Introduction Let H be the class of functions analytic in the unit disk U = {z : |z| &lt

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65, 3 (2013), 299–305 September 2013

research paper

ON CERTAIN UNIVALENT CLASS ASSOCIATED WITH FUNCTIONS OF NON-BAZILEVI ˇC TYPE

Rabha W. Ibrahim

Abstract. In this work, we study certain differential inequalities and first order differential subordinations. As their applications, we obtain some sufficient conditions for univalence, which generalize and refine some previous results.

1. Introduction

Let H be the class of functions analytic in the unit disk U = {z : |z| < 1}

and fora ∈C (set of complex numbers) andn ∈N (set of natural numbers), let H[a, n] be the subclass ofHconsisting of functions of the form f(z) =a+anzⁿ+ an+1zⁿ⁺¹+· · ·. Let A be the class of functionsf, analytic in U and normalized by the conditionsf(0) =f⁰(0)−1 = 0.

Letf be analytic inU, g analytic and univalent inU andf(0) =g(0). Then, by the symbolf(z)≺g(z) (f subordinate tog) inU, we shall meanf(U)⊂g(U).

Letφ:C²→Cand lethbe univalent inU. Ifpis analytic inU and satisfies the differential subordinationφ(p(z)), zp⁰(z))≺h(z) thenpis called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, p≺q. Ifpand φ(p(z)), zp⁰(z)) are univalent in U and satisfy the differential superordination h(z)≺φ(p(z)), zp⁰(z)) thenpis called a solution of the differential superordination. An analytic function qis called subordinant of the solution of the differential superordination ifq≺p.

The functionf ∈ Ais called Φ-like if

<n zf⁰(z) Φ(f(z))

o

>0, z∈U.

This concept was introduced by Brickman [2] and established that a functionf ∈ A is univalent if and only iff is Φ-like for some Φ.

2010 Mathematics Subject Classification: 30C45.

Keywords and phrases: Univalent functions; starlike functions; convex functions; close-to- convex functions; differential subordination; subordination; superordination; unit disk; Φ-like functions; non-Bazileviˇc type; Dziok-Srivastava linear operator; sandwich theorem.

299

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Definition 1. Let Φ be analytic function in a domain containingf(U), Φ(0) = 0, Φ⁰(0) = 1 and Φ(ω)6= 0 forω∈f(U)− {0}. Letq(z) be a fixed analytic function inU,q(0) = 1. The functionf ∈ Ais called Φ-like with respect toqif

zf⁰(z)

Φ(f(z)) ≺q(z), z∈U.

Ruscheweyh [12] investigated this general class of Φ-like functions.

In the present paper, we consider another new classH^µ¡

λ; Φ1(f(z)),Φ2(f(z))¢ involving two different types of Φ-like functions, Φ1 and Φ2, which are defined by

(1 +λ) zf⁰(z) Φ1(f(z))

³ z f(z)

´_µ

−λ zf⁰(z) Φ2(f(z))

³ z f(z)

´_µ

≺F(z), (1) whereµ, λ∈R,F is the conformal mapping of the unit diskU withF(0) = 1 and Φ1and Φ2 satisfy Definition 1.1.

Remark 1. As special cases of the class H^µ¡

λ; Φ1(f(z)),Φ2(f(z))¢

and for different type ofF, are the following well known classes: H⁰¡

0; Φ(f(z))¢

(see [12]);

H^µ¡ 0;z¢

(see [11]); H^µ¡

λ;zf⁰(z), f(z)¢

(see [15]) when F(z) := ^1+Az_1+Bz. Also this class reduces to the classes of starlike functions, convex functions and close-to- convex functions.

Recently, many authors studied the non-Bazileviˇc type of functions (see [5, 6, 7, 16, 17]). In order to obtain our results, we need the following lemmas.

Lemma 1. [8]Letq(z)be univalent in the unit diskU andθ andφbe analytic in a domain D containing q(U) with φ(w) 6= 0 when w ∈ q(U). Set Q(z) :=

zq⁰(z)φ(q(z)),h(z) :=θ(q(z)) +Q(z). Suppose that 1. Q(z)is starlike univalent in U, and

2. <{^zh_Q(z)⁰^(z)}>0 forz∈U.

