Paper2-Acta25-2011. 133KB Jun 04 2011 12:03:16 AM

CERTAIN DIFFERENTIAL SUPERORDINATIONS USING A

GENERALIZED S ˘AL ˘AGEAN AND RUSCHEWEYH OPERATORS

Alb Lupas¸ Alina

Abstract. In the present paper we define a new operator using the generalized S˘al˘agean and Ruscheweyh operators. Denote by DRm_λ the Hadamard product of the generalized S˘al˘agean operator D_λm and the Ruscheweyh operator Rm, given by DRm_λ : A → A, DRm_λf(z) = (Dm_λ ∗Rm)f(z) and An = {f ∈ H(U), f(z) = z+an+1zn+1+. . . , z∈U}is the class of normalized analytic functions withA1 =A. We study some differential superordinations regarding the operator DRm_λ.

2000Mathematics Subject Classification: 30C45, 30A20, 34A40.

Keywords: Differential superordination, convex function, best subordinant, dif-ferential operator.

1. Introduction and definitions

Denote byU the unit disc of the complex plane U ={z∈C_:|z|<1}andH(U) the space of holomorphic functions in U.

Let

An={f ∈ H(U), f(z) =z+an+1zn+1+. . . , z∈U} with A1=Aand

H[a, n] ={f ∈ H(U), f(z) =a+a_nzn+a_n+1zn+1+. . . , z ∈U} fora∈C_{and n∈}N_.

Iff andgare analytic functions inU, we say thatf is superordinate tog, written

g≺f, if there is a function wanalytic inU, withw(0) = 0,|w(z)|<1, for allz∈U

such that g(z) =f(w(z)) for all z ∈U. If f is univalent, theng ≺f if and only if

f(0) =g(0) and g(U)⊆f(U).

Let ψ : C2 _×U → C _and h analytic in U. If p and ψ(p(z), zp′

(z) ;z) are univalent inU and satisfies the (first-order) differential superordination

h(z)≺ψ(p(z), zp′

then pis called a solution of the differential superordination. The analytic function

q is called a subordinant of the solutions of the differential superordination, or more simply a subordinant, if q ≺p for all p satisfying (1). An univalent subordinant qe that satisfiesq ≺qefor all subordinantsq of (1) is said to be the best subordinant of (1). The best subordinant is unique up to a rotation of U.

Definition 1 (Al Oboudi [4]) For f ∈ A, λ ≥ 0 and m ∈ N_{, the operator D}m λ is

defined by Dm_λ :A → A,

D0_λf(z) = f(z)

D1_λf(z) = (1−λ)f(z) +λzf′

(z) =D_λf(z)

...

Dm_λf(z) = (1−λ)Dm−₁

λ f(z) +λz(Dλmf(z))

′

=Dλ D_λm−1f(z), f or z ∈U.

Remark 1If f ∈ Aand f(z) =z+P∞

j=2ajzj, thenD_λmf(z) =z+

P∞

j=2[1 + (j−1)λ]majzj, forz∈U.

Remark 2 For λ = 1 in the above definition we obtain the S˘al˘agean differential operator [7].

Definition 2 (Ruscheweyh [6]) For f ∈ A, m ∈N, the operator Rm is defined by

Rm :A → A,

R0f(z) = f(z)

R1f(z) = zf′

(z)

...

(m+ 1)Rm+1f(z) = z(Rmf(z))′

+mRmf(z), z∈U.

Remark 3 Iff ∈ A,f(z) =z+P∞

j=2ajzj, then Rmf(z) =z+ P∞

j=2Cmm+j−1ajz

j_, z∈U.

Definition 3([5]) We denote byQthe set of functions that are analytic and injective on U\E(f), where E(f) = {ζ ∈ ∂U : lim

z→_ζ

f(z) = ∞}, and f′

(ζ) 6= 0 for ζ ∈

∂U\E(f). The subclass of Q for which f(0) =a is denoted byQ(a).

We will use the following lemmas.

Lemma 1 (Miller and Mocanu [5]) Let h be a convex function with h(0) =a, and let γ ∈C_{\{0} be a complex number withRe γ ≥0. Ifp∈ H[a, n]∩Q, p(z) +}1

γzp

′

(z)

is univalent in U and

h(z)≺p(z) + 1

γzp

′

(z), for z∈U,

then

where q(z) = γ

nzγ/n Rz

0 h(t)tγ/n

−1

dt, for z ∈ U. The function q is convex and is the best subordinant.

