ShahramNajafzadeh,S.R.KulkarniandG.Murugusundaramoorthy Certainclassof p -valentFunctionsdefinedbyDziok-SrivastavaLinearOperator

Certain class of p-valent Functions defined by Dziok-Srivastava Linear Operator

Shahram Najafzadeh, S. R. Kulkarni and G.

Murugusundaramoorthy

Abstract

In this paper, we introduce a new class of multivalent func- tions defined by Dziok-Srivastava operator to study some of the interesting properties like coefficient estimates, distortion bounds and to prove the class is closed under convolution product and integral representation.

2000 Mathematical Subject Classification: 30C45, 30C50.

Keywords: p-valent and hypergeometric functions, convolution, distortion bounds, closure theorem

1Received February 20, 2006

Accepted for publication (in revised form) March 11, 2006

65

1 Introduction

LetA_pbe the class ofp-valent analytic functions with positive coefficients of the form

(1) f(z) =z^p+ X∞

k=p+1

a_kz^k, z ∈∆ = {z :|z|<1}.

For functions f(z) given by (1) and

(2) g(z) =z^p+

X∞

k=p+1

b_kz^k,

the Hadamard product (or convolution) of f(z) and g(z) denoted by (f∗g)(z) = (g∗f)(z) is defined by

(3) (f∗g)(z) =z^p +

X∞

k=p+1

akbkz^k.

For {α₁, α₂,· · ·, α_m} ⊆ C and {β₁, β₂,· · ·, β_n} ⊆ C − {0,−1,−2,· · ·}

the generalized hypergeometric function _mF_n(α₁,· · ·, α_m;β₁,· · ·, β_n;z) is defined by

(4)

mF_n(α₁,· · ·, α_m;β₁,· · ·, β_n;z) = P^∞

k=0

(α1)k···(αm)kz^k (β1)k···(βn)kk!

(m ≤n+ 1, m, n,∈IN0 ={0,1,2,· · ·}) where (λ)_k is the pochhammer symbol defined by

(5) (λ)_k= Γ(λ+k) Γ(k) =





1 k = 0

λ(λ+ 1)· · ·(λ+k−1) k ∈IN

Using Dziok - Srivastava operator [2] , f(z)∈ A_p we have (6) DS_p^m,n = DS_p^(m,n)(α₁,· · ·, α_m;β₁,· · ·, β_n)f(z)

= h_p(α₁,· · ·, α_m;β₁,· · ·, β_n;z)∗f(z)

= z^p+ X∞

k=p+1

(α₁)_k−p· · ·(α_m)_k−pa_kz^k (β₁)_k−p· · ·(β_n)_k−p(k−p)!

where

hp(α1,· · ·, αm;β1,· · ·, βn;z) =z^p mFn(α1,· · ·, αm;β1,· · ·, βn;z).

For 1 < γ < 1 + _2p¹, z ∈ ∆ and let g(z) given by (2) we define the class

Ap(g(z), α1,· · ·, αm;β1, β2,· · ·, βn, γ) =A^g(z)_p (m, n, γ) by

A^g(z)p (m, n, γ) = n

f(z)∈ Ap :Re n

1 + ^z(DS_(DSp^pm,nm,n_(f^(f∗g)(z))_∗g)(z))0⁰⁰

o

< pγ,

(7) (1< γ <1 + 1

2p, z∈∆)

2 Main Results

In this section we obtain a necessary and sufficient condition for functions to be in the class A^g(z)p (m, n, γ).

Theorem 2.1. f(z)∈ A^g(z)p (m, n, γ) if and only if (8)

X∞

k=p+1

k(k−pγ)

p²(γ−1) θ(k, p) a_kb_k ≤1.

where

θ(k, p) = (α₁)_k−p· · ·(α_m)_k−p (β₁)_k−p· · ·(β_n)_k−p(k−p)!.

Proof. Iff(z)∈ A^g(z)p (m, n, γ), then by using (6) and (7) we obtain

Re







 1 +

z(p(p−1)z^p−2+ P^∞

k=p+1

θ(k, , p)k(k−1)akbkz^k−2 pz^p−1+ P^∞

k=p+1

θ(k, p)kakbkz^k−1









< pγ.

Choosing values of z on real axis and letting z →1⁻ we have p²+ P^∞

k=p+1

θ(k, p)k²a_kb_k p+ P^∞

k=p+1

θ(k, p)ka_kb_k

< pγ

or equivalently X∞

k=p+1

k(k−pγ)θ(k, p)a_kb_k ≤p²(γ−1).

To prove the “if” part, let (8) holds true, so

z(DS_p^m,n(f∗g)(z))⁰⁰−(p−1)(DS_p^m,n(f∗g)(z))⁰ z(DSp^m,n(f ∗g)(z)⁰⁰−[2p(1−γ)−1 +p](DSp^m,n(f ∗g)(z))⁰

≤

P∞ k=p+1

k(k−p)a_kb_k 2p²(γ−1)− P^∞

k=p+1

[k(k−p)(1−2(1−γ)))]a_kb_k

≤1

or equivalently f(z)∈ A^g(z)p (m, n, γ).

