Inclusion relations fork−uniformly starlike functions under the Dziok-Srivastava operator are established

http://jipam.vu.edu.au/

Volume 6, Issue 2, Article 52, 2005

THE DZIOK-SRIVASTAVA OPERATOR AND k−UNIFORMLY STARLIKE FUNCTIONS

R. AGHALARY AND GH. AZADI UNIVERSITY OFURMIA

URMIA, IRAN

raghalary@yahoo.com azadi435@yahoo.com

Received 04 January, 2005; accepted 12 April, 2005 Communicated by H.M. Srivastava

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.

Key words and phrases: Starlike, Convex, Linear operators.

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

1. INTRODUCTION

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

n=2a_nzⁿwhich are analytic in the open unit discU ={z :|z|<1}. A functionf ∈Ais said to be inU ST(k, γ), the class of k−uniformly starlike functions of orderγ, 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 functionf 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 ,

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

003-05

withp(z) = ^zf_f(z)^0(z) orp(z) = 1 + ^zf_f0⁰⁰(z)^(z) and considering the functions which mapU on to the conic domainΩ_k,γ, such that1∈Ω_k,γ, we may rewrite the conditions (1.1) or (1.2) in the form

(1.3) p(z)≺q_k,γ(z).

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

1−z , and q_1,γ(z) = 1 +2(1−γ) π²

log1 +√ 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 andK is such thatk = cosh^πK_4K(z)^0(z).

By virtue of (1.3) and the properties of the domainsΩ_k,γ we 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).

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 orderγ and typeβ andU QC(0, γ, β)is the class of quasi-convex functions of orderγ and typeβ.

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

Forαj ∈C(j = 1,2,3, . . . , l)andβj ∈C− {0,−1,−2, . . .}(j = 1,2, . . . m), the general- ized hypergeometric function is defined by

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

∞

X

n=0

(α1)n· · ·(αl)n

(β₁)_n· · ·(β_m)_n · zⁿ n!, (l≤m+ 1;l, m∈N₀ ={0,1,2, . . .}),

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

Corresponding to the functionh(α₁, . . . , α_l;β₁, . . . , β_m;z) = z _lF_m(α₁, . . . , α_l;β₁, . . . , β_m) 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· · ·(β_m)_n−1 · a_nzⁿ (n−1)!.

where “∗” stands for convolution.

It is well known [3] that

(1.5) α₁H_m^l (α₁+ 1, . . . , α_l;β₁, . . . , β_m)f(z)

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

+ (α1−1)H_m^l (α1, . . . , αl;β1, . . . , βm)f(z).

To make the notation simple, we write,

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

2. MAINRESULTS

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

Theorem 2.1. Let <α1 > ^1−γ_k+1, and f ∈ A. If H_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. Letβ, γ be complex constants andhbe univalently convex in the unit disk U with h(0) =cand<(βh(z) +γ)>0. Letg(z) = c+P∞

n=1p_nzⁿbe analytic inU. Then g(z) + zg⁰(z)

βg(z) +γ ≺h(z)⇒g(z)≺h(z).

Proof of Theorem 2.1. Settingp(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). From this and the argument given in Section 1 we may write

p(z) + zp⁰(z)

p(z) + (α₁−1) ≺q_k,γ(z).

Therefore the theorem follows by Lemma A and 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, and f ∈ A. If H_m^l [α₁ + 1]f ∈ U CV(k, γ) thenH_m^l [α₁]f ∈ U CV(k, γ).

Proof. By virtue of (1.1), (1.2) and Theorem 2.1 we 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, andf ∈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].

Lemma B. Let hbe convex in the unit diskU and letE ≥ 0. Suppose B(z)is analytic in U with<B(z)≥E. Ifgis analytic inU andg(0) =h(0). Then

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

Proof of Theorem 2.3. SinceH_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).

Lettingh(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 thathandHare analytic in U andh(0) =H(0) = 1.Now, by Theorem 2.1,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).

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

H_m^l [α₁+ 1]g(z) = H_m^l [α₁+ 1](zf⁰)(z) H_m^l [α₁+ 1]g(z) (2.5)

= 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)^Hm_H^{l [α}l ^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. Henceh(z)≺ qkγ(z)and

so the proof is complete.

Using a similar argument to that in Theorem 2.2 we 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 gener- alized Bernardi-Libera-Livingston integral operatorL_c(f)which is defined by

Lc(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_Hl^{ml[α1]Lc(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 Lemma A, since<(q_k,γ(z) +c)>0.

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^mll^{[α1]Lc(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 Theorem 2.5, we haveL_c(H_m^l [α₁]g)∈U ST(k, β).

Letting ^z(H_H^ml^{l[α1]Lc(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 (2.11) z(H_m^l [α₁](zL_c(f))⁰)⁰(z)

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

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

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 ≺qk,γ(z).

LettingB(z) = _H(z)+c¹ in (2.12) we note that<(B(z))>0ifc >−^k+β_k+1. Now forE = 0andB as described we conclude the proof since the required conditions of Lemma B are satisfied.

A similar argument yields

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

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