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Some inclusion properties of a certain family of integral operators
J. Math. Anal. Appl. 276 (2002) 432–445 www.elsevier.com/locate/jmaa
Some inclusion properties of a certain family of integral operators Jae Ho Choi,a,1 Megumi Saigo,a and H.M. Srivastava b,∗ a Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan b Department of Mathematics and Statistics, University of Victoria, Victoria,
British Columbia V8W 3P4, Canada Received 6 June 2002 Submitted by Steven G. Krantz
Abstract The authors introduce several new subclasses of analytic functions, which are defined by means of a general integral operator Iλ,µ , and investigate various inclusion properties of these subclasses. Many interesting applications involving these and other families of integral operators are also considered. 2002 Elsevier Science (USA). All rights reserved.
Keywords: Analytic functions; Univalent functions; Starlike functions; Convex functions; Close-to-convex functions; Subordination; Hadamard product (or convolution); Integral operators; Fractional derivatives
* Corresponding author.
E-mail addresses: [email protected] (J.H. Choi), [email protected] (M. Saigo), [email protected] (H.M. Srivastava). 1 Present adrress: Department of Mathematics Education, Daegu National University of Education, 1797-6 Daemyung 2-Dong, Namgu, Daegu 705-715, Republic of Korea. 0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 2 4 7 X ( 0 2 ) 0 0 5 0 0 - 0
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1. Introduction and definitions Let A denote the class of functions f (z) normalized by f (z) = z +
∞
ak z k ,
(1.1)
k=2
which are analytic in the open unit disk U = z: z ∈ C and |z| < 1 . Also let S, C, S ∗ (α) and K(α) denote, respectively, the subclasses of A consisting of functions which are univalent, close-to-convex, starlike of order α, and convex of order α in U. In particular, the classes S ∗ (0) = S ∗
and K(0) = K
are the familiar classes of starlike and convex functions in U, respectively. Given two functions f and g, which are analytic in U with f (0) = g(0), the function f is said to be subordinate to g in U if there exists a function w, analytic in U, such that w(0) = 0, w(z) < 1 (z ∈ U), and f (z) = g w(z) (z ∈ U). We denote this subordination by f (z) ≺ g(z)
in U.
We also observe that f (z) ≺ g(z)
in U
if and only if f (0) = g(0)
and f (U) ⊂ g(U)
whenever g is univalent in U. Let M be the class of analytic functions φ(z) in U normalized by φ(0) = 1, and let N be the subclass of M consisting of those functions φ which are univalent in U and for which φ(U) is convex and {φ(z)} > 0 (z ∈ U). Making use of the principle of subordination between analytic functions, Ma and Minda [10] and Kim et al. [6] investigated the subclasses S∗ (φ), K(φ), and C(φ, ψ) of the class A for φ, ψ ∈ N , which are defined by zf (z) ∗ ≺ φ(z) in U , S (φ) := f : f ∈ A and f (z) zf
(z) K(φ) := f : f ∈ A and 1 + ≺ φ(z) in U , f (z)
434
and
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f (z) ≺ ψ(z) in U . C(φ, ψ) := f : f ∈ A and ∃h ∈ K(φ) s.t. h (z)
Obviously, for special choices for the functions φ and ψ involved in these definitions, we have the following relationships:
1+z 1+z 1+z 1+z = S ∗, = K, C = C, , K S∗ 1−z 1−z 1−z 1−z and S
∗
1 + (1 − 2α)z 1−z
= S ∗ (α)
(0 α < 1).
Furthermore, for the function classes S ∗ [A, B] and K[A, B] investigated (among others) by Janowski [5] and Goel and Mehrok [4], it is easily seen that
1 + Az S∗ = S ∗ [A, B] (−1 B < A 1) 1 + Bz and
1 + Az K = K[A, B] 1 + Bz
(−1 B < A 1).
