Classes of Multivalent Functions Defined by Dziok ...

KYUNGPOOK Math. J. 46(2006), 97-109

Classes of Multivalent Functions Defined by Dziok-Srivastava Linear Operator and Multiplier Transformation S. Sivaprasad Kumar and H. C. Taneja Department of Applied Mathematics, Delhi College of Engineering, Delhi-110042, India e-mail : [email protected] and [email protected] V. Ravichandran School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia e-mail : [email protected] Abstract. In this paper, the authors introduce new classes of p-valent functions defined by Dziok-Srivastava linear operator and the multiplier transformation and study their properties by using certain first order differential subordination and superordination. Also certain inclusion relations are established and an integral transform is discussed.

1. Preliminaries Let H be the class of functions analytic in ∆ := {z ∈ C : |z| < 1} and H(a, n) be the subclass of H consisting of functions of the form f (z) = a+an z n +an+1 z n+1 +· · · . Let Ap denote the class of all analytic functions of the form (1.1)

f (z) = z p +

∞ X

ak z k

(z ∈ ∆)

k=p+1

and let A := A1 . For two functions f (z) given by (1.1) and g(z) = z p + the Hadamard product (or convolution) of f and g is defined by (1.2)

(f ∗ g)(z) := z p +

∞ X

P∞

k=p+1 bk z

k

,

ak bk z k =: (g ∗ f )(z).

k=p+1

For αj ∈ C (j = 1, 2, · · · , l) and βj ∈ C \ {0, −1, −2, · · · } (j = 1, 2, · · · , m), the generalized hypergeometric function l Fm (α1 , · · · , αl ; β1 , · · · , βm ; z) is defined by the Received September 30, 2004, and, in revised form, March 7, 2005. 2000 Mathematics Subject Classification: 30C80, 30C45. Key words and phrases: analytic function, univalent function, differential subordinations, differential superordinations, Dziok-Srivastava linear operator, multiplier transformation.

97

98

S. Sivaprasad Kumar, V. Ravichandran and H. C. Taneja

infinite series l Fm (α1 , · · · , αl ; β1 , · · · , βm ; z) :=

∞ X (α1 )n · · · (αl )n z n (β1 )n · · · (βm )n n! n=0

(l ≤ m + 1; l, m ∈ N0 := {0, 1, 2, · · · }) where (λ)n is the Pochhammer symbol defined by ½ Γ(λ + n) 1, (λ)n := = λ(λ + 1)(λ + 2) · · · (λ + n − 1), Γ(λ)

(n = 0); (n ∈ N := {1, 2, 3 · · · }).

Corresponding to the function hp (α1 , · · · , αl ; β1 , · · · , βm ; z) := z p l Fm (α1 , · · · , αl ; β1 , · · · , βm ; z), (l,m)

the Dziok-Srivastava operator [7] (see also [18]) Hp defined by the Hadamard product (1.3)

(α1 , · · · , αl ; β1 , · · · , βm ) is

Hp(l,m) (α1 , · · · , αl ; β1 , · · · , βm )f (z) := hp (α1 , · · · , αl ; β1 , · · · , βm ; z) ∗ f (z) ∞ X (α1 )n−p · · · (αl )n−p an z n = zp + . (β1 )n−p · · · (βm )n−p (n − p)! n=p+1

Special cases of the Dziok-Srivastava linear operator includes the Hohlov linear operator [8], the Carlson-Shaffer linear operator [4], the Ruscheweyh derivative operator [16], the generalized Bernardi-Libera-Livingston integral operator (cf. [1], [9], [11]) and the Srivastava-Owa fractional derivative operators (cf. [14], [15]). A function f ∈ Ap is said to be in the class Ha,c,p (A, B) if it satisfies the following subordination: Lp (a + 1, c)f (z) A−B z ≺1+ Lp (a, c)f (z) a 1 + Bz

(z ∈ ∆; a 6= 0; −1 ≤ B < A ≤ 1)

where Lp (a, c) is the familiar Carlson and Shaffer operator [4] given by Lp (a, c) := (2,1) Hp (a, 1; c). This class Ha,c,p (A, B) was introduced by Liu and Owa [10] and they have proved the following: Theorem 1.1 (Liu and Owa [10, Theorem 1, p.1715]). Let a ≥

A−B 1−B ,

then

Ha+1,c,p (A, B) ⊂ Ha,c,p (A, B).

