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volume 6, issue 2, article 52, 2005.

Received 04 January, 2005;

accepted 12 April, 2005.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

THE DZIOK-SRIVASTAVA OPERATOR AND k−UNIFORMLY STARLIKE FUNCTIONS

R. AGHALARY AND GH. AZADI

University of Urmia Urmia, Iran

EMail:raghalary@yahoo.com EMail:azadi435@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 003-05

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The Dziok-Srivastava Operator andk−Uniformly Starlike

Functions R. Aghalary and Gh. Azadi

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J. Ineq. Pure and Appl. Math. 6(2) Art. 52, 2005

Abstract

Inclusion relations fork−uniformly starlike functions under the Dziok-Srivastava operator are established. These results are also extended tok−uniformly convex functions, close-to-convex, and quasi-convex functions.

2000 Mathematics Subject Classification:Primary 30C45; Secondary 30C50.

Key words: Starlike, Convex, Linear operators.

1. Introduction

LetA denote the class of functions of the formf(z) =z +P∞

n=2a_nzⁿ which are analytic in the open unit disc U = {z : |z| < 1}. A function f ∈ A is said to be in U ST(k, γ), the class of k−uniformly starlike functions of order γ, 0≤γ <1,iff satisfies the condition

(1.1) <

zf⁰(z) f(z)

> k

zf⁰(z) f(z) −1

+γ, k≥0.

Replacingf in (1.1) byzf⁰ we obtain the condition

(1.2) <

1 + zf⁰⁰(z) f⁰(z)

> k

zf⁰⁰(z) f⁰(z)

+γ, k≥0

required for the function f to be in the subclassU CV(k, γ) ofk−uniformaly convex functions of orderγ.

Uniformly starlike and convex functions were first introduced by Goodman [5] and then studied by various authors. For a wealth of references, see Ronning [13].

Setting

Ω_k,γ =n

u+iv;u > kp

(u−1)²+v²+γo ,

with p(z) = ^zf_f_(z)⁰^(z) or p(z) = 1 + ^zf_f0⁰⁰(z)^(z) and considering the functions which map U on to the conic domain Ω_k,γ, such that1 ∈ Ω_k,γ, we may rewrite the conditions (1.1) or (1.2) in the form

(1.3) p(z)≺qk,γ(z).

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We note that the explicit forms of functionq_k,γ fork = 0andk = 1are q_0,γ(z) = 1 + (1−2γ)z

1−z , and q_1,γ(z) = 1 +2(1−γ) π²

log 1 +√ z 1−√

z 2

.

For0< k <1we obtain q_k,γ(z) = 1−γ

1−k² cos 2

π(arccosk)ilog 1 +√ z 1−√

z

− k²−γ 1−k², and ifk > 1,thenq_k,γ has the form

q_k,γ(z) = 1−γ

k²−1sin π 2K(k)

Z ^u(z)^√

k

0

dt p1−t²√

1−k²t²

!

+k²−γ k²−1, whereu(z) = ^z−

√ k 1−√

kz andKis such thatk = cosh^πK_4K(z)⁰^(z).

By virtue of (1.3) and the properties of the domainsΩ_k,γwe have (1.4) <(p(z))><(q_k,γ(z))> k+γ

k+ 1.

DefineU CC(k, γ, β)to be the family of functionsf ∈Asuch that

<

zf⁰(z) g(z)

≥k

zf⁰(z) g(z) −1

+γ, k≥0, 0≤γ <1 for someg ∈U ST(k, β).

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Similarly, we defineU QC(k, γ, β)to be the family of functionsf ∈Asuch that

<

(zf⁰(z))⁰ g⁰(z)

≥k

(zf⁰(z))⁰ g⁰(z) −1

+γ, k ≥0, 0≤γ <1 for someg ∈U CV(k, β).

We note thatU CC(0, γ, β)is the class of close-to-convex functions of order γ and typeβ andU QC(0, γ, β)is the class of quasi-convex functions of order γ and typeβ.

The aim of this paper is to study the inclusion properties of the above men- tioned classes under the following linear operator which is defined by Dziok and Srivastava [3].

