ACTA UNIVERSITATIS APULENSIS Special Issue

A NOTE ON STRONG DIFFERENTIAL SUPERORDINATIONS USING A MULTIPLIER TRANSFORMATION AND

RUSCHEWEYH OPERATOR

Alb Lupas¸ Alina and Georgia Irina Oros

Abstract. In the present paper we establish several strong differential su- perordinations regardind the new operator IR^m_λ,l defined by convolution prod- uct of the extended multiplier transformation and the extended Ruscheweyh derivative, IR^m_λ,l:A^∗_nζ → A^∗_nζ, IR^m_λ,lf(z, ζ) = (I(m, λ, l)∗R^m)f(z, ζ), z ∈U, ζ ∈U ,where R^mf(z, ζ) denote the extended Ruscheweyh derivative,

I(m, λ, l)f(z, ζ) is the extended multiplier transformation and A^∗_nζ = {f ∈ H(U ×U), f(z, ζ) = z +a_n+1(ζ)zⁿ⁺¹ +. . . , z ∈ U, ζ ∈ U} is the class of normalized analytic functions.

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

Keywords: strong differential superordination, convex function, best sub- ordinant, extended differential operator, convolution product.

1. Introduction

Denote by U the unit disc of the complex plane U = {z ∈ C : |z| < 1}, U ={z ∈C: |z| ≤1}the closed unit disc of the complex plane andH(U×U) the class of analytic functions inU ×U.

Let

A^∗_nζ ={f ∈ H(U×U), f(z, ζ) =z+a_n+1(ζ)zⁿ⁺¹+. . . , z ∈U, ζ ∈U}, where ak(ζ) are holomorphic functions in U for k≥2, and

H^∗[a, n, ζ] ={f ∈ H(U×U), f(z, ζ) = a+a_n(ζ)zⁿ+a_n+1(ζ)zⁿ⁺¹+. . . , z∈U, ζ ∈U}, for a∈C, n∈N, a_k(ζ) are holomorphic functions in U for k≥n.

We also extend the known differential operators to the new class of analytic functions A^∗_nζ introduced in [15].

Definition No. 1 [7] For n ∈ N, m ∈ N∪ {0}, λ, l ≥ 0, f ∈ A^∗_nζ, f(z, ζ) = z+P∞

j=n+1a_j(ζ)z^j, the operatorI(m, λ, l)f(z, ζ) is defined by the following infinite series

I(m, λ, l)f(z, ζ) =z+

∞

X

j=n+1

1 +λ(j−1) +l l+ 1

m

a_j(ζ)z^j, z∈U, ζ ∈U . Remark No. 1 [7] It follows from the above definition that

(l+ 1)I(m+ 1, λ, l)f(z, ζ) = [l+ 1−λ]I(m, λ, l)f(z, ζ)+λz(I(m, λ, l)f(z, ζ))⁰_z, z ∈U, ζ ∈U .

Definition No. 2 [4] For f ∈ A^∗_nζ, n, m∈N, the operator R^m is defined by R^m :A^∗_nζ → A^∗_nζ,

R⁰f(z, ζ) = f(z, ζ), R¹f(z, ζ) = zf_z⁰ (z, ζ), ...,

(m+ 1)R^m+1f(z, ζ) = z(R^mf(z, ζ))⁰_z+mR^mf(z, ζ), z ∈U, ζ ∈U . Remark No. 2 [4] If f ∈ A^∗_nζ, f(z, ζ) = z + P∞

j=n+1a_j(ζ)z^j, then R^mf(z, ζ) =z+P∞

j=n+1C_m+j−1^m a_j(ζ)z^j, z ∈U, ζ ∈U .

As a dual notion of strong differential subordination G.I. Oros has intro- duced and developed the notion of strong differential superordinations in [14].

Definition No. 3 [14] Let f(z, ζ), H(z, ζ) analytic in U×U . The func- tion f(z, ζ) is said to be strongly superordinate to H(z, ζ) if there exists a function w analytic in U, with w(0) = 0 and |w(z)|<1, such that H(z, ζ) = f(w(z), ζ),for allζ ∈U. In such a case we writeH(z, ζ)≺≺f(z, ζ), z ∈U, ζ ∈U .

Remark No. 3 [14] (i) Since f(z, ζ) is analytic in U ×U, for all ζ ∈U , and univalent in U, for all ζ ∈ U, Definition 3 is equivalent to H(0, ζ) = f(0, ζ), for all ζ ∈U , and H U ×U

⊂f U×U .

(ii) If H(z, ζ) ≡ H(z) and f(z, ζ) ≡ f(z), the strong superordination becomes the usual notion of superordination.

Definition No. 4 [9] We denote by Q^∗ the set of functions that are ana- lytic and injective on U×U\E(f, ζ), where E(f, ζ) = {y∈∂U : lim

z→yf(z, ζ) =

∞}, and are such that f_z⁰(y, ζ)6= 0 for y ∈∂U ×U\E(f, ζ). The subclass of Q^∗ for which f(0, ζ) =a is denoted byQ^∗(a).

