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volume 6, issue 1, article 22, 2005.

Received 12 December, 2004;

accepted 27 January, 2005.

Communicated by:H. Silverman

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Journal of Inequalities in Pure and Applied Mathematics

POLYNOMIALS AND CONVEX SEQUENCE INEQUALITIES

ROSIHAN M. ALI, M. HUSSAIN KHAN, V. RAVICHANDRAN, AND K.G. SUBRAMANIAN

School of Mathematical Sciences Universiti Sains Malaysia 11800 USM, Penang, Malaysia.

EMail:rosihan@cs.usm.my Department of Mathematics Islamiah College

Vaniambadi 635 751, India.

EMail:khanhussaff@yahoo.co.in School of Mathematical Sciences Universiti Sains Malaysia 11800 USM, Penang, Malaysia.

EMail:vravi@cs.usm.my Department of Mathematics Madras Christian College

Tambaram, Chennai- 600 059, India.

EMail:kgsmani@vsnl.net

c

2000Victoria University ISSN (electronic): 1443-5756 237-04

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A Class of Multivalent Functions with Positive Coefficients Defined by

Convolution

Rosihan M. Ali, M. Hussain Khan, V. Ravichandran and

K.G. Subramanian

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Abstract

For a givenp-valent analytic functiongwith positive coefficients in the open unit disk∆, we study a class of functionsf(z) =z^p+P∞

n=manzⁿ,an≥0satisfying 1

p<

z(f∗g)⁰(z) (f∗g)(z)

< α

z∈∆; 1< α <m+p 2p

.

Coefficient inequalities, distortion and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases.

2000 Mathematics Subject Classification:30C45

Key words: Starlike function, Ruscheweyh derivative, Convolution, Positive coeffi- cients, Coefficient inequalities, Growth and distortion theorems.

The authors R. M. Ali and V. Ravichandran respectively acknowledged support from an IRPA grant 09-02-05-00020 EAR and a post-doctoral research fellowship from Universiti Sains Malaysia

1. Introduction

LetAdenote the class of all analytic functionsf(z)in the unit disk∆ := {z ∈ C :|z|<1}withf(0) = 0 = f⁰(0)−1. The classM(α)defined by

M(α) :=

f ∈ A:<

zf⁰(z) f(z)

< α

1< α < 3

2; z ∈∆

was investigated by Uralegaddi et al. [6]. A subclass ofM(α)was recently investigated by Owa and Srivastava [3]. Motivated byM(α), we introduce a more general classP M_g(p, m, α)of analytic functions with positive coefficients. For two analytic functions

f(z) =z^p+

∞

X

n=m

a_nzⁿ and g(z) = z^p+

∞

X

n=m

b_nzⁿ,

the convolution (or Hadamard product) offandg, denoted byf∗gor(f∗g)(z), is defined by

(f ∗g)(z) :=z^p+

∞

X

n=m

a_nb_nzⁿ.

LetT(p, m)be the class of all analyticp-valent functionsf(z) = z^p−P∞

n=ma_nzⁿ (a_n ≥ 0), defined on the unit disk ∆ and let T := T(1,2). A function f(z) ∈ T(p, m) is called a function with negative coefficients. The subclass ofT consisting of starlike functions of orderα, denoted byT S^∗(α), was studied by Silverman [5]. Several other classes of starlike functions with negative coefficients were studied; for e.g. see [2].

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LetP(p, m)be the class of all analytic functions

(1.1) f(z) =z^p+

∞

X

n=m

a_nzⁿ (a_n≥0)

andP :=P(1,2).

Definition 1.1. Let

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

∞

X

n=m

b_nzⁿ (b_n >0)

be a fixed analytic function in∆. Define the classP M_g(p, m, α)by

P M_g(p, m, α) :=

f ∈P(p, m) : 1 p<

z(f ∗g)⁰(z) (f∗g)(z)

< α,

1< α < m+p

2p ;z ∈∆

. Wheng(z) =z/(1−z),p= 1andm = 2, the classP M_g(p, m, α)reduces to the subclassP M(α) :=P ∩M(α). Wheng(z) = z/(1−z)^λ+1, p= 1and m = 2, the classP Mg(p, m, α)reduces to the class:

P_λ(α) =

f ∈P :<

z(D^λf(z))⁰ D^λf(z)

< α,

λ >−1,1< α < 3

2;z ∈∆

,

whereD^λdenotes the Ruscheweyh derivative of orderλ. When g(z) =z+

∞

X

n=2

n^lzⁿ,

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the class of functionsP M_g(1,2, α)reduces to the classP M_l(α)where P M_l(α) =

f ∈P :<

z(D^lf(z))⁰ D^lf(z)

< α,

1< α < 3

2;l ≥0; z ∈∆

,

whereD^ldenotes the Salagean derivative of orderl. Also we have P M(α)≡P₀(α)≡P M₀(α).