Since f (z) ∈ K(φ)
⇔
zf (z) ∈ S∗ (φ),
⇔
∃g ∈ S∗ (φ) s.t.
we also have f ∈ C(φ, ψ)
zf (z) ≺ ψ(z) in U. g(z)
(1.2)
For the functions f and g given by f (z) =
∞
and g(z) =
ak z k
∞
k=0
bk z k ,
k=0
the Hadamard product (or convolution) f ∗ g is defined, as usual, by (f ∗ g)(z) :=
∞
ak bk zk = (g ∗ f )(z).
k=0
Almost two decades ago, by making use of the Hadamard product (or convolution), Carlson and Shaffer [2] defined a linear operator L(a, c) : A → A by L(a, c)f (z) := ϕ(a, c; z) ∗ f (z)
(f ∈ A),
(1.3)
where ϕ(a, c; z) :=
∞ (a)k k=0
(c)k
zk+1
z ∈ U; c ∈ / Z− 0 := {0, −1, −2, . . .}
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and (λ)k is the Pochhammer symbol defined, in terms of the Gamma function, by Γ (λ + k) 1 (k = 0; λ = 0), = (λ)k := λ(λ + 1) . . . (λ + k − 1) k ∈ N := {1, 2, 3, . . .} . Γ (λ) The Carlson–Shaffer operator L(a, c) maps A onto itself and L(c, a) is the inverse of L(a, c), provided that a ∈ / Z− 0 (see also [15] and [18]). Moreover, it is known that z L(a, c)f (z) = aL(a + 1, c)f (z) − (a − 1)L(a, c)f (z)
(a > 0).
(1.4)
Earlier in 1975, Ruscheweyh [16] introduced another linear operator Dλ : A → A defined by the Hadamard product (or convolution) as follows: z Dλ f (z) := ∗ f (z) (λ > −1; z ∈ U), (1 − z)λ+1 which implies that Dn f (z) =
z(zn−1 f (z))(n) n!
n ∈ N0 := N ∪ {0} .
Clearly, we have D0 f (z) = f (z),
D1 f (z) = zf (z),
and Dλ f (z) = L(λ + 1, 1)f (z) (f ∈ A). Remark 1. The Carlson–Shaffer operator L(a, c) not only contains, as its special cases, the Ruscheweyh operator Dλ defined above, but also the familiar fractional derivative operator Dzλ (see, for details, [14] and [17]). Motivated essentially by the Ruscheweyh operator Dn (n ∈ N0 ), by setting z fn (z) = (n ∈ N0 ) (1 − z)n+1 (†)
and defining fn (z) in terms of the Hadamard product (or convolution): z (z ∈ U), fn (z) ∗ fn(†) (z) = (1 − z)2 Noor [12] considered an integral operator In : A → A defined by (see also [8], [9], and [13]) (†) z (†) ∗ f (z) In f (z) = fn (z) ∗ f (z) = (1 − z)n+1 (f ∈ A; n ∈ N0 ; z ∈ U),
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so that, obviously, I0 f (z) = zf (z)
and I1 f (z) = f (z).
Furthermore, it is easily observed that In f (z) = L(2, n + 1)f (z), which exhibits the (hitherto unnoticed) fact that the integral operator In is a very specialized case of the Carlson–Shaffer operator L(a, c) when (cf. Eq. (1.3)) a = 2 and c = n + 1
(n ∈ N0 ).
Remark 2. Noor and Noor [13] investigated several properties of a certain subclass of close-to-convex functions, which was defined by means of the integral operator In , but many of their notations and assertions seem to be in error. By analogy with the general Ruscheweyh operator Dλ (λ > −1), we now set z (λ > −1) fλ (z) = (1 − z)λ+1 and define fλ,µ by means of the Hadamard product (or convolution): z (fλ ∗ fλ,µ )(z) = (z ∈ U). (1.5) (1 − z)µ Then a natural generalization of In is provided by the integral operator Iλ,µ : A → A, which we define here by Iλ,µ f (z) = (fλ,µ ∗ f )(z) (f ∈ A; λ > −1; µ > 0).