If θ(p(z)) +zp⁰(z)φ(p(z))≺θ(q(z)) +zq⁰(z)φ(q(z))thenp(z)≺q(z)andq(z) is the best dominant.

Definition 2. [9] Denote byQthe set of all functionsf(z) that are analytic and injective on U −E(f) where E(f) := {ζ ∈ ∂U : limz→ζf(z) = ∞} and are such thatf⁰(ζ)6= 0 forζ∈∂U−E(f).

Lemma 2. [3]Let q(z) be convex univalent in the unit diskU andϑandϕbe analytic in a domainD containingq(U). Suppose that

1. zq⁰(z)ϕ(q(z))is starlike univalent in U, and 2. <{^ϑ_ϕ(q(z))⁰^(q(z))}>0 forz∈U.

If p(z)∈ H[q(0),1]∩Q, withp(U)⊆D andϑ(p(z)) +zp⁰(z)ϕ(z)is univalent in U andϑ(q(z)) +zq⁰(z)ϕ(q(z))≺ϑ(p(z)) +zp⁰(z)ϕ(p(z)) then q(z)≺p(z) and q(z)is the best subordinant.

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2. The class H^µ¡

λ; Φ1(f(z)),Φ2(f(z))¢

In this section we introduce subordination results and the sufficient conditions for functionsf to be in the classH^µ¡

λ; Φ1(f(z)),Φ2(f(z))¢ .

Theorem 1. Let q, q(z)6= 0, be a univalent function in U, and g(z)6= 0 be analytic in Csuch that for nonnegative real numbersµ andν

<n

1 +zq⁰⁰(z)

q⁰(z) −zq⁰(z) q(z)

o

>maxn 0,³µ

ν

´

<³

q(z)[1 +g⁰(z) g(z)(q(z)

q⁰(z)+ νz µq(z))]´o

. (2) If p(z)6= 0,z∈U satisfies the differential subordination

g(z)h

µp(z) +νzp⁰(z) p(z)

i

≺g(z)h

µq(z) +νzq⁰(z) q(z)

i

, (3)

thenp≺qandq is the best dominant.

Proof. Define the functionsθ andφas follows:

θ(w(z)) :=µw(z)g(z) and φ(w(z)) := νg(z) w(z).

Obviously, the functionsθ andφare analytic in domainD=C\{0}andφ(w)6= 0 inD. Now, define the functions Qandhas follows:

Q(z) :=zq⁰(z)φ(q(z)) =νg(z)zq⁰(z) q(z) , h(z) :=θ(q(z)) +Q(z) =µq(z)g(z) +νg(z)zq⁰(z)

q(z) .

Then in view of condition (2), we obtainQ is starlike inU and<{^zh_Q(z)⁰^(z)}>0 for z∈U. Furthermore, in view of condition (3) we have

θ(p(z)) +zp⁰(z)φ(p(z))≺θ(q(z)) +zq⁰(z)φ(q(z)).

Therefore, the proof follows from Lemma 1.

As an application of Theorem 1, we pose the sufficient condition for functions inH^µ¡

λ; Φ1(f(z)),Φ2(f(z))¢

. We have the following result:

Corollary 1. If f(z)∈ Asatisfies the conditions (2) and(3) for someg in Theorem 1, thenf ∈H^µ¡

λ; Φ1(f(z)),Φ2(f(z))¢ . 3. Sandwich theorem

By employing the concept of the superordination (Lemma 2), we state the sandwich theorem containing functionsf ∈ A.

Theorem 2. Let q(z)be convex univalent in the unit diskU. Suppose that g is analytic in the unit disk such that

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1. νg(z)^zq_q(z)⁰^(z) is starlike univalent inU, and 2. ^µ_ν<{q(z)q⁰(z)}>0forz∈U.

If p(z)∈ H[q(0),1]∩Q, withp(U)⊆D andg(z)£

µp(z) +ν^zp_p(z)⁰^(z)¤

is univalent inU and

g(z) h

µq(z) +νzq⁰(z) q(z)

i

≺g(z) h

µp(z) +νzp⁰(z) p(z)

i

thenq(z)≺p(z)andq(z)is the best subordinant.