A functionf ∈ A(p, m)is inP P C(p, m, α, β)if 1

p< (1−β)zf⁰(z) + ^β_pz(zf⁰)⁰(z) (1−β)f(z) + ^β_pzf⁰(z)

!

< α

β ≥0; 0≤α < m+p 2p

This class is similar to the class of β-Pascu convex functions of orderαand it unifies the class ofP M(α)and the corresponding convex class.

For the newly defined classP M_g(p, m, α), we obtain coefficient inequalities, distortion and covering theorems, as well as closure theorems. As special cases, we obtain results for the classes Pλ(α), and P Ml(α). Similar results for the class P P C(p, m, α, β) also follow from our results, the details of which are omitted here.

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2. Coefficient Inequalities

Throughout the paper, we assume that the functionf(z)is given by the equation (1.1) and g(z) is given by by (1.2). We first prove a necessary and sufficient condition for functions to be in the classP M_g(p, m, α)in the following:

Theorem 2.1. A functionf ∈P M_g(p, m, α)if and only if

(2.1)

∞

X

n=m

(n−pα)a_nb_n≤p(α−1)

1< α < m+p 2p

.

Proof. Iff ∈P Mg(p, m, α), then (2.1) follows from 1

p<

z(f ∗g)⁰(z) (f ∗g)(z)

< α

by lettingz →1−through real values. To prove the converse, assume that (2.1) holds. Then by making use of (2.1), we obtain

z(f ∗g)⁰(z)−p(f∗g)(z) z(f∗g)⁰(z)−(2α−1)p(f ∗g)(z)

≤

P∞

n=m(n−p)a_nb_n 2(α−1)p−P∞

n=m[n−(2α−1)p]a_nb_n ≤1 or equivalentlyf ∈P M_g(p, m, α).

Corollary 2.2. A functionf ∈P_λ(α)if and only if

∞

X

n=2

(n−α)anBn(λ)≤α−1

1< α < 3 2

,

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where

(2.2) B_n(λ) = (λ+ 1)(λ+ 2)· · ·(λ+n−1)

(n−1)! .

Corollary 2.3. A functionf ∈P Mm(α)if and only if

∞

X

n=2

(n−α)a_nn^m ≤α−1

1< α < 3 2

.

Our next theorem gives an estimate for the coefficient of functions in the classP M_g(p, m, α).

Theorem 2.4. Iff ∈P M_g(p, m, α), then

a_n ≤ p(α−1) (n−pα)b_n with equality only for functions of the form

f_n(z) = z^p+ p(α−1) (n−pα)bn

zⁿ.

Proof. Letf ∈P M_g(p, m, α). By making use of the inequality (2.1), we have

(n−pα)anbn≤

∞

X

n=m

(n−pα)anbn ≤p(α−1)

or

a_n≤ p(α−1) (n−pα)bn

.

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Clearly for

f_n(z) =z^p+ p(α−1)

(n−pα)b_nzⁿ∈P M_g(p, m, α), we have

a_n = p(α−1) (n−pα)b_n.

Corollary 2.5. Iff ∈P_λ(α), then

an≤ α−1 (n−α)B_n(λ) with equality only for functions of the form

fn(z) = z+ α−1

(n−α)B_n(λ)zⁿ, whereB_n(λ)is given by (2.2).

Corollary 2.6. Iff ∈P M_m(α), then

a_n≤ α−1 (n−α)n^m with equality only for functions of the form

f_n(z) =z+ α−1 (n−α)n^mzⁿ.

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3. Growth and Distortion Theorems

We now prove the growth theorem for the functions in the classP M_g(p, m, α).

Theorem 3.1. Iff ∈P Mg(p, m, α), then

r^p− p(α−1)

(m−pα)b_mr^m ≤ |f(z)| ≤r^p+ p(α−1)

(m−pα)b_mr^m, |z|=r <1, providedb_n ≥b_m ≥1. The result is sharp for

(3.1) f(z) =z^p + p(α−1)

(m−pα)b_mz^m.

Proof. By making use of the inequality (2.1) for f ∈ P Mg(p, m, α) together with

(m−pα)b_m ≤(n−pα)b_n, we obtain

b_m(m−pα)

∞

X

n=m

a_n≤

∞

X

n=m

(n−pα)a_nb_n≤p(α−1)

or (3.2)

∞

X

n=m

a_n≤ p(α−1) (m−pα)bm

.

