Some Inclusion Properties for Certain Subclasses

Int. J. Open Problems Complex Analysis, Vol. 2, No. 1, March 2010 c ISSN 2074-2827; Copyright ICSRS Publication, 2010 www.i-csrs.org

Some Inclusion Properties for Certain Subclasses Defined by Generalised Derivative Operator and Subordination Ma‘moun Harayzeh Al-Abbadi and ∗ Maslina Darus School of Mathematical Sciences, Faculty of Science and Technology Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia mamoun [email protected], ∗ [email protected](corresponding author) Abstract The authors in [1] have recently introduced a new generalised derivative operator µn,m λ1 ,λ2 , which generalised many wellknown operators studied earlier by many different authors. By using this operator and differential subordination, we introduce certain new subclasses of analytic function defined in the open unit disk U = {z ∈ C : |z| < 1}, which are defined by means of the Hadamard product (or convolution). The purpose of the present paper is to investigate various inclusion properties of these subclasses. In addition, some integral preserving properties are also considered.

Keywords: Analytic function; Hadamard product (or convolution); Univalent function; Convex function; Starlike function; Subordination; Derivative operator. AMS Mathematics Subject Classification (2000): 30C45.

1

Introduction and Definitions

Let A denote the class of functions of the form ∞ X f (z) = z + ak z k , ak is complex number

(1.1)

k=2

which are analytic in the open unit disc U = {z ∈ C : |z| < 1} on the complex plane C. Let S, S ∗ (α), K(α) , C(α) (0 ≤ α < 1) denote the subclasses of A

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

15

consisting of functions that are univalent, starlike of order α, convex of order α, and close-to-convex of order α in U , respectively. In particular, the classes S ∗ (0) = S ∗ , K(0) = K and C(0) = C are the familiar classes of starlike, convex and close-to-convex functions in U , respectively. ∞ ∞ P P Let be given two functions f (z) = z + ak z k and g(z) = z + bk z k analytic k=2

k=2

in the open unit disc U = {z ∈ C : |z| < 1}. Then the Hadamard product (or convolution) f ∗ g of two functions f , g is defined by f (z) ∗ g(z) = (f ∗ g)(z) = z +

∞ X

ak b k z k .

k=2

Next, we state basic ideas on subordination. If f and g are analytic in U , then the function f is said to be subordinate to g, and can be written as f ≺g

or f (z) ≺ g(z)

(z ∈ U ),

if and only if there exists the Schwarz function w, analytic in U , with w(0) = 0 and |w(z)| < 1 such that f (z) = g(w(z)) (z ∈ U ) . Furthermore, if g is univalent in U , then f ≺ g if and only if f (0) = g(0) and f (U ) ⊂ g(U ).([13],P.36). Now, (x)k denotes the Pochhammer symbol (or the shifted factorial) defined by  1 for k = 0, x ∈ C\{0}, (x)k = x(x + 1)(x + 2)...(x + k − 1) for k ∈ N = {1, 2, 3, ...}and x ∈ C. Let ka (z) =

z (1 − z)a

where a is any real number. It is easy to verify that ka (z) = z + Thus ka ∗ f, denotes the Hadamard product of ka with f that is (ka ∗ f )(z) = z +

∞ X (a)k−1 k=2

(1)k−1

∞ P k=2

(a)k−1 k z . (1)k−1

ak z k .

Let N denotes the class of all functions φ which are analytic, convex and univalent in U , with normalisation φ(0) = 1 and Re(φ(z)) > 0 (z ∈ U ). Making

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Ma‘moun Harayzeh Al-Abbadi and M.Darus

use of the principle of subordination between analytic functions, many authors investigated the subclasses S ∗ (φ), K(φ), and C(φ, ψ) of the class A for φ, ψ ∈ N (cf. [7, 10]), which are defined by   zf 0 (z) ∗ S (φ) := f ∈A: ≺ φ(z) in U , f (z)   zf 00 (z) K(φ) := f ∈A: 1+ 0 ≺ φ(z) in U , f (z)   zf 0 (z) ∗ ≺ ψ(z) in U . C(φ, ψ) := f ∈ A : ∃g ∈ S (φ) s.t. g(z) For φ(z) = ψ(z) = (1 + z)/(1 − z) in the definitions defined above, we have the well-known classes S ∗ , K, and C, respectively. Furthermore, for the function classes S ∗ [A, B] and K[A, B] investigated by Janowski [9] ( also see [8]), it is easily seen that   1 + Az ∗ S = S ∗ [A, B] (−1 ≤ B < A ≤ 1), 1 + Bz   1 + Az = K[A, B] (−1 ≤ B < A ≤ 1). K 1 + Bz The authors in [1] have recently introduced a new generalised derivative operator µn,m λ1 ,λ2 , as the following: Definition 1.1. For f ∈ A the generalised derivative operator µn,m λ1 ,λ2 is defined n,m by µλ1 ,λ2 : A → A µn,m λ1 ,λ2 f (z)