Lemma 2 (Miller and Mocanu [5]) Let q be a convex function in U and let h(z) =

q(z) +_γ1zq′

(z),for z∈U,where Re γ ≥0. If p∈ H[a, n]∩Q, p(z) +1_γzp′

(z) is univalent inU and

q(z) + 1

γzq

′

(z)≺p(z) + 1

γzp

′

(z), for z∈U,

then

q(z)≺p(z), for z∈U,

whereq(z) = γ

nzγ/n Rz

0 h(t)tγ/n

−₁

dt,forz∈U.The functionq is the best subordinant.

2. Main results

Definition 4 ([2]) Let λ≥ 0 and m ∈ N_{. Denote by DR}m

λ the operator given by

the Hadamard product (the convolution product) of the generalized S˘al˘agean operator

Dm_λ and the Ruscheweyh operator Rm, DR_λm:A → A,

DR_λmf(z) = (D_λm∗Rm)f(z).

Remark 4If f ∈ Aand f(z) =z+P∞

j=2ajzj,then DRm_λf(z) =z+P∞

j=2Cmm+j−₁[1 + (j−1)λ]

m_a2

jzj, forz∈U.

Remark 5 For λ = 1 we obtain the Hadamard product SRn [1] of the S˘al˘agean operatorSn and Ruscheweyh operator Rn.

Theorem 1Lethbe a convex function,h(0) = 1.Letλ≥0, m∈N_{, f ∈ Aand}

sup-pose that _(mλm+1_+1)zDR_λm+1f(z)− m(1−_λ)

(mλ+1)zDRmλf(z) is univalent and (DRmλf(z))

′

∈ H[1,1]∩Q. If

h(z)≺ m+ 1 (mλ+ 1)zDR

m+1

λ f(z)−

m(1−λ) (mλ+ 1)zDR

m

λf(z), for z∈U, (2)

then

q(z)≺(DR_λmf(z))′

, for z∈U,

where q(z) = m+ 1 λ

zm+ 1λ Rz

0 h(t)tm

−₁₊1

λdt. The function q is convex and it is the best

subordinant.

Proof. With notationp(z) = (DRm_λf(z))′

= 1+P∞

j=2Cmm+j−1[1 + (j−1)λ]

m_·

ja2_jzj−₁

and p(0) = 1, we obtain forf(z) =z+P∞

p(z) +zp′

(z) = m_λz+1DRm_λ+1f(z)− m−1 +1_λ(DRm_λf(z))′

−m(1_λz−λ)DRm_λf(z) and p(z) +_mλλ₊₁zp′

(z) = _(mλm+1_+1)zDRm_λ+1f(z)−₍m_mλ(1₊₁₎−λ)_zDRm_λf(z).

Evidently p∈ H[1,1]. Then (2) becomes

h(z)≺p(z) + λ

mλ+ 1zp

′

(z), for z∈U.

By using Lemma 1 forγ =m+1_λ and n= 1, we have

q(z)≺p(z), for z∈U, i.e. q(z)≺(DRm_λf(z))′

, for z∈U,

where q(z) = m+ 1 λ

zm+ 1λ Rz

0 h(t)tm

−₁₊1

λdt. The function q is convex and it is the best subordinant.

Corollary 1 ([3]) Let h be a convex function, h(0) = 1. Let n ∈ N_{, f ∈ A}

and suppose that 1_zSRn+1f(z) + _nn₊₁z(SRnf(z))′′

is univalent and (SRnf(z))′

∈ H[1,1]∩Q. If

h(z)≺ 1

zSR

n+1_{f(z) +} n

n+ 1z(SR

n_f(z))′′

, for z∈U, (3)

then

q(z)≺(SRnf(z))′

, for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is convex and it is the best subordinant.

Theorem 2Let q be convex inU and lethbe defined by h(z) =q(z) +_mλλ₊₁zq′

(z),

λ≥0, m∈N_{. Iff ∈ A, suppose that} m+1

(mλ+1)zDR m+1

λ f(z)−

m(1−_λ)

(mλ+1)z· DRm_λf(z) is univalent, (DRm_λf(z))′

∈ H[1,1]∩Q and satisfies the differential su-perordination

h(z) =q(z)+ λ

mλ+ 1zq

′

(z)≺ m+ 1 (mλ+ 1)zDR

m+1

λ f(z)−

m(1−λ) (mλ+ 1)zDR

m

λf(z), (4)

for z∈U,then

q(z)≺(DR_λmf(z))′

, for z∈U,

where q(z) = m+ 1 λ

zm+ 1λ Rz

0 h(t)t

m−1+1

λdt. The functionq is the best subordinant.