Proof. Define functionsθandφas follows:

ϑ(w(z)) :=µw(z)g(z) and ϕ(w(z)) := νg(z) w(z).

Obviously, the functionsϑandϕare analytic in domainD=C\ {0}andϕ(w)6= 0 inD. Hence the assumptions of Lemma 2 are satisfied.

Combining Theorem 1 and Theorem 3 we get the following sandwich theorem:

Theorem 3. Let q1(z), q2 6= 0 be convex and univalent in U respectively.

Suppose thatg is analytic inU such that 1. νg(z)^zq_q⁰¹^(z)

1(z) is starlike univalent in U, and 2. ^µ_ν<{q1(z)q₁⁰(z)}>0 forz∈U and

<n

1+zq⁰⁰₂(z)

q⁰₂(z) −zq₂⁰(z) q2(z)

o

>maxn 0,³µ

ν

´

<³

q2(z)[1+g⁰(z) g(z)(q2(z)

q⁰₂(z)+ νz µq2(z))]´o

. (4) If p(z)6= 0∈ H[q(0),1]∩Q, withp(U)⊆D andg(z)£

µp(z) +ν^zp_p(z)⁰^(z)¤

is univalent inU and

g(z) h

µq1(z) +νzq₁⁰(z) q1(z)

i

≺g(z) h

µp(z) +νzp⁰(z) p(z)

i

≺g(z) h

µq2(z) +νzq⁰₂(z) q2(z)

i

then

q1(z)≺p(z)≺q2(z), (z∈U)

andq1(z), q2(z)are the best subordinant and the best dominant respectively.

By lettingp(z) :=^zf_f(z)⁰^(z) in Theorem 3, we have

Corollary 2. Let the conditions of Theorem 3 on the functions q1 and q2 hold. If for f ∈ A, ^zf_f(z)⁰^(z) 6= 0 ∈ H[q(0),1]∩Q, with (^zf_f⁰)(U) ⊆ D and g(z)£

(µ−ν)^zf_f(z)⁰^(z)+ν(1 + ^zf_f0⁰⁰(z)^(z))¤

is univalent inU and g(z)h

µq1(z) +νzq₁⁰(z) q1(z)

i

≺g(z)h

(µ−ν)zf⁰(z)

f(z) +ν(1 +zf⁰⁰(z) f⁰(z) )i

≺g(z) h

µq2(z) +νzq₂⁰(z) q2(z)

i

(5)

then

q₁(z)≺ zf⁰(z)

f(z) ≺q₂(z), (z∈U) (5) andq1(z), q2(z)are the best subordinant and the best dominant respectively.

Note that Ali et al. [1] have used the results of Bulboacˇa [3] and obtained sufficient conditions for certain normalized analytic functionsf(z) to satisfy (5).

By assumingp(z) := _zf^f(z)0(z) in Theorem 3, we obtain

Corollary 3. Let the conditions of Theorem 3 on the functions q1 and q2

hold. If forf ∈ A, _zf^f(z)0(z) 6= 0∈ H[q(0),1]∩Q, with(_zf^f0(U)⊆Dandg(z)£

µ_zf^f(z)0(z)+ ν(^zf_f(z)⁰^(z)−1−^zf_f0⁰⁰(z)^(z))¤

is univalent in U and g(z)

h

µq1(z) +νzq₁⁰(z) q1(z)

i

≺g(z) h

µ f(z)

zf⁰(z)+ν(zf⁰(z)

f(z) −1−zf⁰⁰(z) f⁰(z) )

i

≺g(z)h

µq2(z) +νzq₂⁰(z) q2(z)

i

then

q1(z)≺ f(z)

zf⁰(z) ≺q2(z), (z∈U) (6) andq1(z), q2(z)are the best subordinant and the best dominant respectively.

Note that Shanmugam et al. [13] posed sufficient conditions for certain normalized analytic functionsf(z) to satisfy (6).