(1.6)
In particular, by taking λ = n (n ∈ N0 )
and µ = 2
in (1.6), we have the following relationship In,2 f (z) = In f (z)
(n ∈ N0 )
with the integral operator In (n ∈ N0 ) considered by Noor [12] (see also [8], [9], and [13]). Also, in view of the definitions (1.3) and (1.6), we have Iλ,µ f (z) = L(µ, λ + 1)f (z) (λ > −1; µ > 0).
(1.7)
By applying (1.4), (1.5), (1.7), and certain well-known identities involving Dλ f , we obtain z Iλ+1,µ f (z) = (λ + 1)Iλ,µ f (z) − λIλ+1,µ f (z) (f ∈ A; λ > −1; µ > 0) and
(1.8)
z Iλ,µ f (z) = µIλ,µ+1 f (z) − (µ − 1)Iλ,µ f (z) (f ∈ A; λ > −1; µ > 0).
(1.9)
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Next, by using the general operator Iλ,µ , we introduce the following (presumably new) classes of analytic functions for φ, ψ ∈ N , λ > −1, and µ > 0: S∗λ,µ (φ) := f : f ∈ A and Iλ,µ f (z) ∈ S∗ (φ) , Kλ,µ (φ) := f : f ∈ A and Iλ,µ f (z) ∈ K(φ) , and Cλ,µ (φ, ψ) = f : f ∈ A and Iλ,µ f (z) ∈ C(φ, ψ) . We also note that f (z) ∈ Kλ,µ (φ)
⇔
zf (z) ∈ S∗λ,µ (φ).
(1.10)
In particular, we set
1+z ∗ Sn,2 = Sn∗ , 1−z
1 + Az ∗ ∗ Sλ,µ [A, B] (−1 B < A 1), = Sλ,µ 1 + Bz and
Kλ,µ
1 + Az 1 + Bz
= Kλ,µ [A, B]
(−1 B < A 1).
In this paper, we investigate several inclusion properties of the classes S∗λ,µ (φ), Kλ,µ (φ), and Cλ,µ (φ, ψ) associated with the general integral operator Iλ,µ . Some applications involving these and other families of integral operators are also considered.
2. Inclusion properties involving Iλ,µ The following results will be required in our investigation. Lemma 1 (Eenigenburg et al. [3]). Let h be convex univalent in U with h(0) = 1 and βh(z) + γ > 0 (β, γ ∈ C). If p(z) is analytic in U with p(0) = 1, then p(z) +
zp (z) ≺ h(z) βp(z) + γ
implies that p(z) ≺ h(z) in U.
in U
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Lemma 2 (Miller and Mocanu [11]). Let h be convex in U with h(0) = 1. Suppose also that Q(z) is analytic in U with {Q(z)} 0 (z ∈ U). If p(z) is analytic in U with p(0) = 1, then p(z) + Q(z)zp (z) ≺ h(z)
in U
implies that p(z) ≺ h(z) in U. We begin by proving our first inclusion relation given by Theorem 1. Let λ 0 and µ 1. Then S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) ⊂ S∗λ+1,µ (φ)
(φ ∈ N ).
Proof. First of all, we show that S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) Let
f (z) ∈ S∗λ,µ+1 (φ)
(φ ∈ N ; λ 0; µ 1).
and set
z(Iλ,µ f (z)) = p(z), Iλ,µ f (z)
(2.1)
where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. Applying (1.9) and (2.1), we obtain µ
Iλ,µ+1 f (z) = p(z) + µ − 1. Iλ,µ f (z)
(2.2)
By using the logarithmic differentiation on both side of (2.2), we have zp (z) z(Iλ,µ+1 f (z)) z(Iλ,µ f (z)) = + Iλ,µ+1 f (z) Iλ,µ f (z) p(z) + µ − 1
zp (z) . = p(z) + p(z) + µ − 1 Since µ 1, φ(z) ∈ N , and f (z) ∈ S∗λ,µ+1 (φ), from (2.3) we see that φ(z) + µ − 1 > 0 (z ∈ U) and p(z) +
zp (z) ≺ φ(z) p(z) + µ − 1
in U.