Theorem 1.2 (Liu and Owa [10, Theorem 2, p.1716]). Let λ be a complex nuumber such that A−B 1−B − p. If f ∈ Ha,c,p (A, B), then the function Z λ + p z λ−1 t f (t)dt F (z) = zλ 0

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Classes of Multivalent Functions

belongs to Ha,c,p (A, B). Theorem 1.3 (Liu and Owa [10, Theorem 3, p.1717]). Let f (z) ∈ Ap . Then f ∈ Ha,c,p (A, B) if and only if Z z a ta−p−1 f (t)dt ∈ Ha+1,c,p (A, B). F (z) = a−p z 0 For two analytic functions f and F , we say that F is superordinate to f if f is subordinate to F . Recently Miller and Mocanu [13] considered certain second order differential superordinations. Using the results of Miller and Mocanu [13], Bulboac˘a have considered certain classes of first order differential superordinations [3] and superordination-preserving integral operators [2]. In the present investigation, we generalize the above-stated results of Liu and Owa [10] to a more general classes of p-valent functions which we define below using subordination and superordination. Definition 1. A function f ∈ A(p, n) is said to be in the class A(p, n, α1 · · · αl ; β1 · · · βm ; ϕ) if it satisfies the following subordination: Hpl,m [α1 + 1]f (z)

(1.4)

Hpl,m [α1 ]f (z)

≺ ϕ(z),

and is said to be in A(p, n, α1 · · · αl ; β1 · · · βm ; ϕ) if f satisfies the following superordination: (1.5)

ϕ(z) ≺

Hpl,m [α1 + 1]f (z) Hpl,m [α1 ]f (z)

,

where ϕ(z) is analytic in ∆ and ϕ(0) = 1 and Hpl,m [α1 ]f (z) := Hp(l,m) (α1 , · · · , αl ; β1 , · · · , βm )f (z). To make the notation simple, we also write A(p, n, α1 ; ϕ) := A(p, n, α1 · · · αl ; β1 · · · βm ; ϕ) and A(p, n, α1 ; ϕ) := A(p, n, α1 · · · αl ; β1 · · · βm ; ϕ). Also we define the class A(p, n, α1 ; ϕ1 , ϕ2 ) by the following: A(p, n, α1 ; ϕ1 , ϕ2 ) := A(p, n, α1 ; ϕ1 ) ∩ A(p, n, α1 ; ϕ2 ). For ϕ(z) = 1 +

A−B z a 1 + Bz

(z ∈ ∆; −1 ≤ B < A ≤ 1),

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the class A(p, n, a, 1; c; ϕ) reduces to the class Ha,c,p (A, B), introduced and studied by Liu and Owa [10]. Motivated by the multiplier transformation on A, we define the operator Ip (r, λ) on Ap by the following infinite series (1.6)

¶r ∞ µ X k+λ ak z k Ip (r, λ)f (z) := z + p+λ p

(λ ≥ 0).

k=p+1

The operator Ip (r, λ) is closely related to the Sˇalˇagean derivative operators [17]. The operator Iλr := I1 (r, λ) was studied recently by Cho and Srivastava [5] and Cho and Kim [6]. The operator Ir := I1 (r, 1) was studied by Uralegaddi and Somanatha [19]. Definition 2. A function f ∈ A(p, n) is said to be in the class A(p, n, r, λ; ϕ) if it satisfies the following subordination: (1.7)

Ip (r + 1, λ)f (z) ≺ ϕ(z) (f ∈ A(p, n)), Ip (r, λ)f (z)

and is said to be in A(p, n, r, λ; ϕ) if f satisfies the following superordination: (1.8)

ϕ(z) ≺

Ip (r + 1, λ)f (z) Ip (r, λ)f (z)

(f ∈ A(p, n)),

where ϕ(z) is analytic in ∆ and ϕ(0) = 1. Also we define the class A(p, n, r, λ; ϕ1 , ϕ2 ) by the following: A(p, n, r, λ; ϕ1 , ϕ2 ) := A(p, n, r, λ; ϕ1 ) ∩ A(p, n, r, λ; ϕ2 ). In our present investigation of the above-defined classes, we need the following: Definition 3 [13, Definition 2, p. 817]. Denote by Q, the set of all functions f (z) that are analytic and injective on ∆ − E(f ), where E(f ) = {ζ ∈ ∂∆ : lim f (z) = ∞}, z→ζ

and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂∆ − E(f ). Lemma 1 [cf. Miller and Mocanu [12, Theorem 3.4h, p.132]]. Let ψ(z) be univalent in the unit disk 4 and let ϑ and ϕ be analytic in a domain D ⊃ ψ(4) with ϕ(w) 6= 0, when w ∈ ψ(4). Set Q(z) := zψ 0 (z)ϕ(ψ(z)), Suppose that

h(z) := ϑ(ψ(z)) + Q(z).