=z+

∞ X (1 + λ1 (k − 1))m−1 k=2

(1 + λ2 (k − 1))m

c(n, k)ak z k ,

(z ∈ U ),

where n, m ∈ N0 = {0, 1, 2... .} , λ2 ≥ λ1 ≥ 0 and c(n, k) =



n+k−1 n



=

(n+1)k−1 . (1)k−1

Special cases of this operator includes the Ruscheweyh derivative operator in n,m n,0 n the cases µn,1 λ1 ,0 ≡ µ0,0 ≡ µ0,λ2 ≡ R [17], the Salagean derivative operator n µ0,m+1 ≡ S n [18], the generalised Ruscheweyh derivative operator µn,2 1,0 λ1 ,0 ≡ Rλ [5], the generalised Salagean derivative operator introduced by Al-Oboudi µ0,m+1 ≡ Sβn [3], and the generalised Darus and Al-Shaqsi derivative opλ1 ,0 0,m n erator µn,m+1 ≡ Dλ,β [4]. It is easily seen that µ0,1 λ1 ,0 λ1 ,0 f (z) = µ0,0 f (z) = 1,1 1,m 1,0 0 µ0,0 0,λ2 f (z) = f (z) and µλ1 ,0 f (z) = µ0,0 f (z) = µ0,λ2 f (z) = zf (z) and also µλa−1,0 f (z) = µa−1,m f (z) where a = 1, 2, 3, ... . 0,0 1 ,0

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

17

Let us remind the well known Carlson-Shaffer operator L(a, c) [6] associated with the incomplete beta function h(a, c; z), defined by L(a, c) : A → A, L(a, c)f (z) := h(a, c; z) ∗ f (z), (z ∈ U ), where h(a, c; z) = z +

∞ X (a)k−1 k=2

(c)k−1

zk ,

a is any real number and c ∈ / z0− ; z0− = {0, −1, −2, ...}. It is easily seen that 0,m 0,1 µ0,0 0,λ2 f (z) = µ0,0 f (z) = µλ1 ,0 f (z) = L(0, 0)f (z) = f (z), 1,0 1,1 0 µ1,m 0,0 f (z) = µ0,λ2 f (z) = µλ1 ,0 f (z) = L(2, 1)f (z) = zf (z).

Furthermore, we note that n,0 n,1 n µn,m 0,0 f (z) = µ0,λ2 f (z) = µλ1 ,0 f (z) = L(n + 1, 1)f (z) = D f (z)

(n ∈ N0 ),

where the symbol Dn denotes the familiar Ruscheweyh derivative [17] (also, see [2]) for n ∈ N0 . By using the new generalised derivative operator µn,m λ1 ,λ2 , we introduce the following classes of analytic functions for φ, ψ ∈ N, λ2 ≥ λ1 ≥ 0 and n, m ∈ N0 = {0, 1, 2... .} ,  ∗ Sλn,m (φ) := f ∈ A : µn,m λ1 ,λ2 f (z) ∈ S (φ) , 1 ,λ2  n,m Kλn,m (φ) := f ∈ A : µ f (z) ∈ K(φ) , λ1 ,λ2 1 ,λ2  n,m n,m Cλ1 ,λ2 (φ, ψ) := f ∈ A : µλ1 ,λ2 f (z) ∈ C(φ, ψ) . We note that the class a−1,m a−1,0 S0,0 (φ) = S0,λ (φ) = Sa (φ), 2

was studied by Padmanabhan and Parvatham in [14], a−1,m a−1,0 K0,0 (φ) = K0,λ (φ) = Ka (φ), 2

were studied by Padmanabhan and Manjini in [13]. Obviously, for the special choices function φ and variables n, m we have the following relation ships:       1+z 1+z 1+z 0,m 0,0 ∗ ≡ S0,0 ≡S , S0,λ2 1−z 1−z 1−z       1+z 1+z 1+z 0,0 0,m K0,λ2 ≡ K0,0 ≡K , 1−z 1−z 1−z     1+z 1+z 0,0 C0,λ ,ψ ≡ C ,ψ . 2 1−z 1−z