We have need the following lemmas to study the strong differential super- ordinations.

Lemma No. 1 [9] Let h(z, ζ) be a convex function with h(0, ζ) = a and let γ ∈ C^∗ be a complex number with Re γ ≥ 0. If p ∈ H^∗[a, n, ζ] ∩Q^∗, p(z, ζ) + ¹_γzp⁰_z(z, ζ) is univalent in U ×U and

h(z, ζ)≺≺p(z, ζ) + 1

γzp⁰_z(z, ζ), z ∈U, ζ ∈U , then

q(z, ζ)≺≺p(z, ζ), z ∈U, ζ ∈U , where q(z, ζ) = ^γ

nz^γn

Rz

0 h(t, ζ)t^nγ−1dt, z ∈ U, ζ ∈ U . The function q is convex and is the best subordinant.

Lemma No. 2 [9] Let q(z, ζ) be a convex function in U × U and let h(z, ζ) =q(z, ζ) + ¹_γzq_z⁰(z, ζ), z ∈U, ζ ∈U , where Re γ ≥0.

If p∈ H^∗[a, n, ζ]∩Q^∗, p(z, ζ) + _γ¹zp⁰_z(z, ζ) is univalent in U ×U and q(z, ζ) + 1

γzq⁰_z(z, ζ)≺≺p(z, ζ) + 1

γzp⁰_z(z, ζ), z ∈U, ζ ∈U , then

q(z, ζ)≺≺p(z, ζ), z ∈U, ζ ∈U , where q(z, ζ) = ^γ

nz^γn

Rz

0 h(t, ζ)t^γn−1dt, z∈U, ζ ∈U . The function q is the best subordinant.

2. Main results

Definition No. 5[5] Let λ, l≥0andm∈N. Denote byIR^m_λ,lthe operator given by the Hadamard product (the convolution product) of the extended mul- tiplier transformation I(m, λ, l) and the extended Ruscheweyh operator R^m, IR^m_λ,l:A^∗_nζ → A^∗_nζ,

IR_λ,l^mf(z, ζ) = (I(m, λ, l)∗R^m)f(z, ζ). Remark No. 4 [5] If f ∈ A^∗_nζ, f(z, ζ) = z+P∞

j=n+1aj(ζ)z^j, then IR^m_λ,lf(z, ζ) = z+P∞

j=n+1

1+λ(j−1)+l l+1

m

C_m+j−1^m a²_j(ζ)z^j, z ∈U, ζ ∈U .

Remark No. 5 For l = 0, λ ≥ 0, we obtain the extended Hadamard productDRⁿ_λ ([6], [2], [12], [13]) of the extended generalized S˘al˘agean operator Dⁿ_λ and the extended Ruscheweyh operator Rⁿ.

For l = 0 and λ = 1, we obtain the extended Hadamard product SRⁿ ([1], [3], [10], [11]) of the extended S˘al˘agean operator Sⁿ and the extended Ruscheweyh operator Rⁿ.

Theorem No. 1 Leth(z, ζ)be a convex function inU×U withh(0, ζ) = 1. Letm∈N, λ, l≥0, f(z, ζ)∈ A^∗_nζ, F(z, ζ) = I_c(f) (z, ζ) = _z^c+2c+1

Rz

0 t^cf(t, ζ)dt, z ∈ U, ζ ∈ U, Rec > −2, and suppose that IR^m_λ,lf(z, ζ)0

z is univalent in U ×U, IR^m_λ,lF(z, ζ)0

z ∈ H^∗[1, n, ζ]∩Q^∗ and h(z, ζ)≺≺ IR_λ,l^mf(z, ζ)⁰

z, z ∈U, ζ ∈U , (1)

then

q(z, ζ)≺≺ IR^m_λ,lF(z, ζ)0

z, z ∈U, ζ ∈U , where q(z, ζ) = ^c+2

nz^c+2n

Rz

0 h(t, ζ)t^{c+2n −1}dt. The function q is convex and it is the best subordinant.

Proof. We have

z^c+1F (z, ζ) = (c+ 2) Z z

0

t^cf(t, ζ)dt

and differentiating it, with respect toz, we obtain (c+ 1)F (z, ζ)+zF_z⁰(z, ζ) = (c+ 2)f(z, ζ) and

(c+ 1)IR^m_λ,lF (z, ζ)+z IR^m_λ,lF(z, ζ)0

z = (c+ 2)IR^m_λ,lf(z, ζ), z ∈U, ζ ∈U . Differentiating the last relation with respect to z we have

IR^m_λ,lF (z, ζ)0

z+ 1

c+ 2z IR^m_λ,lF(z, ζ)00

z² = IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U . (2) Using (2), the strong differential superordination (1) becomes

h(z, ζ)≺≺ IR^m_λ,lF(z, ζ)0

z+ 1

c+ 2z IR^m_λ,lF (z, ζ)00

z². (3)

Denote

p(z, ζ) = IR^m_λ,lF (z, ζ)0

z, z∈U, ζ ∈U . (4)

Replacing (4) in (3) we obtain h(z, ζ)≺≺p(z, ζ) + 1

c+ 2zp⁰_z(z, ζ) , z ∈U, ζ ∈U . Using Lemma 1 for γ =c+ 2, we have

q(z, ζ)≺≺p(z, ζ), z ∈U, ζ ∈U , i.e. q(z, ζ)≺≺ IR_λ,l^mF (z, ζ)⁰

z,z ∈U, ζ ∈U , where q(z, ζ) = ^c+2

nz^c+2n

Rz

0 h(t, ζ)t^{c+2n −1}dt. The function q is convex and it is the best subordinant.