Thus, by using Lemma 1 and (2.1), we observe that p(z) ≺ φ(z)
in U,
so that f (z) ∈ S∗λ,µ (φ).
(2.3)
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This implies that S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ). To prove the second part, let f (z) ∈ S∗λ,µ (φ) (λ 0; µ 1) and put z(Iλ+1,µ f (z)) = q(z), Iλ+1,µ f (z) where q(z) = 1 + d1 z + d2 z2 + · · · is analytic in U and q(z) = 0 for all z ∈ U. Then, by using arguments similar to those detailed above with (1.8), it follows that q(z) ≺ φ(z)
in U,
which implies that f (z) ∈ S∗λ+1,µ (φ). Hence we conclude that S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) ⊂ S∗λ+1,µ (φ), which completes the proof of Theorem 1.
✷
Remark 3. By putting λ=n
(n ∈ N0 ),
µ = 2,
1+z (z ∈ U) 1−z (n ∈ N0 ), which was asserted earlier by
and φ(z) =
∗ in Theorem 1, we obtain Sn∗ ⊂ Sn+1 Noor [12].
Theorem 2. Let λ 0 and µ 1. Then Kλ,µ+1 (φ) ⊂ Kλ,µ (φ) ⊂ Kλ+1,µ (φ)
(φ ∈ N ).
Proof. Applying (1.10) and Theorem 1, we observe that f (z) ∈ Kλ,µ+1 (φ) ⇔ Iλ,µ+1 f (z) ∈ K(φ) ⇔ z Iλ,µ+1 f (z) ∈ S∗ (φ) ⇔ Iλ,µ+1 zf (z) ∈ S∗ (φ) ⇔ zf (z) ∈ S∗λ,µ+1 (φ) ⇒ zf (z) ∈ S∗λ,µ (φ) ⇔ Iλ,µ zf (z) ∈ S∗ (φ) ⇔ z Iλ,µ f (z) ∈ S∗ (φ) ⇔ Iλ,µ f (z) ∈ K(φ) ⇔ f (z) ∈ Kλ,µ (φ) and f (z) ∈ Kλ,µ (φ) ⇔ zf (z) ∈ S∗λ,µ (φ)
⇒ zf (z) ∈ S∗λ+1,µ (φ) ⇔ z Iλ+1,µ f (z) ∈ S∗ (φ) ⇔ Iλ+1,µ f (z) ∈ K(φ) ⇔ f (z) ∈ Kλ+1,µ (φ),
which evidently proves Theorem 2.
✷
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Taking 1 + Az (−1 B < A 1; z ∈ U) 1 + Bz in Theorems 1 and 2, we have φ(z) =
Corollary 1. Let λ 0, µ 1, and −1 B < A 1. Then ∗ ∗ ∗ [A, B] ⊂ Sλ,µ [A, B] ⊂ Sλ+1,µ [A, B] Sλ,µ+1
and Kλ,µ+1 [A, B] ⊂ Kλ,µ [A, B] ⊂ Kλ+1,µ [A, B]. Next, by using Lemma 2, we obtain the following inclusion theorem for the class Cλ,µ (φ, ψ). Theorem 3. Let λ 0 and µ 1. Then Cλ,µ+1 (φ, ψ) ⊂ Cλ,µ (φ, ψ) ⊂ Cλ+1,µ (φ, ψ)
(φ, ψ ∈ N ).
Proof. We begin by proving that Cλ,µ+1 (φ, ψ) ⊂ Cλ,µ (φ, ψ)
(λ 0; µ 1; φ, ψ ∈ N ).
Let f (z) ∈ Cλ,µ+1 (φ, ψ). Then, in view of (1.2), there exists a function q(z) ∈ S∗ (φ) such that z(Iλ,µ+1 f (z)) ≺ ψ(z) q(z)
in U.