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Ma‘moun Harayzeh Al-Abbadi and M.Darus

And 

   1 + (1 − 2α)z 1 + (1 − 2α)z 0,m ≡ S0,0 ≡ S ∗ (α), 1−z 1−z     1 + (1 − 2α)z 1 + (1 − 2α)z 0,0 0,m K0,λ2 ≡ K0,0 = K(α) (0 ≤ α < 1). 1−z 1−z 0,0 S0,λ 2

In particular, we set   1 + Az n,m [A, B] (−1 ≤ B < A ≤ 1), Sλ1 ,λ2 = Sλn,m 1 ,λ2 1 + Bz   1 + Az n,m Kλ1 ,λ2 = Kλn,m [A, B] (−1 ≤ B < A ≤ 1). 1 ,λ2 1 + Bz In this paper, we investigate several inclusion properties of the classes Sλn,m (φ), 1 ,λ2 n,m n,m Kλ1 ,λ2 (φ) and Cλ1 ,λ2 (φ, ψ). The integral preserving properties in connection with the operator µn,m λ1 ,λ2 are also considered. Furthermore, relevant connection of the results presented here with those obtained in earlier works are pointed out. We first state some preliminary lemmas which shall be used in our investigation.

2

Preliminary Results

To establish our results, we recall the following: Lemma 2.1 (Ruscheweyh and Sheil-Small ([16], p.54). If f ∈ K, g ∈ S ∗ , then for each analytic function φ in U , (f ∗ φg) (U ) ⊂ coφ(U ), (f ∗ g) (U ) where coφ(U ) denotes the closed convex hull of φ(U ). Lemma 2.2 (Ruscheweyh [15]). Let 0 < a ≤ c. If c ≥ 2 or a + c ≥ 3, then the function ∞ X (a)k−1 k h(a, c; z) = z + z (z ∈ U ), (c) k−1 k=2 belongs to the class K of convex functions. Lemma 2.3 ([11]). Let φ be analytic, univalent, convex in U , with φ(0) = 1 and Re(βφ(z) + γ) > 0

(β, γ ∈ C; z ∈ U ).

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

19

If p(z) is analytic in U , with p(0) = φ(0), then p(z) +

3

zp0 (z) ≺ φ(z) ⇒ p(z) ≺ φ(z). βp(z) + γ

Inclusion Properties Involving the Operator µn,m λ1 ,λ2

Our main results, are the following: Theorem 3.1. f (z) ∈ Kλn,m (φ) if and only if zf 0 (z) ∈ Sλn,m (φ). 1 ,λ2 1 ,λ2 Proof. Consider n,m µn,m λ1 ,λ2 f (z) = hλ1 ,λ2 (z) ∗ f (z),

where hn,m λ1 ,λ2 (z)

=z+

∞ X (1 + λ1 (k − 1))m−1 k=2

(1 + λ2 (k − 1))

m

(3.1)

c(n, k)z k .

(3.2)

By the definition of the class Kλn,m (φ) and using the well-known property of 1 ,λ2 0 0 convolution z(f ∗ g) (z) = (f ∗ zg )(z), we have 00 z(µn,m λ1 ,λ2 f (z)) ≺ φ(z), 0 (µn,m λ1 ,λ2 f (z))  n,m 0 z(µλ1 ,λ2 f (z))0 ≺ φ(z), 0 (µn,m λ1 ,λ2 f (z)) h  0 i0 z z hn,m (z) ∗ f (z) λ1 ,λ2  n,m 0 ≺ φ(z), z hλ1 ,λ2 (z) ∗ f (z)  0 0 z hn,m (z) ∗ zf (z) λ1 ,λ2 ≺ φ(z), 0 hn,m λ1 ,λ2 (z) ∗ zf (z)  0 0 z µn,m λ1 ,λ2 zf (z) ≺ φ(z), 0 µn,m λ1 ,λ2 zf (z)

f ∈ Kλn,m (φ) ⇔ 1 + 1 ,λ2 ⇔

⇔ ⇔ ⇔

⇔ zf 0 (z) ∈ Sλn,m (φ). 1 ,λ2 Theorem 3.2. Let φ ∈ N, λ2 ≥ λ1 ≥ 0, n2 ≥ n1 ≥ 0 and n1 , n2 , m ∈ N0 . If n2 ≥ 1 or n1 + n2 ≥ 1, then ,m ,m Sλn12,λ (φ) ⊂ Sλn11,λ (φ). 2 2

