .

Hencef1∗f2∗ · · · ∗fm(z)∈Σ^∗_2m−1(p, α, β).

Theorem 2.2. Let the functionsf_i(z)belong to the classΣS₀^∗(p, α, β)for everyi= 1,2, . . . , m.

Then the Hadamard productf₁∗f₂∗ · · · ∗f_m(z)belongs to the classΣ^∗_m−1(p, α, β).

Proof. Sincef_i(z)∈ΣS₀^∗(p, α, β), we have

(2.3)

∞

X

n=1

{[(n+ 2p−1) +β(n+ 2α−1)]a_p+n−1,i} ≤2β(p−α)a_p−1,i,

fori= 1,2, . . . , m. Therefore, ap+n−1,i ≤

2β(p−α)

[(n+ 2p−1) +β(n+ 2α−1)]

ap−1,i,

and hence

(2.4) ap+n−1,i ≤

p+n−1 p

−1

ap−1,i, for everyi= 1,2, . . . , m.

Using (2.4) fori= 1,2, . . . , m−1, and (2.3) fori=m, we get

∞

X

n=1

(

p+n−1 p

m−1

[(n+ 2p−1) +β(n+ 2α−1)]

m

Y

i=1

ap+n−1,i

)

≤

∞

X

n=1

(

p+n−1 p

m−1

[(n+ 2p−1) +β(n+ 2α−1)]

×

p+n−1 p

−(m−1)

·

m−1

Y

i=1

a_p−1,i

!

a_p+n−1,m )

=

"_m−1 Y

i=1

ap−1,i

# _∞ X

n=1

{[(n+ 2p−1) +β(n+ 2α−1)]an+p−1,m}

≤2β(p−α)

" _m Y

i=1

ap−1,i

# .

Hencef₁∗f₂∗ · · · ∗f_m(z)∈Σ^∗_m−1(p, α, β).

Theorem 2.3. Let the functionsf_i(z)belong to the classΣC₀^∗(p, α, β)for everyi= 1,2, . . . , m, and let the functionsg_j(z)belong to the classΣS₀^∗(p, α, β)for everyj = 1,2, . . . , q. Then the Hadamard productf₁∗f₂∗ · · · ∗f_m∗g₁∗g₂∗ · · · ∗g_q(z)belongs to the classΣ^∗_2m+q−1(p, α, β).

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Proof. It is sufficient to show that

∞

X

n=1

(

p+n−1 p

2m+q−1

[(n+ 2p−1) +β(n+ 2α−1)]

×

m

Y

i=1

ap+n−1,i·

q

Y

i=1

bp+n−1,i

!)

≤2β(p−α)

m

Y

i=1

ap−1,i q

Y

i=1

bp−1,i

! .

Since f_i(z) ∈ ΣC₀^∗(p, α, β), the inequalities (2.1) and (2.2) hold for every i = 1,2, . . . , m.

Further, sinceg_j(z)∈ΣS₀^∗(p, α, β), we have (2.5)

∞

X

n=1

{[(n+ 2p−1) +β(n+ 2α−1)]bp+n−1,j} ≤2β(p−α)bp−1,j,

for everyj = 1,2, . . . , q. Whence we obtain

(2.6) bp+n−1,j ≤

p+n−1 p

−1

bp−1,j , for everyj = 1,2, . . . , q.

Using (2.2) fori= 1,2, . . . , m, (2.6) forj = 1,2, . . . , q−1, and (2.5) forj =q, we get

∞

X

n=1

(

p+n−1 p

2m+q−1

[(n+ 2p−1) +β(n+ 2α−1)]

×

m

Y

i=1

ap+n−1,i·

q

Y

j=1

bp+n−1,j

!)

≤

∞

X

n=1

(

p+n−1 p

^2m+q−1

[(n+ 2p−1) +β(n+ 2α−1)]

×

p+n−1 p

−2m

p+n−1 p

−(q−1) m

Y

i=1

ap−1,i q−1

Y

j=1

bp−1,j

!

bp+n−1,q

)

=

m

Y

i=1

ap−1,i q−1

Y

j=1

bp−1,j

! _∞ X

n=1

{[(n+ 2p−1) + β(n+ 2α−1)]bp+n−1,q}

≤2β(p−α)

m

Y

i=1

ap−1,i q

Y

j=1

bp−1,j

! .

Hencef₁∗f₂∗ · · · ∗f_m∗g₁∗g₂∗ · · · ∗g_q(z)∈Σ^∗_2m+q−1(p, α, β).

We note that the required estimate can also be obtained by using (2.2) fori= 1,2, . . . , m−1, (2.6) forj = 1,2, . . . , q, and (2.1) fori=m.

Remark 1. Puttingp= 1in the above results, we obtain the results obtained by Mogra [10].

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