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(1) Q(z) is starlike in 4, and (2)
0 for z ∈ 4.

If q(z) is analytic in ∆, with q(0) = ψ(0), q(∆) ⊂ D and (1.9)

ϑ(q(z)) + zq 0 (z)ϕ(q(z)) ≺ ϑ(ψ(z)) + zψ 0 (z)ϕ(ψ(z)),

then q(z) ≺ ψ(z) and ψ(z) is the best dominant. Lemma 2 ([3]). Let ψ(z) be univalent in the unit disk ∆ and ϑ and ϕ be analytic in a domain D containing ψ(∆). Suppose that (1) < [ϑ0 (ψ(z))/ϕ(ψ(z))] > 0 for z ∈ ∆, (2) zψ 0 (z)ϕ(ψ(z)) is starlike in ∆. If q(z) ∈ H(ψ(0), 1) ∩ Q, with q(∆) ⊆ D, and ϑ(q(z)) + zq 0 (z)ϕ(q(z)) is univalent in ∆, then (1.10)

ϑ(ψ(z)) + zψ 0 (z)ϕ(ψ(z)) ≺ ϑ(q(z)) + zq 0 (z)ϕ(q(z)),

implies ψ(z) ≺ q(z) and ψ(z) is the best subordinant. 2. Results involving Dziok-Srivastava linear operator By making use of Lemma 1, we first prove the following generalization of Theorem 1.1: Theorem 2.1. Let ψ(z) be univalent in ∆, ψ(0) = 1. Assume that zψ 0 /ψ is starlike in ∆ and < {α1 ψ(z)} > 0. Let χ(z) be defined by · ¸ 1 zψ 0 (z) (2.1) χ(z) := α1 ψ(z) + 1 + (α1 6= −1). α1 + 1 ψ(z) If f ∈ A(p, n, α1 + 1; χ), then f ∈ A(p, n, α1 ; ψ). If f ∈ A(p, n, α1 + 1; χ), (2.2) 0 6=

Hpl,m [α1 + 1]f (z) Hpl,m [α1 ]f (z)

∈ H(1, 1) ∩ Q,

Hpl,m [α1 + 2]f (z) Hpl,m [α1 + 1]f (z)

is univalent in ∆,

then f ∈ A(p, n, α1 ; ψ). Proof. Define the function q(z) by (2.3)

q(z) :=

Hpl,m [α1 + 1]f (z) Hpl,m [α1 ]f (z)

.

Then, clearly, q(z) is analytic in ∆. Also by a simple computation, we find from (2.3) that (2.4)

zq 0 (z) q(z)

=

z(Hpl,m [α1 + 1]f (z))0 Hpl,m [α1

+ 1]f (z)

−

z(Hpl,m [α1 ]f (z))0 Hpl,m [α1 ]f (z)

.

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By making use of the identity (2.5)

z(Hpl,m [α1 ]f (z))0 = α1 Hpl,m [α1 + 1]f (z) − (α1 − p)Hpl,m [α1 ]f (z),

we have from (2.4), that (2.6)

Hpl,m [α1 + 2]f (z) Hpl,m [α1 + 1]f (z)

=

1 α1 + 1

µ ¶ zq 0 (z) α1 q(z) + 1 + . q(z)

Since f ∈ A(p, n, α1 + 1; χ), we have from (2.6) that α1 q(z) +

zψ 0 (z) zq 0 (z) ≺ α1 ψ(z) + q(z) ψ(z)

and this can be written as (1.9), by defining ϑ(w) := α1 w and ϕ(w) :=

1 . w

Note that ϕ(w) 6= 0 and ϑ(w), ϕ(w) are analytic in C − {0}. Set (2.7)

Q(z) :=

zψ 0 (z) ψ(z)

(2.8)

h(z) :=

ϑ(ψ(z)) + Q(z) = α1 ψ(z) +

zψ 0 (z) . ψ(z)