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Ma‘moun Harayzeh Al-Abbadi and M.Darus

,m (φ). Then there exists an analytic function Proof. We suppose that f ∈ Sλn12,λ 2 w in U with |w(z)| < 1 (z ∈ U ) and w(0) = 0 such that ,m f (z))0 z(µnλ12,λ 2 = φ(w(z)), (z ∈ U ). ,m (µnλ12,λ f (z)) 2

(3.3)

Where φ is analytic and convex univalent with φ(0) = 1 and Re(φ(z)) > 0, (z ∈ U ). We set ,m ,m f (z) ∗ ψnn21 (z), f (z) = µnλ12,λ µnλ11,λ 2 2

where ψnn21 (z) = z +

∞ X (n1 + 1)k−1 k=2

(n2 + 1)k−1

zk .

We get  ,m 0 ,m z µnλ12,λ z(µnλ11,λ f (z) ∗ ψnn21 (z) f (z))0 2 2 = ,m ,m (µnλ11,λ f (z)) µnλ12,λ f (z) ∗ ψnn21 (z) 2 2   ,m 0 ψnn21 (z) ∗ z(µnλ12,λ f (z)) 2 = ,m ψnn21 (z) ∗ µnλ12,λ f (z) 2 n1 ψn2 (z) ∗ φ(w(z))p(z) = , ψnn21 (z) ∗ p(z)

(3.4)

,m where p(z) = µnλ12,λ f (z). 2

It follows from Lemma 2.2 that ψnn21 (z) ∈ K and it follows from the definition of the class Sλn,m (φ) that p(z) ∈ S ∗ . Therefore applying Lemma 2.1 to (3.4) 1 ,λ2 we obtain  n ψn21 (z) ∗ φ(w(z))p (U ) ⊂ co φ(w(U )) ⊂ φ(U ). (3.5) {ψnn21 (z) ∗ p} (U ) Since φ is analytic and convex univalent. Therefore from the definition of subordination and (3.5), we note that (3.4) is subordinate to φ in U and ,m consequently f (z) ∈ Sλn11,λ (φ). This completes the proof of Theorem 3.2. 2 Theorem 3.3. Let φ ∈ N, λ2 ≥ λ1 ≥ 0, n2 ≥ n1 ≥ 0 and n1 , n2 , m ∈ N0 . If n2 ≥ 1 or n1 + n2 ≥ 1, then ,m ,m Kλn12,λ (φ) ⊂ Kλn11,λ (φ). 2 2 ,m ,m Proof. Let f (z) ∈ Kλn12,λ (φ) we want to show f (z) ∈ Kλn11,λ (φ). Applying 2 2 Theorem 3.1 and Theorem 3.2 we observe that ,m ,m f (z) ∈ Kλn12,λ (φ) ⇔ zf 0 (z) ∈ Sλn12,λ (φ), 2 2 n1 ,m 0 ⇒ zf (z) ∈ Sλ1 ,λ2 (φ), ,m ⇔ f (z) ∈ Kλn11,λ (φ). 2

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

21

Theorem 3.4. Let φ ∈ N, λ2 ≥ λ1 ≥ 0 and n, m ∈ N0 then i) Sλn,m (φ) ⊂ Sλn,m+1 f (z) 1 ,λ2 1 ,λ2 (φ) (φ) ⊂ Kλn,m+1 ii) Kλn,m 1 ,λ2 1 ,λ2 (φ). Using similar arguments as in Proof. i) We suppose that f (z) ∈ Sλn,m 1 ,λ2 the proof of Theorem 3.2, and set n,m λ1 µn,m+1 λ1 ,λ2 f (z) = µλ1 ,λ2 f (z) ∗ ψλ2 (z),

where ψλλ21 (z)

=z+

∞ X 1 + λ1 (k − 1) k=2

1 + λ2 (k − 1)

zk .

We obtain 0 z(µn,m+1 λ1 ,λ2 f (z))

µn,m+1 λ1 ,λ2 f (z)

= = =

 0 λ1 z µn,m λ1 ,λ2 f (z) ∗ ψλ2 (z) λ1 µn,m λ1 ,λ2 f (z) ∗ ψλ2 (z) 0 ψλλ21 (z) ∗ z(µn,m λ1 ,λ2 f (z))

ψλλ21 (z) ∗ µn,m λ1 ,λ2 f (z) ψλλ21 (z) ∗ φ(w(z))p(z) ψλλ21 (z) ∗ p(z)

,

,

.

