A GENERALIZED CLASS OF k-STARLIKE FUNCTIONS WITH VARYING ARGUMENTS OF COEFFICIENTS

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A Class Of Starlike Functions J. Dziok, G. Murugusundaramoorthy

and K. Vijaya vol. 10, iss. 3, art. 66, 2009

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A GENERALIZED CLASS OF k-STARLIKE FUNCTIONS WITH VARYING ARGUMENTS OF

COEFFICIENTS

J. DZIOK

Institute of Mathematics

University of Rzeszow ul. Rejtana 16A PL-35-310 Rzeszow, Poland

EMail:jdziok@univ.rzeszow.pl

G. MURUGUSUNDARAMOORTHY AND K. VIJAYA

School of Science and Humanities, V I T University, Vellore - 632014,T.N., India

EMail:gmsmoorthy@yahoo.com kvavit@yahoo.co.in

Received: 01 October, 2008

Accepted: 26 June, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45.

Key words: Univalent functions, starlike functions, varying arguments, coefficient estimates.

Abstract: In terms of Wright generalized hypergeometric function we define a class of analytic functions. The class generalize well known classes ofk-starlike functions andk-uniformly convex functions. Necessary and sufficient coefficient bounds are given for functions in this class. Further distortion bounds, extreme points and results on partial sums are investigated.

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1. Introduction

LetAdenote the class of functions of the form

(1.1) f(z) =z+

∞

X

n=2

a_nzⁿ

which are analytic in the open unit disc U = {z : |z| < 1}. We denote by S the subclass ofAconsisting of functionsf which are univalent inU.

Also we denote by V, the class of analytic functions with varying arguments (introduced by Silverman [16]) consisting of functionsf of the form (1.1) for which there exists a real numberηsuch that

(1.2) θ_n+ (n−1)η=π(mod 2π), where arg(a_n) =θ_n for all n ≥2.

Letk, γbe real parameters withk ≥0, −1≤γ <1.

Definition 1.1. A functionf ∈Ais said to be in the classU CV(k, γ)ofk-uniformly convex functions of orderγ if it satisfies the condition

Re

1 + zf⁰⁰(z) f⁰(z) −γ

> k

zf⁰⁰(z) f⁰(z)

, z ∈U.

In particular, the classes U CV := U CV (1,0), k−U CV := U CV(k,0)were introduced by Goodman [6] (see also [10,13]), and Kanas and Wisniowska [8] (see also [7]), respectively, where their geometric definition and connections with the conic domains were considered.

Related to the classU CV(k, γ)by means of the well-known Alexander equiva- lence between the usual classes of convex and starlike functions, we define the class SP(k, γ)ofk-starlike functions of orderγ.

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Definition 1.2. A function f ∈ A is said to be in the class SP(k, γ) ofk-starlike functions of orderγ if it satisfies the condition

Re

zf⁰(z) f(z) −γ

> k

zf⁰(z) f(z) −1

, z ∈U.

The classes S_p :=SP(1,0), k−ST :=SP(k,0)were investigated by Rønning [13,14], Kanas and Wisniowska [9], Kanas and Srivastava [7].

Note that the classes

ST :=SP(0,0), CV :=U CV(0,0)

are the well known classes of starlike and convex functions, respectively.

For functionsf ∈Agiven by (1.1) andg ∈Agiven by g(z) = z+

∞

X

n=2

b_nzⁿ, z ∈U,

we define the Hadamard product (or convolution) off andgby (f∗g)(z) = z+

∞

X

n=2

a_nb_nzⁿ, z ∈U.

For positive real parameters α1, A1, . . . , αp, Ap and β1, B1, . . . , βq, Bq (p, q ∈ N = 1,2,3,. . .)such that

(1.3) 1 +

q

X

n=1

Bn−

p

X

n=1

An≥0,

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the Wright generalized hypergeometric function [24]

pΨq[(α1, A1), . . . ,(αp, Ap); (β1, B1), . . . ,(βq, Bq);z]

= _pΨ_q[(α_n, A_n)_1,p; (β_n, B_n)_1,q;z]

is defined by

pΨ_q[(α_t, A_t)_1,p;(β_t, B_t)_1,q;z]

=

∞

X

n=0

( _p Y

t=0

Γ(α_t+nA_t ) ( _q

Y

t=0

Γ(β_t+nB_t )⁻¹

zⁿ

n!, z ∈U.