Corollary No. 1 Let h(z, ζ) = ^{ζ+(2β−ζ)z}_1+z , where β ∈ [0,1). Let m ∈ N, λ, l ≥ 0, f (z, ζ) ∈ A^∗_nζ, F (z, ζ) = I_c(f) (z, ζ) = _z^c+2c+1

Rz

0 t^cf(t, ζ)dt, z ∈ U, ζ ∈ U , Rec > −2, and suppose that IR^m_λ,lf(z, ζ)0

z is univalent in U ×U, IR^m_λ,lF (z, ζ)⁰

z ∈ H^∗[1, n, ζ]∩Q^∗ and h(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z∈U, ζ ∈U , (5)

then

q(z, ζ)≺≺ IR^m_λ,lF (z, ζ)0

z, z∈U, ζ ∈U , where q is given byq(z, ζ) = 2β−ζ+2(c+2)(ζ−β)

nz^c+2n

Rz 0

t^{c+2n −1}

t+1 dt, z ∈U, ζ ∈U . The function q is convex and it is the best subordinant.

Proof. Following the same steps as in the proof of Theorem 1 and con- sidering p(z, ζ) = IR^m_λ,lF(z, ζ)0

z, the strong differential superordination (5) becomes

h(z, ζ) = ζ+ (2β−ζ)z

1 +z ≺≺p(z, ζ) + 1

c+ 2zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ =c+ 2, we haveq(z, ζ)≺≺p(z, ζ), i.e.

q(z, ζ) = c+ 2 nz^c+2n

Z z

0

h(t, ζ)t^{c+2n −1}dt = c+ 2 nz^c+2n

Z z

0

ζ+ (2β−ζ)t

1 +t t^{c+2n −1}dt

= 2β−ζ+2 (c+ 2) (ζ−β) nz^c+2n

Z z

0

t^{c+2n −1}

t+ 1 dt≺≺ IR^m_λ,lF (z, ζ)0

z, z ∈U, ζ ∈U . The function q is convex and it is the best subordinant.

Theorem No. 2Letq(z, ζ)be a convex function inU×U and leth(z, ζ) = q(z, ζ) + _c+2¹ zq⁰_z(z, ζ), where z ∈U, ζ ∈U , Rec >−2.

Letm∈N,λ, l ≥0, f(z, ζ)∈ A^∗_nζ, F(z, ζ) = I_c(f) (z, ζ) = _z^c+2c+1

Rz

0 t^cf(t, ζ)dt, z ∈ U, ζ ∈ U , and suppose that IR_λ,l^mf(z, ζ)0

z is univalent in U × U, IR^m_λ,lF (z, ζ)0

z ∈ H^∗[1, n, ζ]∩Q^∗ and h(z, ζ)≺≺ IR_λ,l^mf(z, ζ)0

z, z ∈U, ζ ∈U , (6)

then

q(z, ζ)≺≺ IR^m_λ,lF (z, ζ)⁰

z, z ∈U, ζ ∈U , where q(z, ζ) = ^c+2

nz^c+2n

Rz

0 h(t, ζ)t^{c+2n −1}dt. The functionq is the best subordinant.

Proof. Following the same steps as in the proof of Theorem 1 and consid- ering p(z, ζ) = IR^m_λ,lF (z, ζ)0

, z ∈ U, ζ ∈U , the strong differential superor- dination (6) becomes

h(z, ζ) =q(z, ζ)+ 1

c+ 2zq_z⁰ (z, ζ)≺≺p(z, ζ)+ 1

c+ 2zp⁰_z(z, ζ) , z ∈U, ζ ∈U . Using Lemma 2 for γ =c+ 2, we have

q(z, ζ)≺≺p(z, ζ), z ∈U, ζ ∈U , i.e. q(z, ζ)≺≺ IR_λ,l^mF (z, ζ)0

z,z ∈U, ζ ∈U , whereq(z, ζ) = ^c+2

nz^c+2n

Rz

0 h(t, ζ)t^{c+2n −1}dt.The functionqis the best subordinant.