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By using (3.2) for the function f(z) = z^p +P∞

n=ma_nzⁿ ∈ P M_g(p, m, α), we have for|z|=r,

|f(z)| ≤r^p+

∞

X

n=m

anrⁿ

≤r^p+r^m

∞

X

n=m

a_n

≤r^p+ p(α−1) (m−pα)bm

r^m, and similarly,

|f(z)| ≥r^p − p(α−1) (m−pα)b_mr^m.

Theorem3.1also shows thatf(∆)for everyf ∈P M_g(p, m, α)contains the disk of radius1− _(m−pα)b^p(α−1)

m. Corollary 3.2. Iff ∈P_λ(α), then

r− α−1

(2−α)(λ+ 1)r² ≤ |f(z)| ≤r+ α−1

(2−α)(λ+ 1)r² (|z|=r).

The result is sharp for

(3.3) f(z) = z+ α−1

(2−α)(λ+ 1)z².

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Corollary 3.3. Iff ∈P M_m(α), then

r− α−1

(2−α)2^mr² ≤ |f(z)| ≤r+ α−1

(2−α)2^mr² (|z|=r).

The result is sharp for

(3.4) f(z) = z+ α−1

(2−α)2^mz².

The distortion estimates for the functions in the classP M_g(p, m, α)is given in the following:

Theorem 3.4. Iff ∈P M_g(p, m, α), then

pr^p−1− mp(α−1)

(m−pα)b_mr^m−1 ≤ |f⁰(z)| ≤pr^p−1+ mp(α−1) (m−pα)b_mr^m−1,

|z|=r <1,

providedb_n ≥b_m. The result is sharp for the function given by (3.1).

Proof. By making use of the inequality (2.1) forf ∈P M_g(p, m, α), we obtain

∞

X

n=m

a_nb_n≤ p(α−1) (m−pα)

and therefore, again using the inequality (2.1), we get

∞

X

n=m

nan≤ mp(α−1) (m−pα)b_m.

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For the functionf(z) =z^p+P∞

n=ma_nzⁿ∈P M_g(p, m, α), we now have

|f⁰(z)| ≤pr^p−1+

∞

X

n=m

na_nrⁿ⁻¹ (|z|=r)

≤pr^p−1+r^m−1

∞

X

n=m

na_n

≤pr^p−1+ mp(α−1) (m−pα)b_mr^m−1 and similarly we have

|f⁰(z)| ≥pr^p−1− mp(α−1) (m−pα)b_mr^m−1.

Corollary 3.5. Iff ∈Pλ(α), then

1− 2(α−1)

(2−α)(λ+ 1)r ≤ |f⁰(z)| ≤1 + 2(α−1)

(2−α)(λ+ 1)r (|z|=r).

The result is sharp for the function given by (3.3) Corollary 3.6. Iff ∈P M_m(α), then

1− 2(α−1)

(2−α)2^mr ≤ |f⁰(z)| ≤1 + 2(α−1)

(2−α)2^mr (|z|=r).

The result is sharp for the function given by (3.4)

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4. Closure Theorems

We shall now prove the following closure theorems for the classP Mg(p, m, α).

Let the functionsF_k(z)be given by (4.1) F_k(z) = z^p+

∞

X

n=m

f_n,kzⁿ, (k= 1,2, . . . , M).

Theorem 4.1. Let λ_k ≥ 0 for k = 1,2, . . . , M and PM

k=1λ_k ≤ 1. Let the function Fk(z) defined by (4.1) be in the class P Mg(p, m, α) for every k = 1,2, . . . , M. Then the functionf(z)defined by

f(z) =z^p +

∞

X

n=m M

X

k=1

λ_kf_n,k

! zⁿ

belongs to the classP M_g(p, m, α).

Proof. SinceF_k(z)∈P M_g(p, m, α), it follows from Theorem2.1that (4.2)

∞

X

n=m

(n−pα)bnfn,k ≤p(α−1)

for everyk = 1,2, . . . , M.Hence

∞

X

n=m

(n−pα)b_n

M

X

k=1

λ_kf_n,k

!

=

M

X

k=1

λ_k

∞

X

n=m

(n−pα)b_nf_n,k

!

≤

M

X

k=1

λkp(α−1)≤p(α−1).

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By Theorem2.1, it follows thatf(z)∈P M_g(p, m, α).

Corollary 4.2. The class P M_g(p, m, α)is closed under convex linear combi- nations.

Theorem 4.3. Let

F_p(z) = z^pandF_n(z) = z^p+ p(α−1) (n−pα)b_nzⁿ

for n = m, m+ 1, . . .. Then f(z) ∈ P Mg(p, m, α)if and only iff(z)can be expressed in the form

(4.3) f(z) = λ_pz^p+

∞

X

n=m

λ_nF_n(z),

where eachλ_j ≥0andλ_p+P∞

n=mλ_n = 1.

