USING THE DZIOK-SRIVASTAVA LINEAR OPERATOR

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USING THE DZIOK-SRIVASTAVA LINEAR OPERATOR

GEORGIA IRINA OROS

By using the properties of the Dziok-Srivastava linear operator we obtain differential superordinations by using function from classA.

AMS 2000 Subject Classification: 30C45, 30A10, 30C80.

Key words: univalent function, starlike function, convex function, differential subordination, differential superordination, Dziok-Srivastava linear operator.

1. INTRODUCTION AND PRELIMINARIES

LetU denote the unit disc of the complex plane:

U ={z∈C:|z|<1} and U ={z∈C:|z| ≤1}.

LetH(U) denote the space of holomorphic functions in U and let A_n={f ∈ H(U), f(z) =z+a_n+1zⁿ⁺¹+. . . , z∈U} with A₁ =A.

Let

H[a, n] ={f ∈ H(U), f(z) =a+a_nzⁿ+a_n+1zⁿ⁺¹+. . . , z∈U}, S ={f ∈A; f is univalent in U}.

Let

K =

f ∈A; Re zf⁰⁰(z)

f⁰(z) + 1>0, z ∈U

denote the class of normalized convex functions in U, and S^∗ =

f ∈A; Re zf⁰(z)

f(z) >0, z ∈U

denote the class of starlike functions inU.

Iff andgare analytic functions inU, then we say thatf is subordinate to g, written f ≺ g, if there is a function w analytic in U, with w(0) = 0,

|w(z)|<1, for all z∈U such that f(z) =g[w(z)], for z∈U.

MATH. REPORTS12(62),1 (2010), 37–44

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Ifgis univalent, then f ≺g if and only iff(0) =g(0) andf(U)⊂g(U).

The method of differential subordinations (also known as the admissible functions method) was introduced by P.T. Mocanu and S.S. Miller in 1978 [2]

and 1981 [3] and developed in [4].

Let Ω and ∆ be any sets in C and let p be an analytic function in the unit disk with p(0) = aand let ψ(r, s, t;z) : C³ ×U → C. The heart of this theory deals with generalizations of the following implication:

(i){ψ(p(z), zp⁰(z), z²p⁰⁰(z);z)|z∈U} ⊂Ω implies p(U)⊂∆.

In [5] the authors introduce the dual problem of the differential subordination which they call differential superordination.

Definition1. Letf, F ∈ H(U) and letF be univalent inU. The function F is said to be superordinate to f, or f is subordinate to F, written f ≺F, iff(0) =F(0) andf(U)⊂F(U).

Let Ω and ∆ be any sets in C and let p be an analytic function in the unit disk and letϕ(r, s, t;z) :C³×U →C. The heart of this theory deals with generalizations of the following implication:

(ii) Ω⊂ {ϕ(z, zp⁰(z), z²p⁰⁰(z);z)|z∈U}implies ∆⊂p(U).

Definition 2 ([5]). Let ψ:C³×U →C and let h be analytic in U. If p andϕ(p(z), zp⁰(z), z²p⁰⁰(z);z) are univalent inU and satisfy the (second-order) differential superordination

(iii) h(z)≺ϕ(p(z), zp⁰(z), z²p⁰⁰(z);z)

then pis called a solution of the differential superordination.

An 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 (iii). A univalent subordinant qethat satisfies q ≺ qefor all subordinants q of (iii) is said to be the best subordinant. (Note that the best subordinant is unique up to a rotation of U.) For Ω a set inC, with ϕand p as given in Definition 2, suppose (iii) is replaced by

(iv) Ω⊂ {ϕ(p(z), zp⁰(z), z²p⁰⁰(z);z)|z∈U}.

Although this more general situation is a “differential containment”, the condition in (iv) will also be referred to as a differential superordination, and the definitions of solution, subordinant and best subordinant as given above can be extended to this generalization.

Definition 3 ([4], Definition 2.2.b, p. 21). We denote by Q the set of functions f that are analytic and injective on U\E(f), where

E(f) = n

ζ ∈∂U; lim

z→ζf(z) =∞o and are such that f⁰(ζ)6= 0 for ζ∈∂U\E(f).

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The subclass ofQ for which f(0) =ais denoted byQ(a).

Definition 4 ([5], Definition 3). Let Ω be a set in C and q ∈ H[a, n]

with q⁰(z) 6= 0. The class of admissible functions φ_n[Ω, q], consists of those functions ϕ:C²×U →Cthat satisfy the admissibility condition:

(A)ϕ(r, s;ζ)∈Ω,

whenever r =q(z), s= ^zq_m⁰^(z), where ζ ∈ ∂U, z∈ U, andm ≥n≥ 1. When n= 1 we write φ₁[Ω, q] asφ[Ω, q].

In order to prove the new results we shall use the following lemma:

Lemma A([5], Lemma 2.2.d, p. 24). Let p∈Ω(a), and let q(z) =a+anzⁿ+. . .

be analytic in U with q(z)6≡aand n≥1.

