In terms of Wright generalized hypergeometric function we define a class of ana- lytic functions

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

J. DZIOK, G. MURUGUSUNDARAMOORTHY, AND K. VIJAYA INSTITUTE OFMATHEMATICS,

UNIVERSITY OFRZESZOW UL. REJTANA16A, PL-35-310 RZESZOW, POLAND

jdziok@univ.rzeszow.pl

SCHOOL OFSCIENCE ANDHUMANITIES, V I T UNIVERSITY, VELLORE- 632014,T.N., INDIA

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

Received 01 October, 2008; accepted 26 June, 2009 Communicated by S.S. Dragomir

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.

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

2000 Mathematics Subject Classification. 30C45.

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 discU = {z : |z| <1}.We denote byS the subclass ofA consisting of functionsf which are univalent inU.

Also we denote byV,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 number ηsuch 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.

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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 classesU 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 equivalence between the usual classes of convex and starlike functions, we define the class SP(k, γ) of k-starlike functions of orderγ.

Definition 1.2. A functionf ∈ Ais said to be in the classSP(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

bnzⁿ, 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 parametersα₁, A₁, . . . , α_p, A_pandβ₁, B₁, . . . , β_q, B_q(p, q ∈N = 1,2,3,. . .) such that

(1.3) 1 +

q

X

n=1

B_n−

p

X

n=1

A_n≥0,

the Wright generalized hypergeometric function [24]

pΨ_q[(α₁, A₁), . . . ,(α_p, A_p); (β₁, B₁), . . . ,(β_q, B_q);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.

Ifp≤q+ 1, A_n = 1 (n= 1, . . . , p)andB_n= 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(α_t)

!−1 q

Y

t=0

Γ(β_t)

! .

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

pφq[(αt, At)1,p; (βt, Bt)1,q;z] = Ωz pΨq[(αt, At)1,p(βt, Bt)1,q;z], z ∈U, and

W =W[(α_n, A_n)_1,p; (β_n, B_n)_1,q] :A →A 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

σnanzⁿ, z ∈U,

where

σn= 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 family SP(γ, k), we define the class W_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 class W_q^p(k, γ) generalizes the classes of k-uniformly convex functions and k-starlike functions. Ifp= 2, q = 1, A₁ =A₂ =B₁ =α₁ =β₁ = 1,then forα₂ = 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 classW_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 classV W_q^p(k, γ).

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2. MAINRESULTS

First we obtain a sufficient condition for functions from the class A to belong to the class W_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).

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

Proof. In view of Theorem 2.1 we 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=2nσ_na_nzⁿ z+P∞

n=2σ_na_nzⁿ −1

≤Re

z+P∞

n=2nσ_na_nzⁿ z+P∞

n=2σ_na_nzⁿ −γ

, or

k

P∞

n=2(n−1)σ_na_nzⁿ⁻¹ 1 +P∞

n=2σ_na_nzⁿ⁻¹

≤Re

(1−γ) +P∞

n=2(n−γ)σ_na_nzⁿ⁻¹ 1 +P∞

n=2σ_na_nzⁿ⁻¹

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

P∞

n=2k(n−1)σ_n|a_n|rⁿ⁻¹ 1−P∞

n=2σ_n|a_n|rⁿ⁻¹ ≤ (1−γ)−P∞

n=2(n−γ)σ_n|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).

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

|a_n| ≤ 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, . . . . 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 σ_n ^∞_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,η of the form (2.3).

Proof. Sincef ∈V W_q^p(k, γ),we apply Theorem 2.2 to 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 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)σ_n}^∞_n=2and_{kn+n−k−γ}

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

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Theorem 2.6. Let f 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) =z andh_n,ηare defined by (2.3).

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)σn

µ_nzⁿ, z ∈U.

Since

∞

X

n=2

(kn+n−k−γ)σ_n 1−γ

(kn+n−k−γ)σn

µ_n

=

∞

X

n=2

µ_n(1−γ) = (1−µ₁)(1−γ)≤1−γ, by Theorem 2.2 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)σn

1−γ , n ≥ 2and µ₁ = 1−P∞

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

3. PARTIALSUMS

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 classV W_q^p(k, γ)and obtain sharp lower bounds for the ratios of the real part off tof_m(z)andf⁰ tof_m⁰ .

Theorem 3.1. Let a function f of the form (1.1) belong to the class V 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) dn:= kn+n−k−γ

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

(3.5) d_n+1 > d_n >1, n= 2,3, . . . . Thus by Theorem 2.1 we 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) = dm+1

f(z) f_m(z)−

1− 1 d_m+1

= 1 + d_m+1P∞

n=m+1a_nzⁿ⁻¹ 1 +Pm

n=2a_nzⁿ⁻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|an| −dm+1P∞

n=m+1|an| ≤1, z ∈U, which readily yields the assertion (3.2) of Theorem 3.1. 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|a_n| −(1−d_m+1)P∞

n=m+1|a_n| ≤1, z ∈U,

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 function f of the form (1.1) belong to the class V 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) +dm+1]

f_m⁰ (z)

f⁰(z) − d_m+1 m+ 1 +d_m+1

, z ∈U,

the proof is analogous to that of Theorem 3.1, and we omit the details.

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Concluding Remarks: Observe that, if we specialize the parameters of the classV 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], Murugusundaramoorthy et al. [22, 23], and others.

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