ON A SUBCLASS OF UNIVALENT FUNCTIONS DEFINED BY A GENERALIZED DIFFERENTIAL OPERATOR

OF UNIVALENT FUNCTIONS DEFINED BY A GENERALIZED DIFFERENTIAL OPERATOR

DORINA R ˘ADUCANU

In this paper we investigate a new subclass of univalent functions defined by a generalized differential operator. An inclusion result, structural formula, extreme points and other properties of this class of functions are obtained.

AMS 2010 Subject Classification: 30C45.

Key words: analytic functions, differential operator, convex functions, structural formula.

1. INTRODUCTION

LetA denote the class of functionsf of the form

(1) f(z) =z+

∞

X

n=2

anzⁿ

which are analytic in the open unit diskU:={z∈C:|z|<1}.

ByS and C we denote the subclasses of functions inAwhich are univa- lent and convex in U, respectively.

LetPbe the well-known Carath´eodory class of normalized functions with positive real part inUand letP(λ), 0≤λ <1 be the subclass ofP consisting of functions with real part greater than λ.

The Hadamard product or convolution of the functions f(z) =z+

∞

X

n=2

anzⁿ and g(z) =z+

∞

X

n=2

bnzⁿ

is given by

(f∗g) (z) =z+

∞

X

n=2

a_nb_nzⁿ, z∈U.

MATH. REPORTS13(63),2 (2011), 197–203

Letf ∈ A. We consider the following differential operator introduced by R˘aducanu and Orhan [13]:

D_αβ⁰ f(z) =f(z),

D_αβ¹ f(z) =D_αβf(z) =αβz²f⁰⁰(z) + (α−β)zf⁰(z) + (1−α+β)f(z), D^m_αβf(z) =D_αβ D_αβ^m−1f(z)

, (2)

where 0≤β≤α and m∈N:={1,2, . . .}.

If the functionf is given by (1) then, from (2) we see that

(3) D^m_αβf(z) =z+

∞

X

n=2

An(α, β, m)anzⁿ, where

(4) A_n(α, β, m) = [1 + (αβn+α−β)(n−1)]^m.

When α = 1 and β = 0, we get S˘al˘agean differential operator [14]. When β = 0, we obtain the differential operator defined by Al-Oboudi [1].

From (3) it follows thatD^m_αβf(z) can be written in terms of convolution as

(5) D^m_αβf(z) = (f ∗g) (z),

where

(6) g(z) =z+

∞

X

n=2

A_n(α, β, m)zⁿ.

We say that a functionf ∈ Ais in the class R^m(α, β, λ) if [D_αβ^mf(z)]⁰ is in the class P(λ), that is, if

(7) Re

D_αβ^mf(z)0

> λ, z∈U

for 0 ≤λ <1, 0≤β ≤α and m ∈N0 := {0,1,2, . . .}. For β = 0, we obtain the class of functions considered in [1].

The main object of this paper is to present a systematic investigation for the class R^m(α, β, λ). In particular, for this function class, we derive an inclusion result, structural formula, extreme points and other interesting properties.

2. INCLUSION RESULT

In order to obtain the inclusion result for the classR^m(α, β, λ), we need the following lemma due to Miller and Mocanu [12, Theorem 1f, p. 198].

Lemma2.1. Leth∈Cand letA≥0. Suppose thatB andDare analytic in U, with D(0) = 0 and

ReB(z)≥A+ 4

D(z) h⁰(0)

for z∈U. If an analytic functionp, with p(0) =h(0) satisfies Az²p⁰⁰(z) +B(z)zp⁰(z) +p(z) +D(z)≺h(z), z∈U then p(z)≺h(z), z ∈U.

Note that the symbol “≺” stands for subordination.

Theorem2.1. Let 0≤λ <1, 0≤β ≤α andm∈N0. Then R^m+1(α, β, λ)⊂R^m(α, β, λ).

Proof. Supposef ∈R^m+1(α, β, λ). Then Re

D_αβ^m+1f(z)0

> λ which is equivalent to

(8)

D_αβ^m+1f(z)0

≺h(z), z∈U, where

(9) h(z) := 1 + (1−2λ)z

1−z , z∈U. From (2), we have

D_αβ^m+1f(z) =αβz²

D^m_αβf(z)00

+ (α−β)z

D_αβ^mf(z)0

+ (1−α+β)D^m_αβf(z).

It follows that (10)

D^m+1_αβ f(z)0

=αβz²

D_αβ^mf(z)000

+ (2αβ+α−β)z

D^m_αβf(z)00

+

D_αβ^mf(z)0

. Denote

(11) p(z) :=

D_αβ^mf(z)0

, z∈U.

Making use of (10) and (11), the differential subordination (8) becomes αβz²p⁰⁰(z) + (2αβ+α−β)zp⁰(z) +p(z)≺h(z), z∈U.

It is easy to check that the conditions of Lemma 2.1 with h(z) given by (9), p(z) given by (11), A=αβ,B(z)≡2αβ+α−β and D(z)≡0 are satisfied.

Thus, we obtainp(z)≺h(z) which implies that Re

D^m_αβf(z)0

> λ, z∈U.

Therefore, f ∈R^m(α, β, λ) and the proof of our theorem is completed.

Corollary 2.1. Let 0≤λ <1, 0≤β ≤α andm∈N0. Then R^m(α, β, λ)⊂S.

Proof. Making use of Theorem 2.1, we obtain

R^m(α, β, λ)⊂R^m−1(α, β, λ)⊂ · · · ⊂R⁰(α, β, λ).

The classR⁰(α, β, λ) consists of functionsf ∈ Afor which Re[D⁰_α,βf(z)]⁰ > λ, that is Ref⁰(z) > λ. It is known (see [9] and also [7]) that, if Ref⁰(z) > λ, 0≤λ <1, thenf is univalent. Thus,

R^m(α, β, λ)⊂R⁰(α, β, λ)⊂S.