Again by consideringp(z) := ^z_f²^f2(z)⁰^(z) in Theorem 3, we find

Corollary 4. Let the conditions of Theorem 3 on the functions q1 and q2 hold. If for f ∈ A, ^z_f²^f2(z)⁰^(z) 6= 0 ∈ H[q(0),1]∩Q, with ^z_f²^f2⁰(U) ⊆ D and g(z)£

µ^z_f²^f2(z)⁰^(z)+ν(^zf_f0⁰⁰(z)^(z)+ 2−2^zf_f(z)⁰^(z))¤

is univalent in U and g(z)h

µq1(z) +νzq₁⁰(z) q₁(z)

i

≺g(z))h

µz²f⁰(z)

f²(z) +ν(zf⁰⁰(z)

f⁰(z) + 2−2zf⁰(z) f(z) )i

≺g(z) h

µq2(z) +νzq₂⁰(z) q2(z)

i

then

q1(z)≺ z²f⁰(z)

f²(z) ≺q2(z), (z∈U) (7) andq1(z), q2(z)are the best subordinant and the best dominant respectively.

Note that Shanmugam et al. [13] estimated sufficient conditions for certain normalized analytic functionsf(z) to satisfy (7).

Furthermore, by lettingp(z) :=^z(f∗g)_Φ(f∗g)(z)⁰^(z) in Theorem 3, we pose

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hold. If for f ∈ A, ^z(f∗g)_Φ(f∗g)(z)⁰^(z) 6= 0∈ H[q(0),1]∩Q, with (^z(f∗g)_Φ(f∗g)⁰)(U)⊆D and g(z)

h

µz(f∗g)⁰(z)

Φ(f∗g)(z) −ν(zf⁰⁰(z)

f⁰(z) + 1−zΦ⁰(f∗g)(z) Φ(f ∗g)(z) )

i

is univalent inU and g(z)

h

µq1(z) +νzq₁⁰(z) q1(z)

i

≺g(z) h

µz(f ∗g)⁰(z)

Φ(f∗g)(z)−ν(zf⁰⁰(z)

f⁰(z) + 1−zΦ⁰(f∗g)(z) Φ(f∗g)(z) )

i

≺g(z)h

µq2(z) +νzq₂⁰(z) q2(z)

i

then

q1(z)≺z(f∗g)⁰(z)

Φ(f ∗g)(z) ≺q2(z), (z∈U) (8) andq1(z), q2(z)are the best subordinant and the best dominant respectively.

Note that Shanmugam et al. [14] posed sufficient conditions for certain normalized analytic functionsf(z) to satisfy (8).

Finally, by setting p(z) := (^H^l^m^[α_z¹^]f(z))^δ, where f ∈ A and H_m^l [α1] is the Dziok-Srivastava linear operator [4], in Theorem 3, we have

hold. If for f ∈ A, (^H^m^l^[α_z¹^]f(z))^δ 6= 0∈ H[q(0),1]∩Q, with ¡

(^H^m^l^[α_z¹^]f)^δ¢

(U)⊆D and

g(z) h

µ(H_m^l [α1]f(z)

z )^δ−νδz( z

H_m^l [α1]f(z)−1) i

is univalent inU and g(z)

h

µq1(z) +νzq⁰₁(z) q1(z)

i

≺g(z) h

µ(H_m^l [α1]f(z)

z )^δ−νδz( z

H_m^l [α1]f(z)−1) i

≺g(z) h

µq2(z) +νzq⁰₂(z) q2(z)

i

then

q1(z)≺(H_m^l [α1]f(z)

z )^δ≺q2(z), (z∈U) (9) andq1(z), q2(z)are the best subordinant and the best dominant respectively.

Note that Murugusundaramoorthy and Magesh [10] introduced sufficient conditions for certain normalized analytic functionsf(z) to satisfy (9).

Corollary 7. Let the assumptions of Theorem 3 on the function

p(z) := (1 +λ) zf⁰(z) Φ1(f(z))

³ z f(z)

´_µ

−λ zf⁰(z) Φ2(f(z))

³ z f(z)

´_µ

(7)

hold. Then

q1(z)≺(1 +λ) zf⁰(z) Φ1(f(z))

³ z f(z)

´_µ

−λ zf⁰(z) Φ2(f(z))

³ z f(z)

´_µ

≺q2(z), (z∈U) (10) andq1(z), q2(z)are the best subordinant and the best dominant respectively.

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(received 03.07.2011; in revised form 11.12.2011; available online 01.05.2012)

Institute of Mathematical Sciences, University Malaya, 50603, Kuala Lumpur, Malaysia E-mail:[email protected]