By the hypothesis of Theorem 2.1, Q(z) is starlike and ½ 0 ¾ ½ ¾ zh (z) zψ 00 (z) zψ 0 (z) < = < α1 ψ(z) + 1 + 0 − > 0. Q(z) ψ (z) ψ(z) By an application of Lemma 1, we obtain that q(z) ≺ ψ(z) or Hpl,m [α1 + 1]f (z) Hpl,m [α1 ]f (z)

≺ ψ(z),

which shows that f ∈ A(p, n, α1 ; ψ). The other half of the Theorem 2.1 follows by a similar application of Lemma 2. ¤ Using Theorem 2.1, we obtain the following “sandwich result”: Corollary 1. Let ψi (z) be univalent in ∆, ψi (0) = 1 (i = 1, 2). Further assume that zψi0 (z)/ψi (z) is starlike univalent in ∆ and 0 (i = 1, 2). If f ∈ A(p, n, α1 + 1; χ1 , χ2 ) satisfies (2.2), then f ∈ A(p, n, α1 ; ψ1 , ψ2 ), where · ¸ zψi0 (z) 1 α1 ψi (z) + 1 + (i = 1, 2; α1 6= −1). χi (z) := α1 + 1 ψi (z)

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Theorem 2.2. Let ψ be univalent in ∆, ψ(0) = 1, and λ be a complex number. Assume that zψ 0 /(λ + p − α1 + α1 ψ) is starlike in ∆ and < {λ + p − α1 + α1 ψ(z)} > 0. Define the functions F and h by Z λ + p z λ−1 (2.9) F (z) := t f (t)dt, zλ 0 h(z) := ψ(z) +

zψ 0 (z) . λ + p − α1 + α1 ψ(z)

If f ∈ A(p, n, α1 ; h), then F ∈ A(p, n, α1 ; ψ). If f ∈ A(p, n, α1 ; h), (2.10) Hpl,m [α1 + 1]F (z) Hpl,m [α1 + 1]f (z) 0 6= ∈ H(1, 1) ∩ Q and is univalent in ∆, Hpl,m [α1 ]F (z) Hpl,m [α1 ]f (z) then the function F ∈ A(p, n, α1 ; ψ). Proof. From the definition of F (z) and (2.5), we obtain that (λ + p)Hpl,m [α1 ]f (z) = (2.11)

=

λHpl,m [α1 ]F (z) + z(Hpl,m [α1 ]F (z))0 α1 Hpl,m [α1 + 1]F (z) + (λ + p − α1 )Hpl,m [α1 ]F (z).

Define the function q(z) by (2.12)

q(z) :=

Hpl,m [α1 + 1]F (z) Hpl,m [α1 ]F (z)

.

Then, clearly, q(z) is analytic in ∆. Using (2.11) and (2.12), we have (2.13)

(λ + p)

Hpl,m [α1 ]f (z) Hpl,m [α1 ]F (z)

= λ + p − α1 + α1 q(z).

Upon logarithmic differentiation of (2.13) and using (2.5), and (2.12), we get (2.14)

Hpl,m [α1 + 1]f (z) Hpl,m [α1 ]f (z)

= q(z) +

zq 0 (z) . λ + p − α1 + α1 q(z)

Since f ∈ A(p, n, α1 ; h), we have, from (2.14), q(z) +

zψ 0 (z) zq 0 (z) ≺ ψ(z) + λ + p − α1 + α1 q(z) λ + p − α1 + α1 ψ(z)

and this can be written as (1.9), by defining ϑ(w) := w and ϕ(w) :=

1 . λ + p − α1 + α1 w

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Note that ϕ(w) 6= 0 and ϑ(w), ϕ(w) are analytic in C − { α1 −p−λ }. Set α1 Q(z) :=

zψ 0 (z) λ + p − α1 + α1 ψ(z)

h(z) :=

ϑ(ψ(z)) + Q(z) = ψ(z) +

zψ 0 (z) . λ + p − α1 + α1 ψ(z)

By the hypothesis of Theorem 2.2, Q(z) is starlike and ½
0. ψ (z) λ + p − α1 + α1 ψ(z)

By an application of Lemma 1, we obtain that q(z) ≺ ψ(z) or

Hpl,m [α1 + 1]F (z) Hpl,m [α1 ]F (z)