3. STRUCTURAL FORMULA

In this section a structural formula, extreme points and coefficient bounds for functions in R^m(α, β, λ) are obtained.

Theorem 3.1. A function f ∈ Ais in the classR^m(α, β, λ) if and only if it can be expressed as

(12) f(z) =

"

z+

∞

X

n=2

1

A_n(α, β, m)zⁿ

#

∗ Z

|ζ|=1

"

z+2(1−λ) ¯ζ

∞

X

n=2

(ζz)ⁿ n

# dµ(ζ), where µ is a positive Borel probability measure defined on the unit circle T= {ζ ∈C:|ζ|= 1}.

Proof. From (5) it follows that,f ∈R^m(α, β, λ) if and only if D_αβ^mf(z)0

−λ 1−λ ∈ P.

Using Herglotz integral representation of functions in Carath´eodory class P (see [8] and also [10]), there exists a positive Borel probability measure µ such that

D_αβ^mf(z)0

−λ

1−λ =

Z

|ζ|=1

1 +ζz

1−ζz dµ(ζ), z∈U which is equivalent to

D^m_αβf(z)0

= Z

|ζ|=1

1 + (1−2λ)ζz 1−ζz dµ(ζ).

Integrating this last equality, we obtain D^m_αβf(z) =

Z z

0

"

Z

|ζ|=1

1 + (1−2λ)ζu 1−ζu dµ(ζ)

# du=

= Z

|ζ|=1

Z z

0

1 + (1−2λ)ζu 1−ζu du

dµ(ζ) that is

(13) D^m_αβf(z) = Z

|ζ|=1

"

z+ 2(1−λ) ¯ζ

∞

X

n=2

(ζz)ⁿ n

# dµ(ζ).

From (5), (6) and (13) it follows that f(z) =

"

z+

∞

X

n=2

1

A_n(α, β, m)zⁿ

#

∗ Z

|ζ|=1

"

z+ 2(1−λ) ¯ζ

∞

X

n=2

(ζz)ⁿ n

# dµ(ζ).

Since this deductive process can be converse, we have proved our theorem.

Corollary 3.1.The extreme points of the class R^m(α, β, λ) are (14) f_ζ(z) =z+ 2(1−λ) ¯ζ

∞

X

n=2

(ζz)ⁿ

nA_n(α, β, m), z∈U, |ζ|= 1.

Proof. Consider the functions

g_ζ(z) =z+ 2(1−λ) ¯ζ

∞

X

n=2

(ζz)ⁿ n and

gµ(z) = Z

|ζ|=1

g_ζ(z)dµ(ζ).

Since the map µ → g_µ is one-to-one, making use of (5), (6) and (13), the assertion follows from (12) (see [5]).

From Corollary 3.1 we can obtain coefficient bounds for the functions in the class R^m(α, β, λ).

Corollary 3.2. If f ∈R^m(α, β, λ) is given by (1) then

|a_n| ≤ 2(1−λ)

nAn(α, β, m), n≥2.

The result is sharp.

Proof. The coefficient bounds are maximized at an extreme point. There- fore, the result follows from (14).

Corollary 3.3. If f ∈R^m(α, β, λ) then, for |z|=r <1 r−2(1−λ)r²

∞

X

n=2

1

nA_n(α, β, m) ≤ |f(z)| ≤r+ 2(1−λ)r²

∞

X

n=2

1 nA_n(α, β, m) and

1−2(1−λ)r

∞

X

n=2

1

An(α, β, m) ≤ |f⁰(z)| ≤1 + 2(1−λ)r

∞

X

n=2

1 An(α, β, m).

4. CONVOLUTION PROPERTY

In order to prove a convolution property for the class R^m(α, β, λ), we need the following result.

Lemma 4.1[15]. Ifp(z) is analytic inU,p(0) = 1andRep(z)> ¹₂ then, for any function F analytic inU, the functionF∗ptakes values in the convex hull of F(U).

Theorem4.1.The classR^m(α, β, λ)is closed under the convolution with a convex function. That is, iff∈R^m(α, β, λ)andg∈C thenf∗g∈R^m(α, β, λ).

Proof. Letg∈C. Then (see [12])

Reg(z) z > 1

2.

Suppose f ∈R^m(α, β, λ). Making use of the convolution properties, we have Re

D^m_αβ(f∗g)(z)0

= Re

D^m_αβf(z)0

∗ g(z) z

.

By applying Lemma 4.1, the result follows.

Corollary 4.1.The class R^m(α, β, λ) is invariant under Bernardi in- tegral operator [4]:

Fc(f)(z) = 1 +c z^c

Z z

0

t^c−1f(t)dt, Rec >0.

Proof. Assume f ∈ R^m(α, β, λ). It is easy to check that Fc(f)(z) = (f∗g)(z), where

g(z) =

∞

X

n=1

1 +c

n+czⁿ= 1 +c z^c

Z z

0

t^c

1−tdt, z∈U, Rec >0.

Since the function φ(z) = _1−z^z , z ∈ U is convex, it follows (see [11]) that the function g is also convex. From Theorem 4.1 we obtain F_c(f) ∈R^m(α, β, λ).

Therefore, Fc[R^m(α, β, λ)]⊂R^m(α, β, λ).

Acknowledgement.The author would like to thank the referee for his/hers helpful remarks and suggestions.

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Received 15 September 2009 “Transilvania” University of Bra¸sov Faculty of Mathematics and Computer Science

Str. Iuliu Maniu 50, 50091 Bra¸sov, Romania dorinaraducanu@yahoo.com