Choose the function g(z) such that Iλ,µ+1 g(z) = q(z). Then g(z) ∈ S∗λ,µ+1 (φ) and z(Iλ,µ+1 f (z)) ≺ ψ(z) in U. (2.4) Iλ,µ+1 g(z) Now let z(Iλ,µ f (z)) = p(z), Iλ,µ g(z)
(2.5)
where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. Thus, by using (1.9), we obtain z(Iλ,µ+1 f (z)) Iλ,µ+1 (zf (z)) = Iλ,µ+1 g(z) Iλ,µ+1 g(z) z(Iλ,µ (zf (z))) + (µ − 1)Iλ,µ (zf (z)) = z(Iλ,µ g(z)) + (µ − 1)Iλ,µ g(z)
J.H. Choi et al. / J. Math. Anal. Appl. 276 (2002) 432–445
=
z(Iλ,µ (zf (z))) I (zf (z)) + (µ − 1) λ,µ Iλ,µ g(z) Iλ,µ g(z) . z(Iλ,µ g(z)) Iλ,µ g(z) + µ − 1
441
(2.6)
Since g(z) ∈ S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) (φ ∈ N ), by Theorem 1, we set z(Iλ,µ g(z)) = G(z), Iλ,µ g(z) where G(z) ≺ φ(z) in U for φ ∈ N . Then, by virtue of (2.5) and (2.6), we observe that Iλ,µ zf (z) = p(z)Iλ,µ g(z) (2.7) and z(Iλ,µ+1 f (z)) = Iλ,µ+1 g(z)
z(Iλ,µ (zf (z))) Iλ,µ g(z)
+ (µ − 1)p(z)
G(z) + µ − 1
.
(2.8)
Upon differentiating both sides of (2.7), we have z(Iλ,µ (zf (z))) = G(z)p(z) + zp (z). Iλ,µ g(z)
(2.9)
Making use of (2.4), (2.8), and (2.9), we get z(Iλ,µ+1 f (z)) zp (z) = p(z) + ≺ ψ(z) Iλ,µ+1 g(z) G(z) + µ − 1
(z ∈ U).
(2.10)
Since µ 1 and G(z) ≺ φ(z) in U, G(z) + µ − 1 > 0 (z ∈ U). Hence, by taking Q(z) =
1 G(z) + µ − 1
in (2.10), and applying Lemma 2, we can show that p(z) ≺ ψ(z)
in U,
so that f (z) ∈ Cλ,µ (φ, ψ)
(φ, ψ ∈ N ).
For the second part, by using arguments similar to those detailed above with (1.8), we obtain Cλ,µ (φ, ψ) ⊂ Cλ+1,µ (φ, ψ)
(φ, ψ ∈ N ).
The proof of Theorem 3 is thus completed. ✷
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Remark 4. In its special case when 1+z (z ∈ U), 1−z Theorem 3 would reduce immediately to a known result due to Noor and Noor [13, p. 343, Theorem 3.1]. µ = 2,
λ = n (n ∈ N0 ),
and φ(z) = ψ(z) =
3. Inclusion properties involving Jσ In this section, we consider the generalized Bernardi–Libera–Livingston integral operator Jσ (σ > −1) defined by (cf. [1], [7], and [18]) σ +1 Jσ (f )(z) := σ z
z
t σ −1 f (t) dt
(f ∈ A; σ > −1).
(3.1)
0
We first prove Theorem 4. Let σ 0, λ > −1, and µ > 0. If f (z) ∈ S∗λ,µ (φ) (φ ∈ N ), then Jσ (f )(z) ∈ S∗λ,µ (φ)
(φ ∈ N ).