If p ≤ q + 1, An = 1 (n = 1, . . . , p) and Bn = 1 (n = 1, . . . , q), we have the relationship:

(1.4) Ω_pΨ_q[(α_n,1)_1,p; (β_n,1)_1,q;z] = _pF_q(α₁, . . . , α_p; β₁, . . . , β_q;z), z ∈U, where_pF_q(α₁, . . . , α_p;β₁, . . . , β_q;z)is the generalized hypergeometric function and

(1.5) Ω =

p

Y

t=0

Γ(αt)

!⁻¹ _q Y

t=0

Γ(βt)

! .

In [3] Dziok and Raina defined the linear operator by using Wright generalized hypergeometric function. Let

pφ_q[(α_t, A_t)_1,p; (β_t, B_t)_1,q;z] = Ωz _pΨ_q[(α_t, A_t)_1,p(β_t, B_t)_1,q;z], z ∈U, and

W =W[(αn, An)1,p; (βn, Bn)1,q] :A →A

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be a linear operator defined by

Wf(z) :=z_pφ_q[(α_t, A_t)_1,p; (β_t, B_t)_1,q;z]∗f(z), z ∈U.

We observe that, forf of the form (1.1), we have

(1.6) Wf(z) = z+

∞

X

n=2

σ_na_nzⁿ, z ∈U, where

σ_n = Ω Γ(α1+A1(n−1))· · ·Γ(αp+Ap(n−1)) (n−1)!Γ(β₁ +B₁(n−1))· · ·Γ(β_q+B_q(n−1)) , andΩis given by (1.5).

In view of the relationship (1.4), the linear operator (1.6) includes the Dziok- Srivastava operator (see [5]) and other operators. For more details on these operators, see [1], [2], [4], [11], [12], [15] and [19].

Motivated by the earlier works of Kanas and Srivastava [7], Srivastava and Mishra [20] and Vijaya and Murugusundaramoorthy [23], we define a new class of functions based on generalized hypergeometric functions.

Corresponding to the familySP(γ, k), we define the classW_q^p(k, γ)for a functionf of the form (1.1) such that

(1.7) Re

z(Wf(z))⁰ Wf(z) −γ

≥k

z(Wf(z))⁰ Wf(z) −1

, z∈U.

We also let

V W_q^p(k, γ) = V ∩W_q^p(k, γ).

The classW_q^p(k, γ)generalizes the classes ofk-uniformly convex functions and k-starlike functions. Ifp = 2, q = 1, A1 = A2 = B1 = α1 = β1 = 1,then for

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α₂ = 2we have

W₁²(k,0) =k−U CV, and forα₂ = 1we have

W₁²(k,0) =k−ST.

In this paper we obtain a sufficient coefficient condition for functionsf given by (1.1) to be in the class W_q^p(k, γ) and we show that it is also a necessary condition for functions to belong to this class. Distortion results and extreme points for functions in V W_q^p(k, γ)are obtained. Finally, we investigate partial sums for the class V W_q^p(k, γ).

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2. Main Results

First we obtain a sufficient condition for functions from the classAto belong to the classW_q^p(k, γ).

Theorem 2.1. Letf be given by (1.1). If

(2.1)

∞

X

n=2

(kn+n−k−γ)σ_n|a_n| ≤1−γ, thenf ∈W_q^p(k, γ).

Proof. By definition of the classW_q^p([α₁], γ),it suffices to show that k

z(Wf(z))⁰ Wf(z) −1

−Re

z(Wf(z))⁰ Wf(z) −1

≤1−γ, z ∈U.

Simple calculations give k

z(Wf(z))⁰ Wf(z) −1

−Re

z(Wf(z))⁰ Wf(z) −γ

≤(k+ 1)

z(Wf(z))⁰ Wf(z) −1

≤(k+ 1) P∞

n=2(n−1)σ_n|a_n||z|ⁿ⁻¹ 1−P∞

n=2σ_n|a_n||z|ⁿ⁻¹ . Now the last expression is bounded above by(1−γ)if (2.1) holds.

In the next theorem, we show that the condition (2.1) is also necessary for functions from the classV W_q^p(k, γ).

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Theorem 2.2. Letf be given by (1.1) and satisfy (1.2).Then the functionf belongs to the classV W_q^p(k, γ)if and only if (2.1) holds.