If q is not subordinate to p, then there exists points z₀ =r₀e^iθ⁰ ∈U and ζ0∈∂U\E(p), and an m≥n≥1 for which q(Ur0)⊂p(U).

(i)q(z₀) =p(ζ₀);

(ii)z0q⁰(z0) =mζ0p⁰(ζ0).

The Dziok-Srivastava operator was defined in paper [1] as follows:

(1) H_m^l (α1, α2, . . . , α_l;β1, β2, . . . , βm)f(z) =

=z+

∞

X

n=2

(α₁)n−1(α₂)n−1. . .(α_l)n−1

(β1)n−1(β2)n−1. . .(βm)n−1

·a_n· zⁿ (n−1)!, where α_i ∈C,i= 1,2, . . . , l,β_j ∈C\ {0,−1,−2, . . .},j= 1,2, . . . , m.

For simplicity, we write

(2) H_m^l [α1]f(z) =H_m^l (α1, α2, . . . , αl;β1, β2, . . . , βm)f(z).

For this operator we have the property

(3) α₁H_m^l [α₁+ 1]f(z) =z[H_m^l [α₁]f(z)]⁰+ (α₁−1)H_m^l [α₁]f(z).

2. MAIN RESULTS

Theorem 1.LetΩ⊂C,Ω6=C,q∈ H[a, n],α₁>0and letϕ∈φ_n[Ω, q].

If ^H^m^l ^[α_z¹^]f(z) ∈Q(1)andψ

H_m^l [α1]f(z)

z ,^H^m^l ^[α¹_z^+1]f(z);z

is univalent in U, then

(4) Ω⊂

ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

implies

q(z)≺ H_m^l [α1]f(z)

z , z∈U, where H_m^l [α1]f(z) is given by(1).

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Proof. Let

(5) p(z) = H_m^l [α₁]f(z)

z =

z+

∞

P

n=2

(α1)n−1(α2)n−1...(αl)n−1

(β1)n−1(β2)n−1...(βm)n−1 ·an·_(n−1)!^zⁿ z

= 1 +p1z+p2z²+. . . , p(0) = 1, p∈ H[1,1].

Differentiating (5), we obtain

(6) [H_m^l [α1]f(z)]⁰=p(z) +zp⁰(z), z∈U.

Using (3) and (6), we have (7) H_m^l [α1+1]f(z)

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

α₁z =p(z)+ 1

α₁zp⁰(z), and using (5) and (7), we obtain

(8) ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

=ϕ

p(z), p(z) + 1

α₁zp⁰(z);z

.

Then (4) becomes

(9) Ω⊂

ϕ

p(z), p(z) + 1

α₁zp⁰(z);z

, z∈U.

Assume q(z) ⊀ p(z). By Lemma A, there exist points z0 = r0e^iθ⁰ ∈ U andζ₀∈∂I\E(p) andm≥n≥1 that satisfy conditions (i)–(ii) of Lemma A.

Using these conditions with r=p(ζ0),s=ζ0p⁰(ζ0) and ζ =ζ0 in Definition 4 we obtain

ϕ

p(ζ0), p(ζ0) + 1 α1

ζ0p⁰(ζ0);ζ0

∈Ω.

Since this contradicts (9) we must have q(z)≺p(z) = H_m^l [α₁]f(z)

z , z∈U.

Next, we consider the special situation when h is analytic on U and h(U) = Ω6=C. In this case, the class φ_n[h(U), q] isφ_n[h, q] and the following result is an immediate consequence of Theorem 1.

Theorem 2. Let q ∈ H[a, n], α1 > 0, h be analytic in U, and ϕ ∈ φ₁[h, q]. If ^H^m^l^[α_z¹^]f(z) ∈ Q(1) and ϕ_Hl

m[α1]f(z)

z ,^H^l^m^[α¹_z^+1]f(z);z

(10) h(z)≺ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

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implies

q(z)≺ H_m^l [α₁]f(z)

z , z∈U.

From Theorem 1 and Theorem 2 we see that we can obtain subordinants of a differential superordination of the form (4) or (10) by merely checking that the function ϕ is an admissible function. The following theorem proves the existence of the best subordinant of (10) for certain ϕand also provides a method for finding the best subordinant.

Theorem 3. Let h be analytic in U and ϕ:C²×U →C. Suppose that the differential equation

(11) ϕ(q(z), zq⁰(z);z) =h(z), z∈U

has a solution q ∈ Q(a). If ϕ ∈ φ₁[h, q], ^H^m^l ^[α_z¹^]f^(z) ∈ Q(z), α₁ > 0 and ϕ_Hl

m[α1]f(z)

z ,^H^m^l^[α¹_z^+1]f^(z);z

(12) h(z)≺ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

implies

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

Proof. Since ϕ ∈ φ1[h, q], by applying Theorem 2 we deduce that q is a subordinant of (12). Since q also satisfies (11), it is also a solution of the differential superordination (12) and therefore all subordinants of (12) will be subordinate to q. Henceq will be the best subordinant of (12).