Proof. Letp(z) = (DRm_λf(z))′

= 1 +P∞

j=2Cmm+j−₁[1 + (j−1)λ]

m_ja2

jzj

−₁

. Differentiating, we obtainp(z) +zp′

(z) = m_λz+1DRm_λ+1f(z)− m−1 +1_λ· (DRm_λf(z))′

− m(1_λz−λ)DRn_λf(z) and p(z) + _mλλ₊₁zp′

m(1−_λ)

(mλ+1)zDRmλf(z),forz∈U and (4) becomes q(z) + λ

mλ+ 1zq

′

(z)≺p(z) + λ

mλ+ 1zp

′

(z), for z∈U.

Using Lemma 2 forγ =m+_λ1 and n= 1, we have

q(z)≺p(z), for z∈U, i.e. q(z) = m+ 1

λ zm+λ1

Z z

0

h(t)tm−1+1

λdt≺(DRm

λf(z))

′

,

forz∈U,and q is the best subordinant.

Corollary 2 ([3]) Let q be convex in U and let h be defined by h(z) = q(z) +

zq′

(z).Ifn∈N_{,f ∈ A, suppose that} 1

zSRn+1f(z) +nn+1z(SRnf(z))

′′

is univalent,

(SRnf(z))′

∈ H[1,1]∩Qand satisfies the differential superordination

h(z) =q(z) +zq′

(z)≺ 1

zSR

n+1_{f(z) +} n

n+ 1z(SR

n_f(z))′′

, for z∈U, (5)

then

q(z)≺(SRnf(z))′

, for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is the best subordinant.

Theorem 3 Let h be a convex function, h(0) = 1. Let λ ≥0, m ∈ N, f ∈ A and suppose that (DR_λmf(z))′

is univalent and DRmλf(z)

z ∈ H[1,1]∩Q. If h(z)≺(DRm_λf(z))′

, for z∈U, (6)

then

q(z)≺ DR

m λf(z)

z , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is convex and it is the best subordinant. Proof. Considerp(z) = DRmλf(z)

z =

z+P∞

j=2Cm+jm −1[1+(j−1)λ] m_a2

jzj

z =

1 +P∞

j=2Cmm+j−₁[1 + (j−1)λ]

m_a2

jzj

−₁

.Evidently p∈ H[1,1]. We have p(z) +zp′

(z) = (DRm_λf(z))′

, forz∈U. Then (6) becomes

h(z)≺p(z) +zp′

(z), for z∈U.

By using Lemma 1 forγ = 1 and n= 1, we have

q(z)≺p(z), for z∈U, i.e. q(z)≺ DR

m λf(z)

where q(z) = 1_zR₀zh(t)dt. The functionq is convex and it is the best subordinant. Corollary 3 ([3]) Let h be a convex function, h(0) = 1. Let n ∈ N_{, f ∈ A and}

suppose that (SRnf(z))′

is univalent and SRn_zf(z) ∈ H[1,1]∩Q. If

h(z)≺(SRnf(z))′

, for z∈U, (7)

then

q(z)≺ SR

n_f(z)

z , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is convex and it is the best subordinant.

Theorem 4 Let q be convex in U and let h be defined by h(z) =q(z) +zq′

(z).If

λ≥0,m∈N_{,f ∈ A, suppose that(DR}m λf(z))

′

is univalent, DRλmf(z)

z ∈ H[1,1]∩Q

and satisfies the differential superordination

h(z) =q(z) +zq′

(z)≺(DRm_λf(z))′

, for z∈U, (8)

then

q(z)≺ DR

m λf(z)

z , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is the best subordinant. Proof. Letp(z) = DRmλf(z)

z =

z+P∞

j=2Cm+jm −1[1+(j−1)λ] m_a2

jzj

z =

1 +P∞

j=2Cmm+j−₁[1 + (j−1)λ]

m_a2

jzj

−1_.

Evidently p∈ H[1,1]. Differentiating, we obtain p(z) +zp′

(z) = (DRm_λf(z))′

, for z ∈ U and (8) be-comes

q(z) +zq′

(z)≺p(z) +zp′

(z), for z∈U.