Proof. Let f (z) ∈ S∗λ,µ (φ) for φ ∈ N , and set z(Iλ,µ Jσ (f )(z)) = p(z), Iλ,µ Jσ (f )(z)
(3.2)
where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. From (3.1) we obtain z Iλ,µ Jσ (f )(z) = (σ + 1)Iλ,µ f (z) − σ Iλ,µ Jσ (f )(z) (z ∈ U). (3.3) By applying (3.2) and (3.3), we have (σ + 1)
Iλ,µ f (z) = p(z) + σ. Iλ,µ Jσ (f )(z)
(3.4)
Making use of the logarithmic differentiation on both sides in (3.4), we get z(Iλ,µ f (z)) zp (z) = p(z) + . Iλ,µ f (z) p(z) + σ
(3.5)
Since σ 0, φ(z) ∈ N , and f (z) ∈ S∗λ,µ (φ), from (3.5) we obtain φ(z) + σ > 0
and p(z) +
zp (z) ≺ φ(z) p(z) + σ
Hence, by virtue of Lemma 1, we conclude that p(z) ≺ φ(z)
in U,
(z ∈ U).
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which implies that Jσ (f )(z) ∈ S∗λ,µ (φ)
(φ ∈ N ).
✷
Next we derive an inclusion property involving Jσ , which is given by Theorem 5. Let σ 0, λ > −1, and µ > 0. If f (z) ∈ Kλ,µ (φ) (φ ∈ N ), then Jσ (f )(z) ∈ Kλ,µ (φ)
(φ ∈ N ).
Proof. By applying Theorem 4, it follows that f (z) ∈ Kλ,µ (φ) ⇔ zf (z) ∈ S∗λ,µ (φ) ⇒ Jσ zf (z) ∈ S∗λ,µ (φ) ⇔ z Jσ (f )(z) ∈ S∗λ,µ (φ) ⇔ Jσ (f )(z) ∈ Kλ,µ (φ) which proves Theorem 5.
(φ ∈ N ),
✷
Finally, we prove Theorem 6. Let σ 0, λ > −1, and µ > 0. If f (z) ∈ Cλ,µ (φ, ψ) (φ, ψ ∈ N ), then Jσ (f )(z) ∈ Cλ,µ (φ, ψ)
(φ, ψ ∈ N ).
Proof. Let f (z) ∈ Cλ,µ (φ, ψ) for φ, ψ ∈ N . Then, in view of (1.2), there exists a function g(z) ∈ S∗λ,µ (φ) such that z(Iλ,µ f (z)) ≺ ψ(z) Iλ,µ g(z)
in U.
(3.6)
Thus we set z(Iλ,µ Jσ (f )(z)) = p(z), Iλ,µ Jσ (g)(z) where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. Applying (3.3), we get z(Iλ,µ f (z)) Iλ,µ (zf (z)) z(Iλ,µ Jσ (zf (z))) + σ Iλ,µ Jσ (zf (z)) = = Iλ,µ g(z) Iλ,µ g(z) z(Iλ,µ Jσ (g)(z)) + σ Iλ,µ Jσ (g)(z) =
z(Iλ,µ Jσ (zf (z))) Iλ,µ Jσ (zf (z)) Iλ,µ Jσ (g)(z) + σ Iλ,µ Jσ (g)(z) z(Iλ,µ Jσ (g)(z)) Iλ,µ Jσ (g)(z) + σ
.
(3.7)
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Since g(z) ∈ S∗λ,µ (φ) (φ ∈ N ), by virtue of Theorem 4, we have Jσ (g)(z) ∈ S∗λ,µ (φ). Let us now put z(Iλ,µ Jσ (g)(z)) = H (z), Iλ,µ Jσ (g)(z) where H (z) ≺ φ(z) in U for φ ∈ N . Then, by using the same techniques as in the proof of Theorem 3, we conclude from (3.6) and (3.7) that z(Iλ,µ f (z)) zp (z) = p(z) + ≺ ψ(z) Iλ,µ g(z) H (z) + σ
in U.
(3.8)
Hence, upon setting Q(z) =
1 H (z) + σ
(z ∈ U)
in (3.8), if we apply Lemma 2, we obtain p(z) ≺ ψ(z)
in U,
which yields Jσ (f )(z) ∈ Cλ,µ (φ, ψ)
(φ, ψ ∈ N ).
