DIFFERENTIAL SUBORDINATIONS ASSOCIATED WITH THE DZIOK-SRIVASTAVA OPERATOR

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WITH THE DZIOK-SRIVASTAVA OPERATOR

GEORGIA IRINA OROS, GHEORGHE OROS, IN HWA KIM and NAK EUN CHO

The purpose of the present paper is to investigate differential subordination pro- perties associated with the Dziok-Srivastava operator. Moreover, we determine dominants and best dominants of differential subordinations presented here.

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

Key words: univalent function, analytic function, differential subordination, Dziok-Srivastava operator.

1. INTRODUCTION AND PRELIMINARIES

Let H(U) denote the space of holomorphic functions in the open unit disk U ={z∈C: |z|<1} and let

An={f ∈ H(U), f(z) =z+an+1zⁿ⁺¹+. . .} with A1 =A. Also, for a positive integer nand a∈C, let

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

S ={f ∈A; f is univalent in U}.

Iff andgare analytic functions inU, then we say thatf is subordinate tog, writtenf ≺gorf(z)≺g(z), if there is a functionwanalytic inU, with w(0) = 0 and|w(z)|<1, and such thatf(z) =g[w(z)]. Ifgis univalent, then f ≺g if and only if f(0) =g(0) andf(U)⊂g(U).

The method of differential subordinations (also known as the admissible functions method) was introduced by Mocanu and Miller [2], [3] and deve- loped in [5].

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 implication

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

MATH. REPORTS13(63),1 (2011), 57–64

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If ∆ is a simply connected domain containing the point a and ∆ 6= C, then there is a conformal mapping q ofU onto ∆ such that q(0) =a. In this case (i) can be rewritten as

(ii){ψ(p(z), zp⁰(z), z²p⁰⁰(z);z)|z∈U} ⊂Ω implies p(z)≺q(z).

If Ω is also a simply connected domain and Ω6=C, then there is a conformal mapping h of U onto Ω such thath(0) =ψ(a,0,0; 0). If in addition, the functionψ(p(z), zp⁰(z), z²p⁰⁰(z);z) is analytic inU, then (i) can be rewritten as

(iii) ψ(p(z), zp⁰(z), z²p⁰⁰(z);z)≺h(z) impliesp(z)≺q(z).

Definition 1 [5, p. 16]. Letψ:C³×U →Cand leth be univalent in U. If pis analytic inU and satisfies the (second-order) differential subordination (iv) ψ(p(z), zp⁰(z), z²p⁰⁰(z);z) ≺ h(z), then p is called a solution of the differential subordination. The univalent function q is called a dominant, if p ≺ q for all p satisfying (iv). A dominant qe that satisfies qe ≺ q for all dominants q of (iv) is said to be the best dominant of (iv). (Note that the best dominant is unique up to a rotation of U.)

Definition 2 [5, Definition 2.2b, p. 21]. We denote by Qthe set of functions f that are analytic and injective onU \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).

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

Definition3 ([3], [4], [5, Definition 2.3a, p. 27]). Let Ω be a set inC,q∈Q and n be a positive integer. The class of admissible functions Ψ_n[Ω, q], consists of those functions ψ : C² ×U → C that satisfy the admissibility condition

(A)ψ(r, s;z)6∈Ω

whenever r=q(ζ), s=mζq⁰(ζ),z ∈U, ζ ∈∂U\E(q) and m ≥n. We write Ψ₁[Ω, q] as Ψ[Ω, q].

In the special case when Ω is a simply connected domain, Ω6=C, and h is a conformal mapping ofU onto Ω, we denote this class by Ψ_n[h, q].

The function q(z) = ^1+z_1−z satisfies q(U) = ∆ = {w ∈ C : Rew > 0}, q(0) = 1, E(q) ={1} and q ∈Q. We set Ψn[Ω,1] = Ψn[Ω, q]. In the special case when Ω = ∆, we denote the class by Ψ_n{1} and admissibility condition (A) becomes

(A⁰)ψ(ρi, σ;z)6∈Ω,

when σ ≤ −ⁿ₂(1 +ρ²),ρ, σ ∈R,n≥1 andz∈U.