Theorem 2.2. If f(z)∈ A^g(z)p (m, n, γ), then

(9) a_k ≤ p²(γ−1)

k(k−pγ)b_kθ(k, p) the result is sharp for functions of the form

f_k(z) =z^p+ p²(γ−1)

k(k−pγ)bkθ(k, p)z^k k=p+ 1, p+ 2,· · ·.

Proof. Since f(z)∈ A^g(z)p (m, n, γ), by (8) we have k(k−pγ)θ(k, p)a_kb_k ≤

X∞

k=p+1

k(k−pγ)θ(k, p)a_kb_k ≤p²(γ−1) or

a_k ≤ p²(γ−1) k(k−pγ)θ(k, p)b_k. The sharpness is trivial and so omitted.

3 Distortion Bounds

In this section we obtain the distortion bounds for f(z)∈ A^g(z)p (m, n, γ).

Theorem 3.1. If f(z)∈ A^g(z)p (m, n, γ), then

(10) r^p− p²(γ−1)

(p+ 1)(p+ 1−pγ)θ(p+ 1, p)b_p+1r^p+1 ≤ |f(z)|

≤r^p+ p²(γ−1)

(p+ 1)(p+ 1−pγ)θ(p+ 1, p)b_p+1r^p+1 where

θ(p+ 1, p) = Qm i=1

α_i Qn j=1

β_j

, |z|=r <1.

The result is sharp for the function

(11) f(z) =z^p+ p²(γ−1)

(p+ 1)(p+ 1−pγ)θ(p+ 1, p)b_p+1z^p+1.

Proof. By using (8), (9) we obtain b_p+1θ(p+1, p)(p+1)(p+1−pγ)

X∞

k=p+1

a_k ≤ X∞

k=p+1

k(k−pγ)θ(k, p)a_kb_k ≤p²(γ−1) or

(12)

X∞

k=p+1

ak ≤ p²(γ−1)

(p+ 1)(p+ 1−pγ)θ(p+ 1, p)b_p+1. For the function f(z) = z^p+ P^∞

k=p+1

akz^k and using (12) and |z| = r we have

|f(z)| ≤ r^p + X∞

k=p+1

a_kr^k

< r^p +r^p+1 X∞

k=p+1

a_k

≤ r^p + p²(γ−1)

(p+ 1)(p+ 1−pγ)θ(p+ 1, p)b_p+1r^p+1, also

|f(z)| ≥ r^p − X∞

k=p+1

akr^k

≥ r^p − p²(γ−1)

(p+ 1)(p+ 1−pγ)θ(p+ 1, p)b_p+1 r^p+1. Hence the proof is complete.

Corollary. If f(z)∈ A^g(z)p (m, n, γ), then pr^p−1− p²(γ−1)

(p+ 1−pγ)θ(p+ 1, p)b_p+1r^p ≤ |f⁰(z)|

≤pr^p−1+ p²(γ−1)

(p+ 1−pγ)θ(p+ 1, p)b_p+1r^p. The result is sharp for the function given by (11).

4 Integral Representation

In this section we obtain integral representation for DS_p^m,n(f ∗g)(z).

Theorem 4.1. If f(z)∈ A^g(z)p (m, n, γ) then DS_p^m,n(f ∗g)(z) = (pγ−1)

Z _z

0

e^{R0zQ(t)t dt}dt.

Proof. By letting DS_p^m,n(f∗g)(z) =M(z) in (7) we have Re

1 + zM⁰⁰(z) M⁰(z)

< pγ.

Thus

zM⁰⁰(z)

M⁰(z) < pγ−1 or

zM⁰⁰(z)

M⁰(z) =Q(z)(pγ−1) where |Q(z)|<1, z∈∆.

So ^M_M⁰⁰0(z)^(z) = ^Q(z)_z (pγ−1), after integration we obtain log(M⁰(z)) =

Z _z

0

Q(t)

t (pγ−1)dt

thus

M⁰(z) = exp Z _z

0

Q(t)

t (pγ−1)dt

.

After integration we have M(z) =DS_p^m,n(f∗g) =

Z _z

0

exp Z _z

0

Q(t)

t (pγ−1)dt

dt and this gives the result.

5 Closure Theorems

In this section, we discuss certain inclusion properties of the classA^g(z)p (m, n, γ).

Theorem 5.1. Let Fj(z) =z^p+ P^∞

k=p+1

ak,jz^k (j = 1,2,· · ·, q) be in the class A^g(z)p (m, n, γ) and ηj ≥0 forj = 1,2,· · ·, q and P^q

j=1

ηj ≤1 then the function

f(z) =z^p+ X∞

k=p+1

Xq

j=1

ηjak,j

! z^k

belongs to A^g(z)p (m, n, γ).

Proof. Since F_j(z) ∈ A^g(z)p (m, n, γ), then from Theorem 2.1 for every j = 1,2,· · ·, q we have

X∞

k=p+1

k(k−pγ)θ(k, p)b_ka_k,j ≤p²(γ−1).