The proof of Theorem 6 is evidently completed. ✷ Remark 5. By comparing the definitions (1.3) and (3.1) rather closely, and applying (1.7), it is not difficult to deduce the following relationship: Jσ (f )(z) = L(σ + 1, σ + 2)f (z) = Iσ +1,σ +1 f (z)
(f ∈ A; σ > −1)
(3.9)
among the Carlson–Shaffer linear operator L(a, c), the general integral operator Iλ,µ , and the generalized Bernardi–Libera–Livingston integral operator Jσ (σ > −1). This relationship (3.9) can alternatively be invoked in order to derive the results of this section from those of the preceding section. The details involved in these alternative derivations are being omitted here.
Acknowledgments The present investigation was initiated during the third-named author’s visit to Fukuoka University in April 2002. This work was supported, in part, by the Science Research Promotion Fund from the Japan Private School Promotion Foundation and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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References [1] S.D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969) 429–446. [2] B.C. Carlson, D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984) 737–745. [3] P. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot–Bouquet differential subordination, in: General Inequalities 3, in: International Series of Numerical Mathematics, Vol. 64, Birkhäuser, Basel, 1983, pp. 339–348. [4] R.M. Goel, B.S. Mehrok, On the coefficients of a subclass of starlike functions, Indian J. Pure Appl. Math. 12 (1981) 634–647. [5] W. Janowski, Some extremal problems for certain families of analytic functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973) 17–25. [6] Y.C. Kim, J.H. Choi, T. Sugawa, Coefficient bounds and convolution properties for certain classes of close-to-convex functions, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000) 95–98. [7] R.J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965) 755– 758. [8] J.-L. Liu, The Noor integral and strongly starlike functions, J. Math. Anal. Appl. 261 (2001) 441–447. [9] J.-L. Liu, K.I. Noor, Some properties of Noor integral operator, J. Natur. Geom. 21 (2002) 81–90. [10] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Z. Li, F. Ren, L. Yang, S. Zhang (Eds.), Proceedings of the Conference on Complex Analysis, International Press, Cambridge, MA, 1992, pp. 157–169. [11] S.S. Miller, P.T. Mocanu, Differential subordinations and inequalities in the complex plane, J. Differential Equations 67 (1987) 199–211. [12] K.I. Noor, On new classes of integral operators, J. Natur. Geom. 16 (1999) 71–80. [13] K.I. Noor, M.A. Noor, On integral operators, J. Math. Anal. Appl. 238 (1999) 341–352. [14] S. Owa, On the distortion theorems. I, Kyungpook Math. J. 18 (1978) 53–59. [15] S. Owa, H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987) 1057–1077. [16] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975) 109–115. [17] H.M. Srivastava, S. Owa (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press/Wiley, New York, 1989. [18] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific, Singapore, 1992.
Some inclusion properties of a certain family of integral operators Jae Ho Choi,a,1 Megumi Saigo,a and H.M. Srivastava b,∗ a Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan b Department of Mathematics and Statistics, University of Victoria, Victoria,
British Columbia V8W 3P4, Canada Received 6 June 2002 Submitted by Steven G. Krantz
Abstract The authors introduce several new subclasses of analytic functions, which are defined by means of a general integral operator Iλ,µ , and investigate various inclusion properties of these subclasses. Many interesting applications involving these and other families of integral operators are also considered. 2002 Elsevier Science (USA). All rights reserved.
Keywords: Analytic functions; Univalent functions; Starlike functions; Convex functions; Close-to-convex functions; Subordination; Hadamard product (or convolution); Integral operators; Fractional derivatives
* Corresponding author.