The Dziok-Srivastava operator was defined in [1] by (1) H_m^l (α1, α2, . . . , αl;β1, β2, . . . , βm)f(z) =

(3)

=z+

∞

X

n=2

(α1)n−1(α2)n−1. . .(α_l)n−1

(β₁)n−1(β₂)n−1. . .(β_m)n−1(n−1)!anzⁿ,

αi ∈C, i= 1,2, . . . , l, βj ∈C\{0,−1,−2, . . .}, j = 1,2, . . . , m), f ∈A, where (ν)_n is the Pochhammer symbol defined (in terms of the Gamma function) by

(ν)n:= Γ(ν+n) Γ(ν) =

1 ifn= 0 andν ∈C\{0},

ν(ν+ 1)· · ·(ν+n−1) ifn∈Nand ν ∈C. For simplicity, we write

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

We note from (1) that we have

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

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

Lemma A [5, Lemma 2.2.d, p. 24]. Let q ∈ Q with q(0) = a, and let p(z) = a+anzⁿ+. . . be analytic in U with p(z) 6≡ a and n ≥1. If p is not subordinate to q, then there exist points z0 =r0e^iθ⁰ ∈U and ζ0 ∈∂U\E(q), and m≥n≥1 for which p(U_r₀)⊂q(U), such that

(i) p(z₀) =q(ζ₀);

(ii)z0p⁰(z0) =mζ0q⁰(ζ0).

Remark1. Let Ω be a set inCandψ(p(z), zp⁰(z);z)∈Ω forz∈U, where ψ(p(z), zp⁰(z);z) may not be analytic inU.

In this paper, we obtain the first order differential subordination impli- cations associated with the Dziok-Srivastava operator defined by (1) applying the concept of admissible functions introduced by Miller and Mocanu [5]. We also provide an useful example as a special case of our results.

2. MAIN RESULTS

Theorem 1. Let ψ∈Ψ{1}, α1 >0. If f ∈A satisfies (4)

ψ

H_m^l [α₁]f(z)

z ,H_m^l [α₁+ 1]f(z) z ;z∈U

⊂Ω ={w∈C: Rew >0}, then

H_m^l [α1]f(z)

z ≺q(z).

Proof. Let

(5) p(z) = H_m^l [α₁]f(z)

z , z∈U.

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From (5) we have

(6) H_m^l [α₁]f(z) =zp(z),

and differentiating (6), we obtain

(7) {H_m^l [α₁]f(z)}⁰=p(z) +zp⁰(z).

Using the property (3) of the Dziok-Srivastava operator, we have (8) H_m^l [α1+ 1]f(z) = z{H_m^l [α1]f(z)}⁰+ (α1−1)H_m^l [α1]f(z)

α1

. Using (7) in (8) we obtain

(9) H_m^l [α1+ 1]f(z)

z =p(z) + zp⁰(z) α₁ . Then (4) becomes

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ψ

H_m^l [α₁]f(z)

z ,H_m^l [α₁+ 1]f(z)

z ;z

=

ψ

p(z), p(z) +zp⁰(z) α₁ ;z

⊂Ω ={w∈C; Rew >0}

which is equivalent to Reψ

p(z), p(z) +zp⁰(z) α1

;z

>0, z∈U.

Assume p ⊀ q. By Lemma A there exist points z0 = r0e^iθ⁰ ∈ U and ζ0 ∈

∂U \E(q), andm≥n≥1 that satisfy

p(z0) =q(ζ0), z0p(z0) =mζ0q⁰(ζ0).

Using these conditions in Definition 3, we obtain ψ

p(z0), p(z0) + 1 α1

z0p⁰(z0);z0

=ψ

q(ζ0),mζ0q⁰(ζ0) α1

+q(ζ0);z0

6∈Ω.

Since this contradicts (10), we must have p≺q, i.e., H_m^l [α1]f(z)

z ≺q(z).

From the proof of Theorem 1, it is easy to see that Theorem 1 also holds if under condition (4), z ∈ U is replaced by a function w(z) which maps U into U and so we have the following corollary.

Corollary 1. Let ψ∈Ψ{1}, α₁ >0. If f ∈A satisfies

ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;w(z)

⊂Ω,

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then

H_m^l [α1]f(z)

z ≺q(z), where w(z) is any function mappingU into U.

For the case when Ω =h(U), wherehis a conformal mapping ofU in Ω, Theorem 1 can be rephrased as the following equivalent theorem.

Theorem 2. Let ψ∈Ψ{1} withq(0) = 1. If ^H^m^l ^[α_z¹^]f(z) ∈ H[1,1], ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

is analytic in U, and

(11) ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

≺h(z), then

H_m^l [α1]f(z)

z ≺q(z).

This result can be extended to those cases in which the behavior ofq on the boundary of U is unknown.

Theorem 3. Let h and q be univalent in U, with q(0) = 1 and set qρ(z) = q(ρz) and hρ(z) = h(ρz). Let ψ : C² ×U → C satisfy one of the following conditions:

(j)ψ∈Ψ[h, qρ], for someρ∈(0,1),

(jj)there exists ρ₀ ∈(0,1) such thatψ∈Ψ[h_ρ, q_ρ], for allρ∈(ρ₀,1).

If ^H^m^l^[α_z¹^]f(z) ∈ H[1,1], ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

is analytic in U, and ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

≺h(z), then

H_m^l [α1]f(z)

z ≺q(z).