(3.6)

Where p(z) = µn,m λ1 ,λ2 f (z) and w is an analytic function in U with |w(z)| < 1 (z ∈ U ) and w(0) = 0. It follows from the definition of the class Sλn,m (φ) 1 ,λ2 ∗ that p(z) ∈ S . And by classical results in the class of convex, the coefficients 1+λ1 (k−1) ≤ 1 since λ2 ≥ λ1 ≥ 0, so we problem for convex: |an | ≤ 1, and here 1+λ 2 (k−1) λ1 find ψλ2 ∈ K. Hence it follows from applying Lemma 2.1 to (3.6) that  λ1 ψλ2 (z) ∗ φ(w(z))p (U )  λ1 (3.7) ⊂ co φ(w(U )) ⊂ φ(U ). ψλ2 (z) ∗ p (U ) Since φ is analytic and convex univalent. And from the definition of subordination and (3.7). Thus (3.6) is subordinate to φ in U and consequently f (z) ∈ Sλn,m+1 f (z). The proof of Theorem 3.4 is complete. 1 ,λ2 ii) Let f (z) ∈ Kλn,m (φ). We shall show f (z) ∈ Kλn,m+1 (φ). By applying 1 ,λ2 1 ,λ2 Theorem 3.1, and Theorem 3.4 (i), we have f (z) ∈ Kλn,m (φ) ⇔ zf 0 (z) ∈ Sλn,m (φ), 1 ,λ2 1 ,λ2 f (z), ⇒ zf 0 (z) ∈ Sλn,m+1 1 ,λ2 ⇔ f (z) ∈ Kλn,m+1 (φ). 1 ,λ2

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Ma‘moun Harayzeh Al-Abbadi and M.Darus

Theorem 3.5. Let φ ∈ N, λ3 ≥ λ2 ≥ λ1 ≥ 0 and n, m ∈ N0 , then Sλn,m (φ) ⊂ Sλn,m (φ). 1 ,λ2 1 ,λ3 Proof. Let f ∈ Sλn,m (φ). Applying the definition of the class Sλn,m (φ), then 1 ,λ2 1 ,λ2 the setting, n,m m µn,m λ1 ,λ3 f (z) = µλ1 ,λ2 f (z) ∗ ψλ2 ,λ3 (z), where ψλm2 ,λ3 (z)

=z+

m ∞  X 1 + λ2 (k − 1) k=2

1 + λ3 (k − 1)

z k , (z ∈ U ).

And by classical results in the class of convex, the coefficients problem for convex: |an | ≤ 1, and here  m 1 + λ2 (k − 1) ≤ 1, since λ3 ≥ λ2 ≥ 0, 1 + λ3 (k − 1) so we get ψλm2 ,λ3 (z) ∈ K. After that, using the same arguments as in the Theorem 3.4 we obtain the desired result. Theorem 3.6. Let φ ∈ N, λ3 ≥ λ2 ≥ λ1 ≥ 0 and n ∈ N0 , then. If m ∈ N then Sλn,m (φ) ⊂ Sλn,m (φ). 2 ,λ3 1 ,λ3 Proof. Let f (z) ∈ Sλn,m (φ). Applying the definition of the class Sλn,m (φ), 2 ,λ3 2 ,λ3 then the setting. n,m m µn,m λ1 ,λ3 f (z) = µλ2 ,λ3 f (z) ∗ ψλ1 ,λ2 (z), where ψλm1 ,λ2 (z)

=z+

m−1 ∞  X 1 + λ1 (k − 1) k=2

1 + λ2 (k − 1)

z k , (z ∈ U ).