Using Lemma 2 forγ = 1 and n= 1, we have

q(z)≺p(z), for z∈U, i.e. q(z) = 1

z

Z z

0

h(t)dt≺ DR

m λf(z)

z , for z∈U,

and q is the best subordinant.

Corollary 4 ([3]) Letq be convex inU and lethbe defined byh(z) =q(z)+zq′

(z).

If n∈ N_{, f ∈ A, suppose that (SR}n_f(z))′ _{is univalent,} SRnf(z)

z ∈ H[1,1]∩Q and

satisfies the differential superordination

h(z) =q(z) +zq′

(z)≺(SRnf(z))′

, for z∈U, (9)

then

q(z)≺ SR

n_f(z)

where q(z) = 1_zR₀zh(t)dt. The function q is the best subordinant.

Theorem 5 Let h(z) = 1+(2β−_1)z

1+z be a convex function in U, where 0 ≤ β < 1.

Let λ≥0, m∈N, f ∈ Aand suppose that (DRm_λf(z))′

is univalent and DRmλf(z)

z ∈

H[1,1]∩Q. If

h(z)≺(DRm_λf(z))′

, for z∈U, (10)

then

q(z)≺ DR

m λf(z)

z , for z∈U,

where q is given by q(z) = 2β−1 + 2(1−β)ln(1+_z z), for z ∈ U. The function q is convex and it is the best subordinant.

Proof. Following the same steps as in the proof of Theorem and considering

p(z) = DRmλf(z)

z , the differential superordination (10) becomes h(z) = 1 + (2β−1)z

1 +z ≺p(z) +zp

′

(z), for z∈U.

By using Lemma 1 forγ = 1 and n= 1, we have q(z)≺p(z), i.e.,

q(z) = 1

z

Z z

0

h(t)dt= 1

z

Z z

0

1 + (2β−1)t

1 +t dt= 2β−1+2(1−β)

1

zln(z+1)≺

DRm_λf(z)

z ,

forz∈U.

The function q is convex and it is the best subordinant.

Theorem 6 Let h be a convex function, h(0) = 1. Let λ ≥0, m ∈ N, f ∈ A and suppose that

zDRm+1_λ f(z)

DRm λf(z)

′

is univalent and DRm+1λ f(z)

DRm

λf(z) ∈ H[1,1]∩Q. If

h(z)≺ zDR

m+1

λ f(z) DRm_λf(z)

!′

, for z∈U, (11)

then

q(z)≺ DR

m+1

λ f(z)

DRm_λf(z) , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is convex and it is the best subordinant. Proof. Considerp(z) = DR

m+1 λ f(z)

DRm

λf(z) =

z+P∞

j=2Cm+jm+1[1+(j−1)λ]m+1a2jzj

z+P∞

j=2Cm+jm −1[1+(j−1)λ] m_a2

jzj = 1+P∞

j=2Cm+jm+1[1+(j−1)λ]m+1a2jzj

−1 1+P∞

j=2Cm+jm −1[1+(j−1)λ] m

a2 jzj

We have p′

(z) = (DR m+1 λ f(z))

′

DRm

λf(z) −p(z)· (DRm

λf(z))

′

DRm λf(z) .

Then p(z) +zp′

(z) =

zDRm+1_λ f(z)

DRm λf(z)

′

.

Then (11) becomes

h(z)≺p(z) +zp′

(z), for z∈U.

By using Lemma 1 forγ = 1 and n= 1, we have

q(z)≺p(z), for z∈U, i.e. q(z)≺ DR

m+1

λ f(z)

DRm_λf(z) , for z∈U, where q(z) = 1_zR₀zh(t)dt. The functionq is convex and it is the best subordinant. Corollary 5 ([3]) Let h be a convex function, h(0) = 1. Let n ∈ N, f ∈ A and suppose that zSR_SRn+1n_f(f_z(₎z)

′

is univalent and SR_SRn+1n_ff_(z(₎z) ∈ H[1,1]∩Q. If

h(z)≺

zSRn+1f(z)

SRn_f(z)

′

, for z∈U, (12)

then

q(z)≺ SR

n+1_f(z)

SRn_f(z) , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is convex and it is the best subordinant.

Theorem 7 Let q be convex in U and let h be defined by h(z) = q(z) +zq′

(z).