Theorem No. 3 Let h(z, ζ) be a convex function, h(0, ζ) = 1. Let λ, l ≥0, m, n∈N, f(z, ζ)∈ A^∗_nζ and suppose that IR^m_λ,lf(z, ζ)⁰

z is univalent and ^IR

m λ,lf(z,ζ)

z ∈ H^∗[1, n, ζ]∩Q^∗. If

h(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U , (7)

then

q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)

z , z ∈U, ζ ∈U , where q(z, ζ) = ¹

nzⁿ¹

Rz

0 h(t, ζ)tⁿ¹⁻¹dt. The functionq is convex and it is the best subordinant.

Proof. Consider p(z, ζ) = ^IR

m λ,lf(z,ζ)

z = ^z+

P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^mCm+j−1^m a²_j(ζ)z^j

z =

1 +P∞ j=n+1

1+λ(j−1)+l l+1

m

C_m+j−1^m a²_j(ζ)z^j−1. Evidently p∈ H^∗[1, n, ζ].

We have p(z, ζ) +zp⁰_z(z, ζ) = IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U. Then (7) becomes

h(z, ζ)≺≺p(z, ζ) +zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ = 1, we have

q(z, ζ)≺≺p(z, ζ), z∈U, ζ ∈U , i.e. q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)

z , z ∈U, ζ ∈U , where q(z, ζ) = ¹

nzⁿ¹

Rz

0 h(t, ζ)tⁿ¹⁻¹dt. The function q is convex and it is the best subordinant.

Corollary No. 2 Let h(z, ζ) = ^{ζ+(2β−ζ)z}_1+z be a convex function in U ×U, where 0 ≤ β < 1. Let λ, l ≥ 0, m, n ∈ N, f(z, ζ) ∈ A^∗_nζ and suppose that

IR^m_λ,lf(z, ζ)0

z is univalent and ^IR

m λ,lf(z,ζ)

z ∈ H^∗[1, n, ζ]∩Q^∗. If h(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U , (8)

then

q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)

z , z ∈U, ζ ∈U , where q is given by q(z, ζ) = 2β −ζ + ^2(ζ−β)

nzⁿ¹

Rz 0

tⁿ¹⁻¹

1+t dt, z ∈ U, ζ ∈ U . The function q is convex and it is the best subordinant.

Proof. Following the same steps as in the proof of Theorem 3 and consid-

ering p(z, ζ) = ^IR

m λ,lf(z,ζ)

z , the strong differential superordination (8) becomes h(z, ζ) = ζ+ (2β−ζ)z

1 +z ≺≺p(z, ζ) +zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ = 1, we have q(z, ζ)≺≺p(z, ζ), i.e.

q(z, ζ) = 1 nz¹ⁿ

Z z

0

h(t, ζ)tⁿ¹⁻¹dt= 1 nzⁿ¹

Z z

0

tⁿ¹⁻¹ζ+ (2β−ζ)t 1 +t dt

= 2β−ζ+2(ζ−β) nz¹ⁿ

Z z

0

tⁿ¹⁻¹

1 +tdt≺≺ IR^m_λ,lf(z, ζ)

z , z ∈U, ζ ∈U . The function q is convex and it is the best subordinant.

Theorem No. 4 Let q(z, ζ) be convex in U ×U and let h be defined by h(z, ζ) = q(z, ζ) +zq⁰_z(z, ζ). If λ, l ≥ 0, m, n ∈ N, f(z, ζ) ∈ A^∗_nζ, suppose that IR^m_λ,lf(z, ζ)⁰

z is univalent, ^IR

m λ,lf(z,ζ)

z ∈ H^∗[1, n, ζ]∩Q^∗ and satisfies the strong differential superordination

h(z, ζ) =q(z, ζ) +zq⁰_z(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U , (9) then

q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)

z , z ∈U, ζ ∈U , where q(z, ζ) = ¹

nzⁿ¹

Rz

0 h(t, ζ)tⁿ¹⁻¹dt. The functionq is the best subordinant.

Proof. Let p(z, ζ) = ^IR

m λ,lf(z,ζ)

z = ^z+

P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^mCm+j−1^m a²_j(ζ)z^j

z =

1 +P∞ j=n+1

_{1+λ(j−1)+l}

l+1

m

C_m+j−1^m a²_j(ζ)z^j−1. Evidently p∈ H^∗[1, n, ζ].

Differentiating with respect toz, we obtainp(z, ζ)+zp⁰_z(z, ζ) = IR_λ,l^mf(z, ζ)0 z, z ∈U, ζ ∈U , and (9) becomes

q(z, ζ) +zq⁰_z(z, ζ)≺≺p(z, ζ) +zp⁰_z(z, ζ), z ∈U, ζ ∈U . Using Lemma 2 for γ = 1, we have

q(z, ζ)≺≺p(z, ζ), z∈U, ζ ∈U , i.e.

q(z, ζ) = 1 nz¹ⁿ

Z z

0

h(t, ζ)tⁿ¹⁻¹dt≺≺ IR^m_λ,lf(z, ζ)

z , z∈U, ζ ∈U ,

and q is the best subordinant.