Proof. Case (j). By applying Theorem 1 we obtain p≺qρ. Since qρ≺q we have p(z)≺q(z), i.e.,

H_m^l [α₁]f(z)

z ≺q(z).

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Case (jj). If we let pρ(z) =p(ρz), then ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

=ψ

pρ(z), pρ(z) + 1 α1

zp⁰_ρ(z);ρz

=

=ψ

p(ρz), p(ρz) + 1 α1

zp⁰(ρz);ρz

∈h_ρ(U).

By using Corollary 1, with w(z) =ρz, we obtain p_ρ(z)≺q_ρ(z), forρ∈(ρ₀,1).

By letting ρ→1 we obtain p(z)≺q(z), i.e., H_m^l [α1]f(z)

z ≺q(z).

The next theorem yields the best dominant of the differential subordination (4).

Theorem 4. Let hbe univalent in U andψ:C²×U →C. Suppose that the differential equation

(12) ψ

q(z), q(z) + 1 α1

zq⁰(z);z

=h(z)

has a solutionq, withq(0) = 1, and one of the following conditions is satisfied:

(k) q∈Q andψ∈Ψ[h, q],

(kk) q is univalent in U and ψ∈Ψ[h, q_ρ], for some ρ∈(0,1), or

(kkk) q is univalent in U and there exists ρ0 ∈ (0,1) such that ψ ∈ Ψ[h_ρ, q_ρ] for allρ∈(ρ₀,1).

If ^H^m^l ^[α_z¹^]f^(z) ∈ H[1,1], ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

is analytic in U, and

(13) ψ

H_m^l [α₁]f(z)

z ,H_m^l [α₁+ 1]f(z)

z ;z

≺h(z), then

H_m^l [α1]f(z)

z ≺q(z), and q is best dominant.

Proof. By applying Theorem 1 and Theorem 3 we deduce that q is a dominant of (13). Since q satisfies (12) it is solution of (13) and therefore q will be the dominant of all dominants of (13). Hence q will be the best dominant of (13).

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Example1. Letf ∈A. If (14) H_m^l [α1+ 1]f(z)

z −H_m^l [α1]f(z)

z ≺h(z), where

(15) h(z) =− 1

α₁ 2z (1−z)² then

H_m^l [α1]f(z)

z ≺ 1 +z 1−z. Proof. Using (5) and (9) we have

ψ

H_m^l [α1]f(z)

z ,H_m^l [α1+ 1]f(z)

z ;z

=ψ

p(z), p(z) + 1 α1

zp⁰(z);z

=

=p(z) + 1

α₁zp⁰(z)−p(z) = 1

α₁zp⁰(z).

Then (14) becomes

(16) 1

α₁zp⁰(z)≺ − 1

α₁ · 2z (1−z)². Since

h(U) = (

w∈C; Rew > 1 2α1sin² ^θ₂

) , where z=e^iθ, (16) is equivalent to

(17) Reψ

p(z), p(z) + 1

α₁zp⁰(z);z

> 1

2α₁sin²^θ₂ >0, z∈U.

Assume p(z) ⊀q(z) = ^1+z_1−z. By Lemma A there exist points z0 =r0e^iθ⁰ ∈U, ζ₀∈∂U\E(q), andm≥n≥1 that satisfy

p(z0) =q(ζ0) =ρi, ρ∈R and z0p⁰(z0) =mζ0q⁰(ζ0) =σ, σ∈R. Then

ψ

p(z0), p(z0) + 1 α1

z0p(z0);z0

=ψ

q(ζ0), q(ζ0) + 1 α1

ζ0q⁰(ζ0);z0

= 1 α1

σ.

Using Definition 3 and condition (A⁰), we calculate Reψ

p(z0), p(z0) + 1

α₁z0p⁰(z0);z0

= Re 1

α₁σ

= 1

α₁σ≤ 1 α₁·−1

2 (1 +ρ²)≤0.

Since this contradicts (17), we must have p(z)≺q(z), i.e., H_m^l [α₁]f(z)

z ≺q(z).

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Remark 1. If we take l = 1, m = 0 and α1 = 1 in Example 1, then we have the following implication. If

f⁰(z)− f(z)

z ≺h(z) =− 2z (1−z)²

then f(z)

z ≺ 1 +z 1−z, and q is the best dominant.

REFERENCES

[1] J. Dziok and H.M. Srivastava,Classes of analytic functions associated with the generalized 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 and inequalities in the complex plane. J. Differential Equations67(1987),2, 199–211.

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

Received 5 June 2009 Georgia Irina Oros and Gheorghe Oros University of Oradea

Department of Mathematics Str. Universit˘at¸ii, no. 1 410087 Oradea, Romania georgia oros [email protected]

gh [email protected] In Hwa Kim and Nak Eun Cho

Pukyong National University Department of Applied Mathematics

Pusan 608-737, Korea

[email protected], [email protected]