Also

X∞

k=p+1

k(k−pγ)θ(k, p)b_k Xq

j=1

η_ja_k,j

!

= Xq

j=1

ηj( X∞

k=p+1

k(k−pγ)θ(k, p)bkak,j)

≤ Xq

j=1

η_jp²(γ−1)

≤p²(γ−1).

So by Theorem 2.1 f(z)∈ A^g(z)p (m, n, γ).

Corollary. The class A^g(z)p (m, n, γ) is closed under convex linear com- bination.

Theorem 5.2. Let F_p(z) = z^p and F_k(z) = z^p+ p²(γ−1)

k(k−pγ)θ(k, p)b_kz^k, (k =p+ 1,· · ·).

Then f(z)∈ A^g(z)p (m, n, γ) if and only if f(z) = ηpz^p+

X∞

k=p+1

ηkFk(z)

where P^∞

k=p

η_k = 1 and η_k ≥0.

Proof. Let f(z)∈ A^g(z)p (m, n, γ), then from Theorem 2.2, we have a_k≤ p²(γ −1)

k(k−pγ)θ(k, p)b_k (k =p+ 1, p+ 2,· · ·) therefore by letting

ηk = k(k−pγ)θ(k, p)b_ka_k

p²(γ−1) (k =p+ 1, p+ 2,· · ·) and η_p = 1− P^∞

k=p+1

η_k.

We conclude the required result.

Conversely, let f(z) = η_pz^p + P^∞

k=p+1

η_kF_k(z), then

f(z) = ηpz^p+ X∞

k=p+1

ηk

z^p+ p²(γ−1)

k(k−pγ)θ(k, p)b_kz^k

= z^p+ X∞

k=p+1

η_kp²(γ−1) k(k−pγ)θ(k, p)b_kz^k. Therefore

X∞

k=p+1

η_kp²(γ−1) k(k−pγ)θ(k, p)b_k

k(k−pγ)

p²(γ−1)θ(k, p)b_k

= X∞

k=p+1

ηk = 1−ηp ≤1.

Hence by Theorem 2.1, we have f(z)∈ A^g(z)p (m, n, γ).

6 Convolution Property and Integral Ope- rator

In this section we show that the classA^g(z)p (m, n, γ) is closed under con- volution and integral operator.

Theorem 6.1. Let h(z) = z^p + P^∞

k=p+1

c_kz^k be analytic in unit disk ∆ and 0 ≤ ck ≤ 1. If f(z) ∈ A^g(z)p (m, n, γ), then (f ∗h)(z) is also in the class A^g(z)p (m, n, γ).

Proof. Since f(z)∈ A^g(z)p (m, n, γ) then by Theorem 2.1 we have X∞

k=p+1

k(k−pγ)θ(k, p)a_kb_k ≤p²(γ−1).

By using the last inequality and the fact that (f∗h)(z) =z^p+

X∞

k=p+1

akckz^k

we have

X∞

k=p+1

k(k−pγ)θ(k, p)a_kc_kb_k

≤ X∞

k=p+1

k(k−pγ)θ(k, p)akbk≤p²(γ−1) and hence by Theorem 2.1 result follows.

Theorem 6.2. If f(z)∈ A^g(z)p (m, n, γ), then F(z) = λ+p

z^λ Z _z

0

t^λ−1f(t)dt (λ >−1; z ∈∆) is also in the class A^g(z)p (m, n, γ).

Proof. Since F(z) = f(z) ∗ z^p+ P^∞

k=p+1 λ+p λ+kz^k

!

and ^λ+p_λ+k ≤ 1, by Theorem 6.1, the proof is trivial.

References

[1] [1] R. M. Ali, M. H. Khan, V. Ravichandran and K. G. Subramanian, A class of multivalent functions with positive coefficients defined by convolution, JIPAM, Vol. 6, Issue 1, (2005).

[2] J. Dziok and H. M. Srivastava, Certain subclass of analytic func- tions associated with the generalized hypergeometric functions, Inte- gral Transforms Spec. Funct. 14 (1), (2003), 7-18.

[3] R. J. Libera, Some classes of regular univalent functions, Proc.

Amer. Math. Soc. 16 (1965), 755-758.

[4] J. L. Liu, On a class of p-valent analytic functions, Chinese Quar.

J. Math., 15(4) (2000), 27-32.

[5] J. L. Liu,Some applications of certain integral operator. Kyungpook Math. J. 43 (2003), 211-219.

[6] M. Nunokawa, On the theory of multivalent functions, Tsukuba J.

Math. 11 (1987), 273-286.

Department of Mathematics,

Fergusson College, Pune University, Pune - 411004, India

Department of Mathematics,

Vellore Institute of Technology,Deemed University, Vellore-632 014 ,T.N., India

Shahram Najafzadeh : [email protected] S. R. Kulkarni : kulkarni₋[email protected]

G.Murugusundaramoorthy: [email protected]