Theorem No. 5 Let h(z, ζ) be a convex function, h(0, ζ) = 1. Let λ, l ≥0, m, n∈N, f(z, ζ)∈ A^∗_nζ and suppose that

zIR^m+1_λ,l f(z,ζ) IR^m_λ,lf(z,ζ)

0

z

is univalent and ^IR

m+1 λ,l f(z,ζ)

IR^m_λ,lf(z,ζ) ∈ H^∗[1, n, ζ]∩Q^∗. If h(z, ζ)≺≺ zIR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ)

!0

z

, z ∈U, ζ ∈U , (10)

then

q(z, ζ)≺≺ IR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ) , z ∈U, ζ ∈U , where q(z, ζ) = ¹

nzⁿ¹

Rz

0 h(t, ζ)t¹ⁿ⁻¹dt. The function q is convex and it is the best subordinant.

Proof. Considerp(z, ζ) = ^IR

m+1 λ,l f(z,ζ) IR^m_λ,lf(z,ζ) = ^z+

P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^m+1Cm+j^m+1a²_j(ζ)z^j z+P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^mCm+j−1^m a²_j(ζ)z^j =

1+P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^m+1Cm+j^m+1a²_j(ζ)z^j−1 1+P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^mCm+j−1^m a²_j(ζ)z^j−1. Evidently p∈ H^∗[1, n, ζ].

We havep⁰_z(z, ζ) = (^IRλ,l^m+1f(z,ζ))⁰_z

IR^m_λ,lf(z,ζ) −p(z, ζ)·(^IRmλ,lf(z,ζ))⁰_z

IR^m_λ,lf(z,ζ) .Thenp(z, ζ)+zp⁰_z(z, ζ) = zIR_λ,l^m+1f(z,ζ)

IR^m_λ,lf(z,ζ)

⁰

z

. Then (10) becomes

h(z, ζ)≺≺p(z, ζ) +zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ = 1, we have

q(z, ζ)≺≺p(z, ζ), z ∈U, ζ ∈U , i.e. q(z, ζ)≺≺ IR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ) , z∈U, ζ ∈U , where q(z, ζ) = ¹

nzⁿ¹

Rz

0 h(t, ζ)tⁿ¹⁻¹dt. The function q is convex and it is the best subordinant.

Corollary No. 3 Let h(z, ζ) = ^{ζ+(2β−ζ)z}_1+z be a convex function in U ×U, where 0 ≤ β < 1. Let λ, l ≥ 0, m, n ∈ N, f(z, ζ) ∈ A^∗_nζ and suppose that zIR_λ,l^m+1f(z,ζ)

IR^m_λ,lf(z,ζ)

0

z

is univalent, ^IR

m+1 λ,l f(z,ζ)

IR^m_λ,lf(z,ζ) ∈ H^∗[1, n, ζ]∩Q^∗. If h(z, ζ)≺≺ zIR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ)

!0

z

, z ∈U, ζ ∈U , (11)

then

q(z, ζ)≺≺ IR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ) , z ∈U, ζ ∈U , where q is given by q(z, ζ) = 2β −ζ + ^2(ζ−β)

nzⁿ¹

Rz 0

tⁿ¹⁻¹

1+t dt, z ∈ U, ζ ∈ U . The function q is convex and it is the best subordinant.

Proof. Following the same steps as in the proof of Theorem 5 and consid- eringp(z, ζ) = ^IR

m+1 λ,l f(z,ζ)

IR^m_λ,lf(z,ζ) , the strong differential superordination (11) becomes h(z, ζ) = ζ+ (2β−ζ)z

1 +z ≺≺p(z, ζ) +zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ = 1, we haveq(z, ζ)≺≺p(z, ζ), i.e.

q(z, ζ) = 1 nz¹ⁿ

Z z

0

h(t, ζ)tⁿ¹⁻¹dt= 1 nzⁿ¹

Z z

0

tⁿ¹⁻¹ζ+ (2β−ζ)t 1 +t dt

= 2β−ζ+2(ζ−β) nzⁿ¹

Z z

0

t¹ⁿ⁻¹

1 +tdt≺≺ IR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ) , z ∈U, ζ ∈U . The function q is convex and it is the best subordinant.

Theorem No. 6 Let q(z, ζ) be convex in U ×U and let h be defined by h(z, ζ) = q(z, ζ) +zq⁰_z(z, ζ). If λ, l ≥ 0, m, n ∈ N, f(z, ζ) ∈ Anζ, suppose that

zIR^m+1_λ,l f(z,ζ) IR^m_λ,lf(z,ζ)

0

z

is univalent, ^IR

m+1 λ,l f(z,ζ)

IR^m_λ,lf(z,ζ) ∈ H^∗[1, n, ζ]∩Q^∗ and satisfies the strong differential superordination

h(z, ζ) =q(z, ζ) +zq⁰_z(z, ζ)≺≺ zIR^m+1_λ,l f(z, ζ) IR^m_λ,lf(z, ζ)

!0

z

, z ∈U, ζ ∈U , (12)

then

q(z, ζ)≺≺ IR^m+1_λ,l f(z, ζ)

IR^m_λ,lf(z, ζ) , z ∈U, ζ ∈U , where q(z, ζ) = ¹

nzⁿ¹

Rz

0 h(t, ζ)tⁿ¹⁻¹dt. The functionq is the best subordinant.