For α_j ∈ C (j = 1,2,3, . . . , l) and β_j ∈ C − {0,−1,−2, . . .} (j = 1,2, . . . m), the generalized hypergeometric function is defined by

lF_m(α₁, . . . , α_l;β₁, . . . , β_m) =

∞

X

n=0

(α₁)_n· · ·(α_l)_n (β₁)_n· · ·(β_m)_n ·zⁿ

n!, (l ≤m+ 1;l, m∈N₀ ={0,1,2, . . .}),

where (a)_n is the Pochhammer symbol defined by (a)_n = ^Γ(a+n)_Γ(a) = a(a + 1)· · ·(a+n−1)forn ∈N={1,2, . . .}and1whenn= 0.

Corresponding to the function

h(α₁, . . . , α_l;β₁, . . . , β_m;z) =z _lF_m(α₁, . . . , α_l;β₁, . . . , β_m)

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the Dziok-Srivastava operator [3],H_m^l (α₁, . . . , α_l;β₁, . . . , β_m)is defined by H_m^l (α₁, . . . , α_l;β₁, . . . , β_m)f(z) = h(α₁, . . . , α_l;β₁, . . . , β_m;z)∗f(z)

=z+

∞

X

n=2

(α₁)n−1· · ·(α_l)n−1

(β1)n−1· · ·(βm)n−1

· a_nzⁿ (n−1)!.

where “∗” stands for convolution.

It is well known [3] that

(1.5) α1H_m^l (α1 + 1, . . . , αl;β1, . . . , βm)f(z)

=z[H_m^l (α₁, . . . , α_l;β₁, . . . , β_m;z)f(z)]⁰

+ (α₁−1)H_m^l (α₁, . . . , α_l;β₁, . . . , β_m)f(z).

To make the notation simple, we write,

H_m^l [α1]f(z) =H_m^l (α1, . . . , αl;β1, . . . , βm;z)f(z).

We note that many subclasses of analytic functions, associated with the Dziok-Srivastava operator H_m^l [α₁] and many special cases, were investigated recently by Dziok-Srivastava [3], Liu [7], Liu and Srivastava [9], [10] and oth- ers. Also we note that special cases of the Dziok-Srivastava linear operator include the Hohlov linear operator [6], the Carlson-Shaffer operator [2], the Ruscheweyh derivative operator [14], the generalized Bernardi-Libera-Livingston linear operator (cf. [1]) and the Srivastava-Owa fractional derivative operators (cf. [11], [12]).

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2. Main Results

In this section we prove some results on the linear operatorH_m^l [α₁]. First is the inclusion theorem.

Theorem 2.1. Let<α₁ > ^1−γ_k+1, andf ∈ A. IfH_m^l [α₁+ 1]f ∈U ST(k, γ)then H_m^l [α₁]f ∈U ST(k, γ).

In order to prove the above theorem we shall need the following lemma which is due to Eenigenburg, Miller, Mocanu, and Read [4].

Lemma A. Let β, γ be complex constants and hbe univalently convex in the unit diskU withh(0) =cand<(βh(z) +γ) >0. Letg(z) = c+P∞

n=1p_nzⁿ be analytic inU. Then

g(z) + zg⁰(z)

βg(z) +γ ≺h(z)⇒g(z)≺h(z).

Proof of Theorem2.1. Setting p(z) = z(H_m^l [α₁]f(z))⁰/(H_m^l [α₁]f(z))in (1.5) we can write

(2.1) α₁H_m^l [α1+ 1]f(z)

H_m^l [α₁]f(z) = z(H_m^l [α1]f(z))⁰

H_m^l [α₁]f(z) + (α₁−1) = p(z) + (α₁−1).

Differentiating (2.1) yields

(2.2) z(H_m^l [α₁+ 1]f(z))⁰

H_m^l [α1+ 1] =p(z) + zp⁰(z) p(z) + (α1−1).

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From this and the argument given in Section1we may write p(z) + zp⁰(z)

p(z) + (α₁−1) ≺q_k,γ(z).

Therefore the theorem follows by Lemma Aand the condition (1.4) sinceq_k,γ is univalent and convex inU and<(q_k,γ)> ^k+γ_k+1.

Theorem 2.2. Let<α₁ > ^1−γ_k+1, andf ∈A. IfH_m^l [α₁ + 1]f ∈U CV(k, γ)then H_m^l [α₁]f ∈U CV(k, γ).