If λ ≥ 0, m ∈ N_{, f ∈ A, suppose that}

zDRm+1_λ f(z)

DRm λf(z)

′

is univalent, DRm+1λ f(z)

DRm

λf(z) ∈ H[1,1]∩Q and satisfies the differential superordination

h(z) =q(z) +zq′

(z)≺ zDR

m+1

λ f(z) DRm_λf(z)

!′

, for z∈U, (13)

then

q(z)≺ DR

m+1

λ f(z)

DRm_λf(z) , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is the best subordinant. Proof. Letp(z) = DRm

+1 λ f(z)

DRm

λf(z) =

z+P∞

j=2Cm+jm+1[1+(j−1)λ]m+1a2jzj

z+P∞

j=2Cm+jm −1[1+(j−1)λ] m_a2

jzj = 1+P∞

j=2Cm +1

m+j[1+(j−1)λ]m +1

a2 jzj

−1 1+P∞

j=2Cmm+j−1[1+(j−1)λ] m_a2

jzj

Differentiating, we obtain p(z) +zp′

(z) =

zDRm+1_λ f(z)

DRm λf(z)

′

, for z ∈ U and (13) becomes

q(z) +zq′

(z)≺p(z) +zp′

(z), for z∈U.

Using Lemma 2 forγ = 1 and n= 1, we have

q(z)≺p(z), for z∈U, i.e. q(z) = 1

z

Z z

0

h(t)dt≺ DR

m+1

λ f(z)

DRm_λf(z) , for z∈U, and q is the best subordinant.

Corollary 6 ([3]) Letq be convex inU and lethbe defined byh(z) =q(z)+zq′

(z).

If n ∈ N_, f ∈ A, suppose that zSR_SRn+1n_f(f_z(₎z) ′

is univalent, SR_SRn+1n_ff_(z(z₎) ∈ H[1,1]∩Q

and satisfies the differential superordination

h(z) =q(z) +zq′

(z)≺

zSRn+1f(z)

SRn_f(z)

′

, for z∈U, (14)

then

q(z)≺ SR

n+1_f(z)

SRn_f(z) , for z∈U,

where q(z) = 1_zR₀zh(t)dt. The function q is the best subordinant.

Theorem 8 Let h(z) = 1+(2β−1)z

1+z be a convex function inU, where 0≤β <1. Let λ ≥ 0, m ∈ N_{, f ∈ A and suppose that}

zDRm+1_λ f(z)

DRm λf(z)

′

is univalent, DRm+1λ f(z)

DRm λf(z) ∈ H[1,1]∩Q. If

h(z)≺ zDR

m+1

λ f(z) DRm_λf(z)

!′

, for z∈U, (15)

then

q(z)≺ DR

m+1

λ f(z)

DRm_λf(z) , for z∈U,

where q is given by q(z) = 2β−1 + 2(1−β)ln(1+_z z), for z ∈ U. The function q is convex and it is the best subordinant.

Proof. Following the same steps as in the proof of Theorem and considering

p(z) = DRm +1 λ f(z)

DRm

λf(z) , the differential superordination (15) becomes

h(z) = 1 + (2β−1)z

1 +z ≺p(z) +zp

′

By using Lemma 1 for γ = 1 and n= 1, we haveq(z)≺p(z), i.e.,

q(z) = 1

z

Z z

0

h(t)dt= 1

z

Z z

0

1 + (2β−1)t

1 +t dt= 2β−1+2(1−β)

1

zln(z+1)≺

DRm_λ+1f(z)

DRm_λf(z) , forz∈U.

The function q is convex and it is the best subordinant. References

[1] Alina Alb Lupa¸s, Adriana C˘ata¸s, Certain differential subordinations using S˘al˘agean and Ruscheweyh operators, Proceedings of International Conference on Complex Analysis and Related Topics, The 11th Romanian-Finish Seminar, Alba Iulia, 2008, (to appear).

[2] Alina Alb Lupa¸s,Certain differential subordinations using a generalized S˘al˘agean operator and Ruscheweyh operator, submitted 2009.

[3] Alina Alb Lupa¸s, Certain differential superordinations using S˘al˘agean and Ruscheweyh operators, Analele Universit˘at¸ii din Oradea, Fascicola Matematica, Tom XVII, Issue no. 2, 2010, 201-208.

[4] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, Ind. J. Math. Math. Sci., 2004, no.25-28, 1429-1436.

[5] S.S. Miller, P.T. Mocanu,Subordinants of Differential Superordinations, Com-plex Variables, vol. 48, no. 10, 2003, 815-826.

[6] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115.

[7] G.St. S˘al˘agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372.

Alb Lupa¸s Alina

Department of Mathematics University of Oradea