Proof. Let p(z, ζ) = ^IR

m+1 λ,l f(z,ζ) IR^m_λ,lf(z,ζ) = ^z+

P∞

j=n+1(^{1+λ(j−1)+l}_l+1 )^m+1Cm+j^m+1a²_j(ζ)z^j z+P∞

j=n+1(^{1+λ(j−1)+l}_l+1 )^mCm+j−1^m a²_j(ζ)z^j =

1+P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^m+1Cm+j^m+1a²_j(ζ)z^j−1 1+P∞

j=n+1(^{1+λ(j−1)+l}l+1 )^mCm+j−1^m a²_j(ζ)z^j−1. Evidently p∈ H^∗[1, n, ζ].

Differentiating with respect toz, we obtainp(z, ζ)+zp⁰_z(z, ζ) =

zIR^m+1_λ,l f(z,ζ) IR^m_λ,lf(z,ζ)

0

z

, z ∈U, ζ ∈U , and (12) becomes

q(z, ζ) +zq⁰_z(z, ζ)≺≺p(z, ζ) +zp⁰_z(z, ζ), z ∈U, ζ ∈U . Using Lemma 2 for γ = 1, we have

q(z, ζ)≺≺p(z, ζ), z∈U, ζ ∈U , i.e.

q(z, ζ) = 1 nz¹ⁿ

Z z

0

h(t, ζ)tⁿ¹⁻¹dt≺≺ IR_λ,l^m+1f(z, ζ)

IR^m_λ,lf(z, ζ) , z ∈U, ζ ∈U , and q is the best subordinant.

Theorem No. 7 Let h(z, ζ) be a convex function, h(0, ζ) = 1. Let λ, l ≥0, m, n∈N, f(z, ζ)∈ A^∗_nζ and suppose that

l+1

[λ(l−m+2)−(l+1)]z·

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ) +

1− λ(l−m+2)−(l+1)^l+1

− 2(l+1)(m−1)−2λm λ(l−m+2)−(l+1)

Rz 0

IR^m_λ,lf(t,ζ)−t

t² dt is univalent and IR^m_λ,lf(z, ζ)⁰

z ∈ H^∗[1, n, ζ]∩Q^∗. If

h(z, ζ)≺≺ l+ 1

[λ(l−m+ 2)−(l+ 1)]z

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ) (13) +

1− l+ 1

λ(l−m+ 2)−(l+ 1)

−2 (l+ 1) (m−1)−2λm λ(l−m+ 2)−(l+ 1)

Z z

0

IR^m_λ,lf(t, ζ)−t

t² dt,

z ∈U, ζ ∈U ,then

q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U , where q(z, ζ) = λ(l−m+2)−(l+1)

λ(l+1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Rz

0 h(t, ζ)t

λ(l−m−nl−n+2)−(l+1)

λ(l+1)n dt. The func- tion q is convex and it is the best subordinant.

Proof. With notation p(z, ζ) = IR^m_λ,lf(z, ζ)0

z = 1 +P∞ j=n+1

_{1+λ(j−1)+l}

l+1

m

C_m+j−1^m ja²_j(ζ)z^j−1 and p(0, ζ) = 1, we obtain for f(z, ζ) =z+P∞

j=n+1a_j(ζ)z^j, p(z, ζ) +zp⁰_z(z, ζ) = 1 +P∞

j=n+1

_{1+λ(j−1)+l}

l+1

m

C_m+j−1^m ja²_j(ζ)z^j−1+ P∞

j=n+1

_{1+λ(j−1)+l}

l+1

m

C_m+j−1^m j(j −1)a²_j(ζ)z^j−1 =

1 z

m+1

λ IR^m+1_λ,l f(z, ζ)− ^m−2_λ IR^m_λ,lf(z, ζ)

+ λ(m−1)−(l+1)

λ(l+1) IR^m_λ,lf(z, ζ)0 z+ 1− ^m−1_l+1 − ²_λ

− 2(l+1)(m−1)−2λm λ(l+1)

Rz 0

IR^m_λ,lf(t,ζ)−t t² dt.

Therefore p(z, ζ) + λ(l−m+2)−(l+1)^λ(l+1) zp⁰_z(z, ζ) =

l+1 [λ(l−m+2)−(l+1)]z

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ) +

1− λ(l−m+2)−(l+1)^l+1

−2(l+1)(m−1)−2λm λ(l−m+2)−(l+1)

Rz 0

IR^m_λ,lf(t,ζ)−t t² dt.