And using the same arguments as in the Theorem 3.4 we obtain the desired result. Corollary 3.1. Let φ ∈ N, λ3 ≥ λ2 ≥ λ1 ≥ 0 and n, m ∈ N0 , then (φ) ⊂ Kλn,m (φ) i) Kλn,m 1 ,λ2 1 ,λ3 ii) If m ∈ N then Kλn,m (φ) ⊂ Kλn,m (φ). 2 ,λ3 1 ,λ3 Taking 1 + Az (−1 ≤ B < A ≤ 1, z ∈ U ). 1 + Bz In Corollary 3.1 and Theorems 3.5 and 3.6 we have the following Corollary. φ(z) =

23

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

Corollary 3.2. Let λ3 ≥ λ2 ≥ λ1 ≥ 0 and n, m ∈ N0 , then Sλn,m [A, B] ⊂ Sλn,m [A, B], (−1 ≤ B < A ≤ 1), 1 ,λ2 1 ,λ3 n,m n,m Kλ1 ,λ2 [A, B] ⊂ Kλ1 ,λ3 [A, B] (−1 ≤ B < A ≤ 1), if m ∈ N then [A, B] (−1 ≤ B < A ≤ 1), [A, B] ⊂ Sλn,m Sλn,m 1 ,λ3 2 ,λ3 n,m n,m Kλ2 ,λ3 [A, B] ⊂ Kλ1 ,λ3 [A, B] (−1 ≤ B < A ≤ 1). Theorem 3.7. If f (z) ∈ Sλn,m (φ) for n, m ∈ N0 . Then Fµ (f ) ∈ Sλn,m (φ). 1 ,λ2 1 ,λ2 Where Fµ is the integral operator defined by µ+1 Fµ (f ) = Fµ (f )(z) := zµ

Zz

tµ−1 f (t)dt (µ ≥ 0).

(3.8)

0

Proof. Let f (z) ∈ Sλn,m (φ) and 1 ,λ2  0 z µn,m λ1 ,λ2 Fµ (f ) . p(z) = µn,m λ1 ,λ2 Fµ (f ) From (3.8), we have z(Fµ (f ))0 (z) + µFµ (f )(z) = (µ + 1)f (z) and so n,m n,m 0 (hn,m λ1 ,λ2 ∗ z(Fµ (f )) )(z) + µ(hλ1 ,λ2 ∗ Fµ (f ))(z) = (µ + 1)(hλ1 ,λ2 ∗ f )(z).

Using the fact n,m 0 0 z(hn,m λ1 ,λ2 ∗ Fµ (f )) (z) = (hλ1 ,λ2 ∗ zFµ (f ))(z),

and setting (3.1), we obtain n,m n,m 0 z(µn,m λ1 ,λ2 Fµ (f )) (z) + µ(µλ1 ,λ2 Fµ (f ))(z) = (µ + 1)µλ1 ,λ2 f (z).

(3.9)

Differentiating (3.9), we have "

# µn,m λ1 ,λ2 f (z) p(z) + µ = (µ + 1) n,m . µλ1 ,λ2 Fµ (f )

(3.10)

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Ma‘moun Harayzeh Al-Abbadi and M.Darus

Making use of the logarithmic differentiation on both sides of (3.10) and multiplying the resulting equation by z, we have 0 z(µn,m zp0 (z) λ1 ,λ2 f (z)) p(z) + . = p(z) + µ µn,m λ1 ,λ2 f (z)

(3.11)

By applying Lemma 2.3 to (3.11), it follows that p ≺ φ in U , that is Fµ (f ) ∈ (φ). Sλn,m 1 ,λ2 Remark 3.1. special case of this operator µn,m λ1 ,λ2 include the Bernardi integral 1 operator Fµ (f ) in two cases, for n = 0, m = 0, and λ1 = 1+µ and also for 1 n = 0, m = 1 and λ2 = 1+µ . Theorem 3.8. If f ∈ Sλn,m (φ). Then 1 ,λ2 Zz ψ=

f (t) dt ∈ Kλn,m (φ). 1 ,λ2 t

0

Proof. Let f ∈ Sλn,m (φ). Now zψ 0 = f that is zψ 0 ∈ Sλn,m (φ), applying 1 ,λ2 1 ,λ2 Theorem 3.1, we see zψ 0 ∈ Sλn,m (φ) ⇔ ψ ∈ Kλn,m (φ). 1 ,λ2 1 ,λ2 This proves the Theorem. To prove the Theorems below, we need the following Lemma. Lemma 3.1. Let φ ∈ N. If f ∈ K and q ∈ S ∗ (φ), then f ∗ q ∈ S ∗ (φ). Proof. Let q ∈ S ∗ (φ). Then there exists an analytic function w in U with |w(z)| < 1, (z ∈ U ) and w(0) = 0 such that zq 0 (z) = φ(w(z)), q(z)

(z ∈ U ).