Proof. By virtue of (1.1), (1.2) and Theorem2.1we have H_m^l [α₁+ 1]f ∈U CV(k, γ)⇔z H_m^l [α₁+ 1]f⁰

∈U ST(k, γ)

⇔H_m^l [α₁+ 1]zf⁰ ∈U ST(k, γ)

⇒H_m^l [α₁]zf⁰ ∈U ST(k, γ)

⇔H_m^l [α₁]f ∈U CV(k, γ).

and the proof is complete.

We next prove

Theorem 2.3. Let<α1 > ^1−γ_k+1, and f ∈ A. IfH_m^l [α1 + 1]f ∈ U CC(k, γ, β) thenH_m^l [α₁]f ∈U CC(k, γ, β).

To prove the above theorem, we shall need the following lemma which is due to Miller and Mocanu [10].

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Lemma B. Lethbe convex in the unit diskU and letE ≥ 0. SupposeB(z)is analytic inU with<B(z)≥E. Ifg is analytic inU andg(0) =h(0). Then

Ez²g⁰⁰(z) +B(z)zg⁰(z) +g(z)≺h(z)⇒g(z)≺h(z).

Proof of Theorem2.3. Since H_m^l [α₁ + 1]f ∈ U CC(k, γ, β), by definition, we can write

z(H_m^l [α₁+ 1]f)⁰(z)

k(z) ≺q_k,γ(z)

for somek(z)∈U ST(k, β).Forg such thatH_m^l [α₁+ 1]g(z) =k(z), we have (2.3) z(H_m^l [α1+ 1]f)⁰(z)

H_m^l [α₁+ 1]g(z) ≺q_k,γ(z).

Letting h(z) = ^z(H_(H^ml^l^[α¹^]f)⁰^(z)

m[α1]g)(z) andH(z) = ^z(H_H^ml^l^[α¹^]g)⁰^(z)

m[α1]g(z) we observe that h and H are analytic inU andh(0) = H(0) = 1.Now, by Theorem2.1, H_m^l [α₁]g ∈ U ST(k, β)and so<H(z)> ^k+β_k+1. Also, note that

(2.4) z(H_m^l [α1]f)⁰(z) = (H_m^l [α1]g(z))h(z).

Differentiating both sides of (2.4) yields z(H_m^l [α₁](zf⁰))⁰(z)

H_m^l [α₁]g(z) = z(H_m^l [α₁]g)⁰(z)

H_m^l [α₁]g(z) h(z) +zh⁰(z) = H(z)h(z) +zh⁰(z).

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Now using the identity (1.5) we obtain z(H_m^l [α1+ 1]f)⁰(z)

H_m^l [α₁+ 1]g(z) (2.5)

= H_m^l [α₁+ 1](zf⁰)(z) H_m^l [α₁ + 1]g(z)

= z(H_m^l [α₁](zf⁰))⁰(z) + (α₁−1)H_m^l [α₁](zf⁰)(z) z(H_m^l [α₁]g)⁰(z) + (α₁−1)H_m^l [α₁]g(z)

=

z(H_m^l [α1](zf⁰))⁰(z)

H_m^l[α1]g(z) + (α₁−1)^H^l^m_H^[αl ¹^](zf⁰^)(z) m[α1]g(z) z(H_m^l[α1]g)⁰(z)

H_m^l[α1]g(z) + (α₁−1)

= H(z)h(z) +zh⁰(z) + (α₁−1)h(z) H(z) + (α₁−1)

=h(z) + 1

H(z) + (α₁−1)zh⁰(z).

From (2.3), (2.4), and (2.5) we conclude that

h(z) + 1

H(z) + (α₁−1)zh⁰(z)≺q_k,γ(z).

On lettingE = 0andB(z) = _H(z)+(α¹

1−1), we obtain

<(B(z)) = 1

|(α₁−1) +H(z)|²<((α₁−1) +H(z))>0.

The above inequality satisfies the conditions required by Lemma B. Hence h(z)≺qkγ(z)and so the proof is complete.

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Using a similar argument to that in Theorem2.2we can prove

Theorem 2.4. Let <α₁ > ^1−γ_k+1, andf ∈ A. IfH_m^l [α₁+ 1]f ∈ U QC(k, γ, β), thenH_m^l [α₁]f ∈U QC(k, γ, β).