Proof. In view of Theorem2.1we need only to show thatf ∈ V W_q^p(k, γ)satisfies the coefficient inequality (2.1). Iff ∈V W_q^p(k, γ)then by definition, we have

k

z+P∞

n=2nσnanzⁿ z+P∞

n=2σ_na_nzⁿ −1

≤Re

z+P∞

n=2nσnanzⁿ z+P∞

n=2σ_na_nzⁿ −γ

, or

k

P∞

n=2(n−1)σ_na_nzⁿ⁻¹ 1 +P∞

n=2σnanzⁿ⁻¹

≤Re

(1−γ) +P∞

n=2(n−γ)σ_na_nzⁿ⁻¹ 1 +P∞

n=2σnanzⁿ⁻¹

. In view of (1.2), we setz =r^iηin the above inequality to obtain

P∞

n=2k(n−1)σn|a_n|rⁿ⁻¹ 1−P∞

n=2σ_n|a_n|rⁿ⁻¹ ≤ (1−γ)−P∞

n=2(n−γ)σn|a_n|rⁿ⁻¹ 1−P∞

n=2σ_n|a_n|rⁿ⁻¹ . Thus

(2.2)

∞

X

n=2

(kn+n−k−γ)σ_n|a_n|rⁿ⁻¹ ≤1−γ, and lettingr →1⁻in (2.2), we obtain the desired inequality (2.1).

Corollary 2.3. If a functionf of the form (1.1) belongs to the classV W_q^p(k, γ),then

|an| ≤ 1−γ

(kn+n−k−γ)σ_n, n= 2,3, . . . . The equality holds for the functions

(2.3) h_n,η(z) =z− (1−γ)e^i(1−n)η

(kn+n−k−γ)σ_nzⁿ, z ∈U; 0≤η <2π, n= 2,3, . . . .

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Next we obtain the distortion bounds for functions belonging to the classV W_q^p(k, γ).

Theorem 2.4. Letf be in the classV W_q^p(k, γ), |z|=r <1.If the sequence {(kn+n−k−γ)σ_n}^∞_n=2

is nondecreasing, then

(2.4) r− 1−γ

(k−γ+ 2)σ₂r² ≤ |f(z)| ≤r+ 1−γ

(k−γ+ 2)σ₂r². If the sequencekn+n−k−γ

n σn

∞

n=2is nondecreasing, then

(2.5) 1− 2(1−γ)

(k−γ+ 2)σ₂r ≤ |f⁰(z)| ≤1 + 2(1−γ) (k−γ+ 2)σ₂r.

The result is sharp. The extremal functions are the functionsh_2,ηof the form (2.3).

Proof. Sincef ∈V W_q^p(k, γ),we apply Theorem2.2to obtain (k−γ+ 2)σ₂

∞

X

n=2

|a_n| ≤

∞

X

n=2

(kn+n−k−γ)σ_n|a_n| ≤1−γ.

Thus

|f(z)| ≤ |z|+|z|²

∞

X

n=2

|a_n| ≤r+ 1−γ

(k−γ+ 2)σ₂r². Also we have

|f(z)| ≥ |z| − |z|²

∞

X

n=2

|a_n| ≥r− 1−γ (k−γ+ 2)σ₂r²

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and (2.4) follows. In similar manner forf⁰, the inequalities

|f⁰(z)| ≤1 +

∞

X

n=2

n|a_n||z|ⁿ⁻¹ ≤1 +|z|

∞

X

n=2

na_n

and ∞

X

n=2

n|a_n| ≤ 2(1−γ) (k−γ+ 2)σ₂ lead to (2.5). This completes the proof.

Corollary 2.5. Letf be in the classV W_q^p(k, γ), |z|=r <1.If

(2.6) p > q, α_q+1 ≥1, α_j ≥β_j and A_j ≥B_j (j = 2, . . . , q), then the assertions (2.4), (2.5) hold true.

Proof. From (2.6) we have that the sequences {(kn+n−k−γ)σn}^∞_n=2 and _{kn+n−k−γ}

n σ_n ^∞_n=2 are nondecreasing. Thus, by Theorem 2.4, we have Corollary 2.5.

Theorem 2.6. Letf be given by (1.1) and satisfy (1.2). Then the functionf belongs to the classV W_q^p(k, γ)if and only iff can be expressed in the form

(2.7) f(z) =

∞

X

n=1

µ_nh_n,η(z), µ_n ≥0 and

∞

X

n=1

µ_n= 1, whereh1(z) = zandhn,ηare defined by (2.3).

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Proof. If a functionf is of the form (2.7), then by (1.2) we have f(z) =z+

∞

X

n=2

(1−γ)e^iθⁿ

(kn+n−k−γ)σ_nµ_nzⁿ, z∈U.