Where φ is analytic and convex univalent with φ(0) = 1 and Re(φ(z)) > 0, (z ∈ U ). Thus we have z (f (z) ∗ q(z))0 f (z) ∗ zq 0 (z) = f (z) ∗ q(z) f (z) ∗ q(z) f (z) ∗ φ(w(z))q(z) = , (z ∈ U ). f (z) ∗ q(z)

(3.12)

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

25

By using similar arguments to those used in the proof of Theorem 3.2, we conclude that (3.12) is subordinate to φ in U and so f ∗ q ∈ S ∗ (φ). Theorem 3.9. Let φ, ψ ∈ N , λ2 ≥ λ1 ≥ 0 and n1 , n2 , m ∈ N0 . If n2 ≥ min{1, 1 − n1 }, then ,m+1 ,m ,m (φ, ψ). (φ, ψ) ⊂ Cλn11,λ (φ, ψ) ⊂ Cλn11,λ Cλn12,λ 2 2 2

Proof. First of all, we show that ,m ,m Cλn12,λ (φ, ψ) ⊂ Cλn11,λ (φ, ψ). 2 2 ,m Let f ∈ Cλn12,λ (φ, ψ). Then there exist a function q ∈ S ∗ (φ) such that 2
0 ,m z µnλ12,λ f (z) 2 ≺ ψ(z) (z ∈ U ). q(z)

(3.13)

(3.14)

Then there exists analytic function w in U with |w(z)| < 1 (z ∈ U ), and w(0) = 0 such that
0 ,m z µnλ12,λ f (z) = ψ(w(z))q(z) (z ∈ U ), 2 where ψ is analytic and convex univalent with ψ(0) = 1 and Re(ψ(z)) > 0, (z ∈ U ). We setting ,m ,m µnλ11,λ f (z) = µnλ12,λ f (z) ∗ ψnn21 (z), 2 2

where ψnn21 (z)

=z+

∞ X (n1 + 1)k−1 k=2

(n2 + 1)k−1

zk .

ψnn21

By virtue of Lemma 2.2, ∈ K and using Lemma 3.1, we see that ψnn21 (z) ∗ ∗ q(z) belongs to S (φ). Then we have  n2 ,m 0
0 ,m n1 z µ f (z) ∗ ψ (z) z µnλ11,λ f (z) n λ1 ,λ2 2 2 = , g(z) ψnn21 (z) ∗ q(z)
0 ,m f (z) ψnn21 (z) ∗ z µnλ12,λ 2 = , ψnn21 (z) ∗ q(z) ψnn21 (z) ∗ ψ(w(z))q(z) . (3.15) = ψnn21 (z) ∗ q(z) By using similar arguments to those used in the proof of Theorem 3.2, we con,m (φ, ψ). clude that (3.15) is subordinated to ψ in U . This implies that f ∈ Cλn11,λ 2 Moreover, the proof of the second part is similar to that of the first part and so we omit the details involved.

26

Ma‘moun Harayzeh Al-Abbadi and M.Darus

4

Convolution Results and Inclusion Properties Involving Various Operators

The next theorem shows that the classes Sλn,m (φ), Kλn,m (φ) and Cλn,m (φ, ψ) 1 ,λ2 1 ,λ2 1 ,λ2 are invariant under convolution with convex functions. Theorem 4.1. Let φ, ψ ∈ N , λ2 ≥ λ1 ≥ 0, n, m ∈ N0 and let g ∈ K. Then i) ii) iii)

f ∈ Sλn,m (φ) ⇒ g ∗ f ∈ Sλn,m (φ), 1 ,λ2 1 ,λ2 n,m n,m f ∈ Kλ1 ,λ2 (φ) ⇒ g ∗ f ∈ Kλ1 ,λ2 (φ), f ∈ Cλn,m (φ, ψ) ⇒ g ∗ f ∈ Cλn,m (φ, ψ). 1 ,λ2 1 ,λ2

Proof. i) We begin by assuming f ∈ Sλn,m (φ) and g ∈ K. In the proof we 1 ,λ2 use the same techniques as in the proof of Theorem 3.2. Let 0 z(µn,m λ1 ,λ2 f (z)) = φ(w(z)), (z ∈ U ). µn,m λ1 ,λ2 f (z)

and p(z) = µn,m λ1 ,λ2 f (z). Using the following equality
0
n,m 0 z hn,m ∗ f (z) = h ∗ zf (z), λ1 ,λ2 λ1 ,λ2 from (3.1) we write  n,m 0  0 z h (z) ∗ (g ∗ f )(z) z µn,m (g ∗ f ) (z) λ1 ,λ2 λ1 ,λ2 = , n,m (g ∗ f )(z) µn,m h λ1 ,λ2 λ1 ,λ2 (z) ∗ (g ∗ f )(z) 0 g(z) ∗ z(µn,m λ1 ,λ2 f (z)) = , g(z) ∗ µn,m λ1 ,λ2 f (z) g(z) ∗ φ(w(z))p(z) ≺ φ(z). = g(z) ∗ p(z)