Finally, we examine the closure properties of the above classes of functions under the generalized Bernardi-Libera-Livingston integral operatorL_c(f) which is defined by

L_c(f) = c+ 1 z^c

Z z

0

t^c−1f(t)dt, c >−1.

Theorem 2.5. Letc > ^−(k+γ)_k+1 .IfH_m^l [α₁]f ∈U ST(k, γ)so isL_c(H_m^l [α₁]f).

Proof. From definition ofL_c(f)and the linearity of operatorH_m^l [α₁]we have (2.6) z(H_m^l [α₁]L_c(f))⁰(z) = (c+ 1)H_m^l [α₁]f(z)−c(H_m^l [α₁]L_c(f))(z).

Substituting ^z(H_H^ml^l ^[α¹^]L^c^(f))⁰^(z)

m[α1]Lc(f)(z) =p(z)in (2.6) we may write (2.7) p(z) = (c+ 1) H_m^l [α1]f(z)

(H_m^l [α₁]L_c(f))(z)−c.

Differentiating (2.7) gives

z(H_m^l [α₁]f)⁰(z)

(H_m^l [α1]f)(z) =p(z) + zp⁰(z) p(z) +c.

Now, the theorem follows by LemmaA, since<(qk,γ(z) +c)>0.

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A similar argument leads to

Theorem 2.6. Letc > ^−(k+γ)_k+1 .IfH_m^l [α₁]f ∈U CV(k, γ)so isL_c(H_m^l [α₁]f).

Theorem 2.7. Letc > ^−(k+γ)_k+1 .IfH_m^l [α₁]f ∈U CC(k, γ, β)so isL_c(H_m^l [α₁]f).

Proof. By definition, there exists a functionk(z) = (H_m^l [α₁]g)(z)∈U ST(k, β) such that

(2.8) z(H_m^l [α₁]f)⁰(z)

(H_m^l [α₁]g)(z) ≺q_k,γ(z) (z∈U).

Now from (2.6) we have z(H_m^l [α₁]f)⁰(z)

(H_m^l [α₁]g)(z) = z(H_m^l [α₁]L_c(zf⁰))⁰(z) +cH_m^l [α₁]L_c(zf⁰)(z) z(H_m^l [α₁]L_c(g(z)))⁰(z) +c(H_m^l [α₁]L_c(g))(z)

=

z(H_m^l [α1]Lc(zf⁰))⁰(z)

(H_m^l[α1]Lc(g))(z) + ^c(H_(H^m^ll^[α¹^]L^c^(zf⁰^))(z) m[α1]Lc(g))(z) z(H_m^l [α1]Lc(g))⁰(z)

(H_m^l[α1]Lc(g))(z) +c . (2.9)

SinceH_m^l [α1]g ∈U ST(k, β), by Theorem2.5, we haveLc(H_m^l [α1]g)∈U ST(k, β).

Letting ^z(H_H^ml^l^[α¹^]L^c^(g))⁰

m[α1]Lc(g) =H(z), we note that<(H(z))> ^k+β_k+1. Now, lethbe defined by

(2.10) z(H_m^l [α₁]L_c(f))⁰ =h(z)H_m^l [α₁]L_c(g).

Differentiating both sides of (2.10) yields z(H_m^l [α₁](zL_c(f))⁰)⁰(z)

(H_m^l [α₁]L_c(g))(z) =zh⁰(z) +h(z)z(H_m^l [α₁]L_c(g))⁰(z) (H_m^l [α₁]L_c(g))(z) (2.11)

=zh⁰(z) +H(z)h(z).

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Therefore from (2.9) and (2.11) we obtain z(H_m^l [α₁]f)⁰(z)

(H_m^l [α₁]g)(z) = zh⁰(z) +h(z)H(z) +ch(z)

H(z) +c .

This in conjunction with (2.8) leads to

(2.12) h(z) + zh⁰(z)

H(z) +c ≺q_k,γ(z).

Letting B(z) = _H(z)+c¹ in (2.12) we note that <(B(z)) > 0 if c > −^k+β_k+1. Now for E = 0 andB as described we conclude the proof since the required conditions of LemmaBare satisfied.

A similar argument yields

Theorem 2.8. Letc > ^−(k+γ)_k+1 .IfH_m^l [α₁]f ∈U QC(k, γ, β)so isL_c(H_m^l [α₁]f).

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