≺ ψ(z),

which shows that F ∈ A(p, n, α1 ; ψ). The second half of the Theorem 2.2 follows by a similar application of Lemma 2. ¤ Using Theorem 2.2, we have the following result: Corollary 2. Let ψi be univalent in ∆, ψi (0) = 1 (i = 1, 2) and λ be a complex number. Assume that zψi0 /(λ + p − α1 + α1 ψi ) is starlike in ∆ and < {λ + p − α1 + α1 ψi (z)} > 0 (i = 1, 2). If f ∈ A(p, n, α1 ; h1 , h2 ) satisfies (2.10), then the function F defined by (2.9) belongs to A(p, n, α1 ; ψ1 , ψ2 ) where hi (z) := ψi (z) +

zψi0 (z) (i = 1, 2). λ + p − α1 + α1 ψi (z)

Theorem 2.3. Let f (z) ∈ A(p, n) and α1 6= −1. Define F by Z z α1 (2.15) F (z) := α1 −p tα1 −p−1 f (t)dt. z 0 ³ ´ 1ϕ Then f ∈ A(p, n, α1 ; ϕ) if and only if F ∈ A p, n, α1 + 1; 1+α . Also f ∈ 1+α1 ´ ³ 1+α1 ϕ A(p, n, α1 ; ϕ) if and only if F ∈ A p, n, α1 + 1; 1+α1 . Proof. From (2.15), we have (2.16)

α1 f (z) = (α1 − p)F (z) + zF 0 (z).


105

By convoluting (2.16) with hp (α1 , · · · , αl ; β1 , · · · , βm ; z) and using the fact that z(f ∗ g)0 (z) = f (z) ∗ zg 0 (z), we obtain α1 Hpl,m [α1 ]f (z) = (α1 − p)Hpl,m [α1 ]F (z) + z(Hpl,m [α1 ]F (z))0 and by using (2.5), we get Hpl,m [α1 ]f (z) = Hpl,m [α1 + 1]F (z)

(2.17) and α1 Hpl,m [α1 + 1]f (z) =

z(Hpl,m [α1 ]f (z))0 + (α1 − p)Hpl,m [α1 ]f (z)

=

z(Hpl,m [α1 + 1]F (z))0 + (α1 − p)Hpl,m [α1 + 1]F (z)

=

(α1 + 1)Hpl,m [α1 + 2]F (z) − (α1 + 1 − p)Hpl,m [α1 + 1]F (z) + (α1 − p)Hpl,m [α1 + 1]F (z)

(2.18)

=

(α1 + 1)Hpl,m [α1 + 2]F (z) − Hpl,m [α1 + 1]F (z).

Therefore, from (2.17) and (2.18), we have # " Hpl,m [α1 + 2]F (z) Hpl,m [α1 + 1]f (z) 1 , = 1 + α1 α1 + 1 Hpl,m [α1 + 1]F (z) Hpl,m [α1 ]f (z) and the desired results follow at once.

¤

Using Theorem 2.3, we have Corollary 3. Let f (z) ∈ A(p, n)³and α1 6= −1. Then f ∈ A(p, ´ n, α1 ; ϕ1 , ϕ2 ) if and 1+α1 ϕ1 1+α1 ϕ2 only if F given by (2.15) is in A p, n, α1 + 1; 1+α1 , 1+α1 . 3. Results involving multiplier transformation By making use of Lemma 1, we prove the following generalization of Theorem 1.1: Theorem 3.1. Let ψ(z) be univalent in ∆, ψ(0) = 1, 0 and zψ 0 /ψ be starlike in ∆. Let χ(z) be defined by · ¸ 1 zψ 0 (z) χ(z) := (p + λ)ψ(z) + . p+λ ψ(z) If f ∈ A(p, n, r + 1, λ; χ), then f ∈ A(p, n, r, λ; ψ). If f ∈ A(p, n, r + 1, λ; χ), (3.1)

0 6=

Ip (r + 1, λ)f (z) Ip (r + 2, λ)f (z) ∈ H(1, 1) ∩ Q, is univalent in ∆, Ip (r, λ)f (z) Ip (r + 1, λ)f (z)

then f ∈ A(p, n, r, λ; ψ).