Then (13) becomes

h(z, ζ)≺≺p(z, ζ) + λ(l+ 1)

λ(l−m+ 2)−(l+ 1)zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ = 1− ^m−1_l+1 − _λ¹, we have

q(z, ζ)≺≺p(z, ζ), z ∈U, ζ ∈U , i.e. q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z∈U, ζ ∈U , where q(z, ζ) = λ(l−m+2)−(l+1)

λ(l+1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Rz

0 h(t, ζ)t

λ(l−m−nl−n+2)−(l+1)

λ(l+1)n dt. The func- tion q is convex and it is the best subordinant.

Corollary No. 4 Let h(z, ζ) = ^{ζ+(2β−ζ)z}_1+z be a convex function in U ×U, where 0≤β <1. Let λ, l ≥0, m, n∈N, f(z, ζ)∈ A^∗_nζ and suppose that

l+1

[λ(l−m+2)−(l+1)]z·

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ) +

1− λ(l−m+2)−(l+1)^l+1

− 2(l+1)(m−1)−2λm λ(l−m+2)−(l+1)

Rz 0

IR^m_λ,lf(t,ζ)−t

t² dt is univalent, IR^m_λ,lf(z, ζ)⁰

z ∈ H^∗[1, n, ζ]∩Q^∗. If

h(z, ζ)≺≺ l+ 1

[λ(l−m+ 2)−(l+ 1)]z

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ) (14) +

1− l+ 1

λ(l−m+ 2)−(l+ 1)

−2 (l+ 1) (m−1)−2λm λ(l−m+ 2)−(l+ 1)

Z z

0

IR^m_λ,lf(t, ζ)−t

t² dt,

z ∈U, ζ ∈U , then

q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)⁰

z, z ∈U, ζ ∈U , where q is given by

q(z, ζ) = 2β−ζ+2(ζ−β) λ(l−m+2)−(l+1) λ(l+1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Rz 0

t

λ(l−m−nl−n+2)−(l+1) λ(l+1)n

1+t dt, z ∈U, ζ ∈U .

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

Proof. Following the same steps as in the proof of Theorem 7 and con- sidering p(z, ζ) = IR_λ,l^mf(z, ζ)0

z, the strong differential superordination (14) becomes

h(z, ζ) = ζ+ (2β−ζ)z

1 +z ≺≺p(z, ζ)+ λ(l+ 1)

λ(l−m+ 2)−(l+ 1)zp⁰_z(z, ζ), z ∈U, ζ ∈U . By using Lemma 1 for γ = λ(l−m+2)−(l+1)

λ(l+1) , we have q(z, ζ)≺≺p(z, ζ), i.e.

q(z, ζ) = λ(l−m+ 2)−(l+ 1) λ(l+ 1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Z z

0

h(t, ζ)t

λ(l−m−nl−n+2)−(l+1) λ(l+1)n dt= λ(l−m+ 2)−(l+ 1)

λ(l+ 1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Z z

0

t

λ(l−m−nl−n+2)−(l+1)

λ(l+1)n ζ+ (2β−ζ)t 1 +t dt =

2β−ζ+2(ζ−β)λ(l−m+ 2)−(l+ 1) λ(l+ 1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Z z

0

t

λ(l−m−nl−n+2)−(l+1) λ(l+1)n

1 +t dt≺≺ IR^m_λ,lf(z, ζ)0 z, z ∈U, ζ ∈U .

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

Theorem No. 8 Let q(z, ζ) be convex in U ×U and let h be defined by h(z, ζ) = q(z, ζ) + λ(l−m+2)−(l+1)^λ(l+1) z· q⁰_z(z, ζ), λ, l ≥ 0, m, n ∈ N. If f(z, ζ) ∈ A^∗_nζ,suppose that [λ(l−m+2)−(l+1)]z^l+1

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR_λ,l^mf(z, ζ) +

1− λ(l−m+2)−(l+1)^l+1

−2(l+1)(m−1)−2λm λ(l−m+2)−(l+1) · Rz

0

IR^m_λ,lf(t,ζ)−t

t² dt is univalent, IR^m_λ,lf(z, ζ)⁰

z ∈ H^∗[1, n, ζ]∩Q^∗ and satisfies the strong differential superor- dination

h(z, ζ) =q(z, ζ)+ λ(l+ 1)

λ(l−m+ 2)−(l+ 1)zq_z⁰ (z, ζ)≺≺ l+ 1

[λ(l−m+ 2)−(l+ 1)]z· (15) (m+ 1)IR_λ,l^m+1f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ)

+

1− l+ 1

λ(l−m+ 2)−(l+ 1)

− 2 (l+ 1) (m−1)−2λm

λ(l−m+ 2)−(l+ 1) Z z

0

IR^m_λ,lf(t, ζ)−t

t² dt, z ∈U, ζ ∈U , then

q(z, ζ)≺≺ IR^m_λ,lf(z, ζ)0

z, z ∈U, ζ ∈U , where q(z, ζ) = λ(l−m+2)−(l+1)

λ(l+1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Rz

0 h(t, ζ)t

λ(l−m−nl−n+2)−(l+1)

λ(l+1)n dt. The func- tion q is the best subordinant.