Since

∞

X

n=2

(kn+n−k−γ)σn

1−γ

(kn+n−k−γ)σ_nµn

=

∞

X

n=2

µ_n(1−γ) = (1−µ₁)(1−γ)≤1−γ, by Theorem2.2 we havef ∈V W_q^p(k, γ).

Conversely, iff is in the classV W_q^p(k, γ),then we may setµ_n = (kn+n−k−γ)σ_n

1−γ ,

n ≥ 2and µ₁ = 1−P∞

n=2µ_n. Then the function f is of the form (2.7) and this completes the proof.

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3. Partial Sums

For a functionf ∈Agiven by (1.1), Silverman [17] and Silvia [18] investigated the partial sumsf₁ andf_mdefined by

(3.1) f₁(z) =z; and f_m(z) = z+

m

X

n=2

a_nzⁿ, (m= 2,3. . .).

We consider in this section partial sums of functions in the class V W_q^p(k, γ) and obtain sharp lower bounds for the ratios of the real part off tofm(z)andf⁰ tof_m⁰ . Theorem 3.1. Let a functionf of the form (1.1) belong to the classV W_q^p(k, γ)and assume (2.6). Then

(3.2) Re

f(z) f_m(z)

≥1− 1

d_m+1, z ∈U, m∈N and

(3.3) Re

f_m(z) f(z)

≥ d_m+1

1 +d_m+1, z ∈U, m∈N, where

(3.4) d_n:= kn+n−k−γ

1−γ σ_n. Proof. By (2.6) it is not difficult to verify that

(3.5) d_n+1> d_n >1, n= 2,3, . . . . Thus by Theorem2.1we have

(3.6)

m

X

n=2

|a_n|+d_m+1

∞

X

n=m+1

|a_n| ≤

∞

X

n=2

d_n|a_n| ≤1.

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Setting

(3.7) g(z) =d_m+1

f(z) fm(z) −

1− 1 dm+1

= 1 + d_m+1P∞

n=m+1a_nzⁿ⁻¹ 1 +Pm

n=2anzⁿ⁻¹ , it suffices to show that

Reg(z)≥0, z ∈U.

Applying (3.6), we find that

g(z)−1 g(z) + 1

≤ d_m+1P∞

n=m+1|a_n| 2−2Pn

n=2|a_n| −d_m+1P∞

n=m+1|a_n| ≤1, z∈U, which readily yields the assertion (3.2) of Theorem3.1. In order to see that

(3.8) f(z) =z+ z^m+1

d_m+1, z∈U, gives sharp the result, we observe that forz =re^iπ/mwe have

f(z)

f_m(z) = 1 + z^m d_m+1

z→1⁻

−→ 1− 1 d_m+1. Similarly, if we take

h(z) = (1 +d_m+1)

f_m(z)

f(z) − d_m+1 1 +d_m+1

= 1− (1 +d_n+1)P∞

n=m+1a_nzⁿ⁻¹ 1 +P∞

n=2a_nzⁿ⁻¹ , z ∈U, and making use of (3.6), we can deduce that

h(z)−1 h(z) + 1

≤ (1 +d_m+1)P∞

n=m+1|a_n| 2−2Pm

n=2|an| −(1−dm+1)P∞

n=m+1|an| ≤1, z ∈U,

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which leads us immediately to the assertion (3.3) of Theorem 3.1. The bound in (3.3) is sharp for eachm ∈ N with the extremal functionf given by (3.8), and the proof is complete.

Theorem 3.2. Let a functionf of the form (1.1) belong to the classV W_q^p(k, γ)and assume (2.6). Then

(3.9) Re

f⁰(z) f_m⁰ (z)

≥1−m+ 1 d_m+1 and

(3.10) Re

f_m⁰ (z) f⁰(z)

≥ d_m+1 m+ 1 +d_m+1, whered_mis defined by (3.4)

Proof. By setting

g(z) =d_m+1

f⁰(z) f_m⁰ (z) −

1− m+ 1 d_m+1

, z ∈U,

and

h(z) = [(m+ 1) +d_m+1]

f_m⁰ (z)

f⁰(z) − d_m+1 m+ 1 +d_m+1

, z∈U,

the proof is analogous to that of Theorem3.1, and we omit the details.

Concluding Remarks: Observe that, if we specialize the parameters of the class V W_q^p(k, γ), we obtain various classes introduced and studied by Goodman [6], Kanas and Srivastava [7], Ma and Minda [10], Rønning [13,14], Murugusundaramoor- thy et al. [22,23], and others.

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