Consequently g ∗ f ∈ Sλn,m (φ). 1 ,λ2 ii) Let f ∈ Kλn,m (φ). Then by Theorem 3.1 and from Theorem 4.1(i), we have 1 ,λ2 f ∈ Kλn,m (φ) ⇔ 1 ,λ2 ⇒ ⇔ ⇔

zf 0 (z) ∈ Sλn,m (φ), 1 ,λ2 0 g ∗ [zf (z)] ∈ Sλn,m (φ), 1 ,λ2 n,m 0 z(g ∗ f ) ∈ Sλ1 ,λ2 (φ), g ∗ f ∈ Kλn,m (φ). 1 ,λ2

27

Some Inclusion Properties for Certain Subclasses Defined by Generalised ...

(φ, ψ). Then there exists a function q ∈ S ∗ (φ) such that iii) Let f ∈ Cλn,m 1 ,λ2
0 z µn,m λ1 ,λ3 f (z) = ψ(w(z))q(z) (z ∈ U ), where w is an analytic function in U with |w(z)| < 1 (z ∈ U ) and w(0) = 0. From Lemma 3.1, we have that g ∗ q ∈ S ∗ (φ). Applying the same method in the proof of Theorem 3.9 and using the fact that z(f ∗ g)0 (z) = (f ∗ zg 0 )(z) we have 0  0 (g ∗ f ) z µn,m (z) g(z) ∗ z(µn,m λ1 ,λ2 λ1 ,λ2 f (z)) = , (g ∗ q)(z) g(z) ∗ q(z) g(z) ∗ ψ(w(z))q(z) = ≺ ψ(z) (z ∈ U ). g(z) ∗ q(z) We obtain (iii). Now we consider the following operators [17, 12] given by ψ1 (z) =

∞ X 1+c k=1

k+c

z k (Re {c} ≥ 0; (z ∈ U ), (4.1) 

ψ2 (z) =

1 1 − xz log 1−x 1−z

 (Log 1 = 0; |x| ≤ 1, x 6= 1; z ∈ U ).

It is well known [17] that the operators ψ1 and ψ2 are convex univalent in U . Therefore, we have the following result, which can be obtained from Theorem 4.1 immediately. Corollary 4.1. Let λ2 ≥ λ1 ≥ 0, n, m ∈ N0 , φ, ψ ∈ N and let ψi (i = 1, 2) be defined by (4.1). Then i) ii) iii)

5

f ∈ Sλn,m (φ) ⇒ ψi ∗ f ∈ Sλn,m (φ), 1 ,λ2 1 ,λ2 n,m n,m f ∈ Kλ1 ,λ2 (φ) ⇒ ψi ∗ f ∈ Kλ1 ,λ2 (φ), f ∈ Cλn,m (φ, ψ) ⇒ ψi ∗ f ∈ Cλn,m (φ, ψ). 1 ,λ2 1 ,λ2

Open Problem

The generalised derivative operator for the three classes Sλn,m (φ), Kλn,m (φ) 1 ,λ2 1 ,λ2 and Cλn,m (φ, ψ) can be applied in the subordination and superordination the1 ,λ2 orem. In fact, basic properties such as the coefficient estimates are yet to be solved for these type of classes. Acknowledgement: This work is fully supported by UKM-GUP-TMK-0702-107, Malaysia.

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Ma‘moun Harayzeh Al-Abbadi and M.Darus

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[14] K. S. Padmanabhan and R.Parvatham, Some applications of differential subordination, Bull. Austral. Math. Soc., 32, (1985), 321-330. [15] S. Ruscheweyh, Convolutions in geometric function theory, Les Presses de I‘Univ.de Montreal., 1982. [16] S. Ruscheweyh and T.Sheil-small, Hadamard product of schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv., 48 (1973), 119 - 135. [17] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., Vol. 49, pp. (1975), 109-115. [18] G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math. (Springer-Verlag)., 1013,(1983), 362-372.