106


Proof. Define the function q(z) by (3.2)

q(z) :=

Ip (r + 1, λ)f (z) . Ip (r, λ)f (z)

Then, clearly, q(z) is analytic in ∆. Also by a simple computation, we find from (3.2) that zq 0 (z) q(z)

(3.3)

=

z(Ip (r + 1, λ)f (z))0 z(Ip (r, λ)f (z))0 − . Ip (r + 1, λ)f (z) Ip (r, λ)f (z)

By making use of the identity (p + λ)Ip (r + 1, λ)f (z) = z[Ip (r, λ)f (z)]0 + λIp (r, λ)f (z),

(3.4)

we have from (3.3), that Ip (r + 2, λ)f (z) 1 = Ip (r + 1, λ)f (z)) p+λ

(3.5)

µ ¶ zq 0 (z) (p + λ)q(z) + . q(z)

Since f ∈ A(p, n, r + 1, λ; χ) and in view of (3.5), we have (p + λ)q(z) +

zq 0 (z) zψ 0 (z) ≺ (p + λ)ψ(z) + . q(z) ψ(z)

The first result follows by an application of Lemma 1. Similarly the second result follows from Lemma 2. ¤ Using Theorem 3.1, we obtain the following “sandwich result”: Corollary 4. Let ψi (z) be univalent in ∆, ψi (0) = 1, 0 and zψi0 (z)/ψi (z) be starlike univalent in ∆ for i = 1, 2. Define · ¸ 1 zψi0 (z) χi (z) := (p + λ)ψi (z) + (i = 1, 2). p+λ ψi (z) If f ∈ A(p, n, r + 1, λ; χ1 , χ2 ) satisfies (3.1), then f ∈ A(p, n, r, λ; ψ1 , ψ2 ). Theorem 3.2. Let ψ be univalent in ∆, ψ(0) = 1, δ be a complex number, zψ 0 /(δ − λ + (p + λ)ψ) be starlike in ∆ and < (δ − λ + (p + λ)ψ(z)) > 0. Define the function F by Z δ + p z δ−1 (3.6) t f (t)dt, F (z) := zδ 0 zψ 0 (z) . h(z) := ψ(z) + δ − λ + (p + λ)ψ(z) If f ∈ A(p, n, r, λ; h), then F ∈ A(p, n, r, λ; ψ). If f ∈ A(p, n, r, λ; h), (3.7)

0 6=

Ip (r + 1, λ)f (z) Ip (r + 1, λ)F (z) ∈ H(1, 1) ∩ Q, is univalent in ∆, Ip (r, λ)F (z) Ip (r, λ)f (z)


107

then F ∈ A(p, n, r, λ; ψ). Proof. From the definition of F (z) and (3.8)

z(Ip (r, λ)F (z))0 = (p + λ)Ip (r + 1, λ)F (z) − λIp (r, λ)F (z),

we have (3.9) (δ + p)Ip (r, λ)f (z)

= δIp (r, λ)F (z) + z(Ip (r, λ)F (z))0 = (p + λ)Ip (r + 1, λ)F (z) + (δ − λ)Ip (r, λ)F (z).

Define the function q(z) by (3.10)

q(z) :=

Ip (r + 1, λ)F (z) . Ip (r, λ)F (z)

Then, clearly, q(z) is analytic in ∆. Using (3.9) and (3.10), we have (3.11)

(δ + p)

Ip (r, λ)f (z) = δ − λ + (p + λ)q(z). Ip (r, λ)F (z)

Upon logarithmic differentiation of (3.11) and using (3.4), (3.8) and (3.10), we get (3.12)

Ip (r + 1, λ)f (z) zq 0 (z) = q(z) + . Ip (r, λ)f (z) δ − λ + (p + λ)q(z)

For f ∈ A(p, n, r, λ, h), we have from (3.11), q(z) +

zq 0 (z) zψ 0 (z) ≺ ψ(z) + . δ − λ + (p + λ)q(z) δ − λ + (p + λ)ψ(z)

The first part of our result now follows by an application of Lemma 1. Similarly the second part follows from Lemma 2. ¤ Using Theorem 2.2, we have the following result: Corollary 5. Let ψi be univalent in ∆, ψi (0) = 1 and δ be a complex number. Assume that zψi0 /(δ−λ+(p+λ)ψi ) is starlike in ∆ and < {δ − λ + (p + λ)ψi (z)} > 0 for i = 1, 2. Define the functions hi by hi (z) := ψi (z) +

zψi0 (z) (i = 1, 2). δ − λ + (p + λ)ψi (z)