Proof. Letf(z)be of the form (4.3). Then f(z) =z^p +

∞

X

n=m

λ_np(α−1) (n−pα)b_nzⁿ

and therefore

∞

X

n=m

λnp(α−1) (n−pα)b_n

(n−pα)bn

p(α−1) =

∞

X

n=m

λ_n= 1−λ_p ≤1.

By Theorem2.1, we havef(z)∈P Mg(p, m, α).

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Conversely, letf(z)∈P M_g(p, m, α). From Theorem2.4, we have a_n≤ p(α−1)

(n−pα)b_n for n=m, m+ 1, . . . . Therefore we may take

λ_n= (n−pα)b_na_n

p(α−1) for n=m, m+ 1, . . . and

λp = 1−

∞

X

n=m

λn. Then

f(z) = λ_pz^p+

∞

X

n=m

λ_nF_n(z).

We now prove that the classP M_g(p, m, α)is closed under convolution with certain functions and give an application of this result to show that the class P Mg(p, m, α)is closed under the familiar Bernardi integral operator.

Theorem 4.4. Leth(z) =z^p +P∞

n=mh_nzⁿbe analytic in∆with0≤h_n≤1.

Iff(z)∈P M_g(p, m, α), then(f∗h)(z)∈P M_g(p, m, α).

Proof. The result follows directly from Theorem2.1.

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The generalized Bernardi integral operator is defined by the following integral:

(4.4) F(z) = c+p z^c

Z z

0

t^c−1f(t)dt (c >−1; z ∈∆).

Since

F(z) = f(z)∗ z^p+

∞

X

n=m

c+p c+nzⁿ

! ,

we have the following:

Corollary 4.5. If f(z) ∈ P M_g(p, m, α), then F(z) given by (4.4) is also in P Mg(p, m, α).

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5. Order and Radius Results

LetP S_h^∗(p, m, β)be the subclass ofP(m, p)consisting of functionsffor which f ∗his starlike of orderβ.

Theorem 5.1. Leth(z) = z^p +P∞

n=mh_nzⁿ withh_n > 0. Let(α−1)nh_n ≤ (n−pα)b_n. Iff ∈P M_g(p, m, α), thenf ∈P S_h^∗(p, m, β),where

β := inf

n≥m

(n−pα)b_n−(α−1)nh_n (n−pα)b_n−(α−1)ph_n

.

Proof. Let us first note that the condition (α −1)nh_n ≤ (n −pα)b_n implies f ∈P S_h^∗(p, m,0). From the definition ofβ, it follows that

β ≤ (n−pα)b_n−(α−1)nh_n (n−pα)b_n−(α−1)ph_n

or (n−pβ)hn

1−β ≤ (n−pα)bn

α−1 and therefore, in view of (2.1),

∞

X

n=m

(n−pβ)

p(1−β)a_nh_n≤

∞

X

n=m

(n−pα)

p(α−1)a_nb_n≤1.

Thus

1

p · z(f ∗h)⁰(z) (f ∗h)(z) −1

≤ P∞

n=m(n/p−1)anhn

1−P∞

n=ma_nh_n ≤1−β and thereforef ∈P S_h^∗(p, m, β).

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Similarly we can prove the following:

Theorem 5.2. If f ∈ P Mg(p, m, α), thenf ∈ P Mh(p, m, β)in|z| < r(α, β) where

r(α, β) := min (

1; inf

n≥m

(n−pα) (n−pα)

(β−1) (α−1)

b_n h_n

_n−p¹ ) .

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References

[1] R.M. ALI, M. HUSSAIN KHAN, V. RAVICHANDRAN AND K.G. SUB- RAMANIAN, A class of multivalent functions with negative coefficients defined by convolution, preprint.

[2] O.P. AHUJA, Hadamard products of analytic functions defined by Ruscheweyh derivatives, in Current Topics in Analytic Function Theory, 13–28, World Sci. Publishing, River Edge, NJ.

[3] S. OWA AND H.M. SRIVASTAVA, Some generalized convolution proper- ties associated with certain subclasses of analytic functions, J. Inequal. Pure Appl. Math., 3(3) (2002), Article 42, 13 pp. [ONLINE:http://jipam.

vu.edu.au/article.php?sid=194]

[4] V. RAVICHANDRAN, On starlike functions with negative coefficients, Far East J. Math. Sci., 8(3) (2003), 359–364.

[5] H. SILVERMAN, Univalent functions with negative coefficients, Proc.

Amer. Math. Soc., 51 (1975), 109–116.

[6] B.A. URALEGADDI, M.D. GANIGI AND S.M. SARANGI, Univalent functions with positive coefficients, Tamkang J. Math., 25(3) (1994), 225–

230.