Remark1. The conclusion of the theorem can be written in the symmetric form

(13) ϕ(q(z), zq⁰(z);z)≺ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

implies

z , z∈U.

Theorem 4. Let h be convex in U, with h(0) = 1, α₁ >0, ^H^m^l ^[α_z¹^]f(z) ∈ H[1,1]∩Q. If ϕ_Hl

m[α1]f(z)

z ,^H^m^l ^[α¹_z^+1]f(z);z

(14) h(z)≺ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

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implies

q(z)≺ H_m^l [α1]f(z)

z , z∈U, where

(15) q(z) = α1

z^α¹ Z z

0

h(t)t^α¹⁻¹dt.

The function q is convex and is the best subordinant.

Proof. Differentiating (15), we obtain (16) h(z) =q(z) + 1

α1

zq⁰(z) =ϕ(q(z), zq⁰(z);z), and (14) becomes

(17) ϕ(q(z), zq⁰(z);z)≺ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

.

Using Theorem 3, we have

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

Example1. Leth(z) = ^1+z_1−z, and α1 >0. From Theorem 4 if h(z)≺ϕ

H_m^l [α₁]f(z)

z ,H_m^l [α₁+ 1]f(z)

z ;z

then

q(z)≺1 +p1z+p2z²+. . . , z∈U, where

q(z) = α1

z^α¹ Z _z

0

1 +t

1−tt^α¹⁻¹dt=−1 +2α1

z^α¹ Z _z

0

t^α¹⁻¹

1−tdt=−1 +2α1

z^α¹σ(α₁), where

σ(α₁) = Z z

0

t^α¹⁻¹

1−tdt, z∈U.

From Theorem 4 we can deduce the following “sandwich-type” result:

Corollary 1. Let h1 and h2 be convex in U, with h1(0) =h2(0) = 1.

Let α₁ >0 and let functions q_i be defined by qi(z) = α1

z^α¹ Z z

0

hi(t)t^α¹⁻¹dt, for i= 1,2.

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If ^H^m^l^[α_z¹^]f(z) ∈ H[1,1]∩Qandϕ

H_m^l [α1]f(z)

z ,^H^m^l ^[α¹_z^+1]f^(z);z

is univalent, then

h1(z)≺ϕ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

≺h2(z) implies

q1(z)≺ H_m^l [α1]f(z)

z ≺q2(z), z∈U.

The functions q₁ and q₂ are convex and they are respectively the best subordinant and best dominant.

Theorem 5. Let h be starlike in U, with h(0) = 0. If H_m^l [α₁]f(z) ∈ H[0,1]∩Q, α₁>0 and

ψ(H_m^l [α1]f(z);H_m^l [α1+ 1]f(z);z) =z[H_m^l [α1]f(z)]⁰ is univalent in U, then

(18) h(z)≺z[H_m^l [α₁]f(z)]⁰, z∈U implies

q(z)≺H_m^l [α1]f(z), z∈U, where

(19) q(z) =

Z z

0

h(t)t⁻¹dt, z∈U.

The function q is convex and is the best subordinant.

Proof. Differentiating (19), we obtain

(20) zq⁰(z) =h(z), z∈U,

and since h is starlike, we deduce that q is convex.

Let

(21) p(z) =H_m^l [α₁]f(z) =z+p₂z²+. . . , p(0) = 0, p∈ H[0,1]∩Q.

Using (21) we have

ϕ[H_m^l [α₁]f(z), H_m^l [α₁+ 1]f(z);z]f(z) =zp⁰(z), z∈U.

Subordination (18) becomes

h(z)≺zp⁰(z), z∈U.

By using Theorem 4, we obtain

q(z)≺H_m^l [α1]f(z) =p(z), where q is given by (19).

Sinceq is a solution of equation (20), from Theorem 3 we have thatq is the best subordinant.

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REFERENCES

[1] J. Dziok and H.M. Srivastava,Classes of analytic functions associated with the genera- lized hypergeometric function. Appl. Math. Comput.103(1999), 1–13.

[2] S.S. Miller and P.T. Mocanu,Second order differential inequalities in the complex plane.

J. Math. Anal. Appl.65(1978), 298–305.

[3] S.S. Miller and P.T. Mocanu,Differential subordinations and univalent functions. Michi- gan Math. J.28(1981), 157–171.

[4] S.S. Miller and P.T. Mocanu,Differential subordinations. Theory and applications. Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2000.

[5] S.S. Miller and P.T. Mocanu,Subordinations of differential superordinations. Complex Variables. 48,10(2003), 815–826.

Received 10 March 2009 University of Oradea

Department of Mathematics Str. Universit˘at¸ii, no. 1 410087 Oradea, Romania georgia oros ro@yahoo.co.uk