If f ∈ A(p, n, r, λ; h1 , h2 ), then F defined by (3.6) belongs to A(p, n, r, λ; ψ1 , ψ2 ). Theorem 3.3. Let f (z) ∈ A(p, n). Then f ∈ A(p, n, r, λ; ϕ) if and only if Z p + λ z λ−1 (3.13) F (z) := t f (t)dt ∈ A (p, n, r + 1, λ; ϕ) . zλ 0

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Also f ∈ A(p, n, r, λ; ϕ) if and only if F ∈ A (p, n, r + 1, λ; ϕ) . Proof. From (3.13), we have (p + λ)f (z) = λF (z) + zF 0 (z). ´r ³ P∞ z k and using the fact By convoluting (3.14) with φp (k, λ; z) := z p + k=p+1 k+λ p+λ (3.14)

that z(f ∗ g)0 (z) = f (z) ∗ zg 0 (z), we obtain (p + λ)Ip (r, λ)f (z) = λIp (r, λ)F (z) + z(Ip (r, λ)F (z))0 and by using (3.4), we get (3.15)

Ip (r, λ)f (z) = Ip (r + 1, λ)F (z)

and (3.16) (p + λ)Ip (r + 1, λ)f (z)

= z(Ip (r, λ)f (z))0 + λIp (r, λ)f (z) = z(Ip (r + 1, λ)F (z))0 + λIp (r + 1, λ)F (z) = (p + λ)Ip (r + 2, λ)F (z).

Therefore, from (3.15) and (3.16), we have Ip (r + 2, λ)F (z) Ip (r + 1, λ)f (z) = , Ip (r + 1, λ)F (z) Ip (r, λ)f (z) and the desired result follows at once.

¤

Using Theorem 3.3, we have Corollary 6. Let f (z) ∈ A(p, n). Then f ∈ A(p, n, r, λ; ϕ1 , ϕ2 ) if and only if F ∈ A (p, n, r + 1, λ; ϕ1 , ϕ2 ) . Acknowledgement. The authors S. Sivaprasad Kumar and H. C. Taneja are thankful to Prof. P. B. Sharma, Principal, Delhi College of Engineering for his support and encouragement. The research of V. Ravichandran is supported by a post-doctoral research fellowship from Universiti Sains Malaysia. The authors are thankful to the referee for his comments.

References [1] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135(1969), 429–446. [2] T. Bulboac˘ a, A class of superordination-preserving integral operators, Indag. Math. (N.S.), 13(3)(2002), 301–311.


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[3] T. Bulboac˘ a, Classes of first-order differential superordinations, Demonstratio Math., 35(2)(2002), 287–292. [4] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15(4)(1984), 737–745. [5] N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling, 37(12)(2003), 39–49. [6] N. E. Cho and T. H. Kim, Multiplier transformations and strongly close-to-convex functions, Bull. Korean Math. Soc., 40(3)(2003), 399–410. [7] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct., 14(1)(2003), 7–18. [8] Ju. E. Hohlov, Operators and operations on the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Mat. 1978, 10(197)(1978), 83–89. [9] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16(1965), 755–758. [10] J.-L. Liu and S. Owa, On a class of multivalent functions involving certain linear operator, Indian J. Pure Appl. Math., 33(11)(2002), 1713–1722. [11] A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 17(1966), 352–357. [12] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, No. 225, Marcel Dekker, New York, (2000). [13] S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Var. Theory Appl., 48(10)(2003), 815–826. [14] S. Owa, On the distortion theorems I, Kyungpook Math. J., 18(1)(1978), 53–59. [15] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39(5)(1987), 1057–1077. [16] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109–115. ¸ . S˘ al˘ agean, Subclasses of univalent functions, in Complex analysis—fifth [17] G. S Romanian-Finnish seminar, Part 1 (Bucharest, 1981), 362–372, Lecture Notes in Math., 1013, Springer, Berlin. [18] H. M. Srivastava, Some families of fractional derivative and other linear operators associated with analytic, univalent, and multivalent functions, in Analysis and its applications (Chennai, 2000), 209–243, Allied Publ., New Delhi. [19] B. A. Uralegaddi and C. Somanatha, Certain classes of univalent functions, in Current topics in analytic function theory, 371–374, World Sci. Publishing, River Edge, NJ, (1992).

Classes of Multivalent Functions Defined by Dziok ...

Classes of Multivalent Functions Defined by Dziok ...

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