Proof. Let

p(z, ζ) = IR^m_λ,lf(z, ζ)0

z = 1 +P∞ j=n+1

_{1+λ(j−1)+l}

l+1

m

C_m+j−1^m ja²_j (ζ)z^j−1. Differentiating with respect to z, we obtain

p(z, ζ) +zp⁰_z(z, ζ) = ¹_z ^m+1_λ IR^m+1_λ,l f(z, ζ)− ^m−2_λ IR^m_λ,lf(z, ζ) +

λ(m−1)−(l+1)

λ(l+1) IR^m_λ,lf(z, ζ)0

z+ 1−^m−1_l+1 −_λ²

− 2(l+1)(m−1)−2λm λ(l+1)

Rz 0

IR^m_λ,lf(t,ζ)−t t² dt and p(z, ζ) + λ(l−m+2)−(l+1)^λ(l+1) zp⁰_z(z, ζ) =

l+1 [λ(l−m+2)−(l+1)]z

(m+ 1)IR^m+1_λ,l f(z, ζ)−(m−2)IR^m_λ,lf(z, ζ) +

1− λ(l−m+2)−(l+1)^l+1

− 2(l+1)(m−1)−2λm λ(l−m+2)−(l+1)

Rz 0

IR^m_λ,lf(t,ζ)−t

t² dt, z ∈U, ζ ∈U , and (15) becomes

q(z, ζ)+ λ(l+ 1)

λ(l−m+ 2)−(l+ 1)zq⁰_z(z, ζ)≺≺p(z, ζ)+ λ(l+ 1)

λ(l−m+ 2)−(l+ 1)zp⁰_z(z, ζ),

z ∈U, ζ ∈U .

Using Lemma 2 for γ = 1− ^m−1_l+1 − _λ¹, we have q(z, ζ) ≺≺ p(z, ζ), z ∈ U, ζ ∈U ,i.e.

q(z, ζ) = λ(l−m+ 2)−(l+ 1) λ(l+ 1)nz

λ(l−m+2)−(l+1) λ(l+1)n

Z z

0

h(t, ζ)t

λ(l−m−nl−n+2)−(l+1)

λ(l+1)n dt≺≺ IR^m_λ,lf(z, ζ)⁰

z, z ∈U, ζ ∈U , and q is the best subordinant.

References

[1] A. Alb Lupa¸s,Certain strong differential subordinations using S˘al˘agean and Ruscheweyh operators, Advances in Applied Mathematical Analysis, Vol- ume 6, Number 1 (2011), 27–34.

[2] A. Alb Lupa¸s, A note on strong differential subordinations using a gen- eralized S˘al˘agean operator and Ruscheweyh operator, submitted 2010.

[3] A. Alb Lupa¸s,A note on strong differential subordinations using S˘al˘agean and Ruscheweyh operators, Libertas Mathematica, submitted 2010.

[4] A. Alb Lupa¸s, G.I. Oros, Gh. Oros,On special strong differential subor- dinations using S˘al˘agean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, 2012 (to appear).

[5] A. Alb Lupa¸s, Certain strong differential subordinations using a multi- plier transformation and Ruscheweyh operator, International Journal of Open Problems in Complex Analysis, Vol. 3, No. 1, 2011, 1-8.

[6] A. Alb Lupa¸s, Certain strong differential subordinations using a gener- alized S˘al˘agean operator and Ruscheweyh operator, submitted 2010.

[7] A. Alb Lupa¸s, On special strong differential subordinations using multi- plier transformation, Applied Mathematics Letter, submitted 2011.

[8] A. Alb Lupa¸s, On special strong differential subordinations using a gen- eralized S˘al˘agean operator and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, 2012 (to appear).

[9] A. Alb Lupa¸s, On special strong differential superordinations using S˘al˘agean and Ruscheweyh operators, submitted 2011.

[10] A. Alb Lupa¸s,Certain strong differential superordinations using S˘al˘agean and Ruscheweyh operators, Acta Universitatis Apulensis, submitted 2011.

[11] A. Alb Lupa¸s, A note on strong differential superordinations using S˘al˘agean and Ruscheweyh operators, Journal of Applied Functional Analysis, 2012 (to appear).

[12] A. Alb Lupa¸s,Certain strong differential superordinations using a gen- eralized S˘al˘agean operator and Ruscheweyh operator, Journal of Applied Func- tional Analysis, 2012 (to appear).

[13] A. Alb Lupa¸s, A note on strong differential superordinations using a generalized S˘al˘agean operator and Ruscheweyh operator, submitted GFTA 2011.

[14] G.I. Oros, Strong differential superordination, Acta Universitatis Apu- lensis, Nr. 19, 2009, 101-106.

[15] G.I. Oros, On a new strong differential subordination, (to appear).

Alb Lupa¸s Alina, Oros Georgia Irina Department of Mathematics

University of Oradea

str. Universit˘at¸ii nr. 1, 410087, Oradea, Romania email: [email protected], georgia oros [email protected]