E-mail addresses: [email protected] (J.H. Choi), [email protected] (M. Saigo), [email protected] (H.M. Srivastava). 1 Present adrress: Department of Mathematics Education, Daegu National University of Education, 1797-6 Daemyung 2-Dong, Namgu, Daegu 705-715, Republic of Korea. 0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 2 4 7 X ( 0 2 ) 0 0 5 0 0 - 0
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1. Introduction and definitions Let A denote the class of functions f (z) normalized by f (z) = z +
∞
ak z k ,
(1.1)
k=2
which are analytic in the open unit disk U = z: z ∈ C and |z| < 1 . Also let S, C, S ∗ (α) and K(α) denote, respectively, the subclasses of A consisting of functions which are univalent, close-to-convex, starlike of order α, and convex of order α in U. In particular, the classes S ∗ (0) = S ∗
and K(0) = K
are the familiar classes of starlike and convex functions in U, respectively. Given two functions f and g, which are analytic in U with f (0) = g(0), the function f is said to be subordinate to g in U if there exists a function w, analytic in U, such that w(0) = 0, w(z) < 1 (z ∈ U), and f (z) = g w(z) (z ∈ U). We denote this subordination by f (z) ≺ g(z)
in U.
We also observe that f (z) ≺ g(z)
in U
if and only if f (0) = g(0)
and f (U) ⊂ g(U)
whenever g is univalent in U. Let M be the class of analytic functions φ(z) in U normalized by φ(0) = 1, and let N be the subclass of M consisting of those functions φ which are univalent in U and for which φ(U) is convex and {φ(z)} > 0 (z ∈ U). Making use of the principle of subordination between analytic functions, Ma and Minda [10] and Kim et al. [6] investigated the subclasses S∗ (φ), K(φ), and C(φ, ψ) of the class A for φ, ψ ∈ N , which are defined by zf (z) ∗ ≺ φ(z) in U , S (φ) := f : f ∈ A and f (z) zf
(z) K(φ) := f : f ∈ A and 1 + ≺ φ(z) in U , f (z)
434
and
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f (z) ≺ ψ(z) in U . C(φ, ψ) := f : f ∈ A and ∃h ∈ K(φ) s.t. h (z)
Obviously, for special choices for the functions φ and ψ involved in these definitions, we have the following relationships:
1+z 1+z 1+z 1+z = S ∗, = K, C = C, , K S∗ 1−z 1−z 1−z 1−z and S
∗
1 + (1 − 2α)z 1−z
= S ∗ (α)
(0 α < 1).
Furthermore, for the function classes S ∗ [A, B] and K[A, B] investigated (among others) by Janowski [5] and Goel and Mehrok [4], it is easily seen that
1 + Az S∗ = S ∗ [A, B] (−1 B < A 1) 1 + Bz and
1 + Az K = K[A, B] 1 + Bz
(−1 B < A 1).
Since f (z) ∈ K(φ)
⇔
zf (z) ∈ S∗ (φ),
⇔
∃g ∈ S∗ (φ) s.t.
we also have f ∈ C(φ, ψ)
zf (z) ≺ ψ(z) in U. g(z)
(1.2)
For the functions f and g given by f (z) =
∞
and g(z) =
ak z k
∞
k=0
bk z k ,
k=0
the Hadamard product (or convolution) f ∗ g is defined, as usual, by (f ∗ g)(z) :=
∞
ak bk zk = (g ∗ f )(z).
k=0
Almost two decades ago, by making use of the Hadamard product (or convolution), Carlson and Shaffer [2] defined a linear operator L(a, c) : A → A by L(a, c)f (z) := ϕ(a, c; z) ∗ f (z)
(f ∈ A),
(1.3)
where ϕ(a, c; z) :=
∞ (a)k k=0
(c)k
zk+1
z ∈ U; c ∈ / Z− 0 := {0, −1, −2, . . .}
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and (λ)k is the Pochhammer symbol defined, in terms of the Gamma function, by Γ (λ + k) 1 (k = 0; λ = 0), = (λ)k := λ(λ + 1) . . . (λ + k − 1) k ∈ N := {1, 2, 3, . . .} . Γ (λ) The Carlson–Shaffer operator L(a, c) maps A onto itself and L(c, a) is the inverse of L(a, c), provided that a ∈ / Z− 0 (see also [15] and [18]). Moreover, it is known that z L(a, c)f (z) = aL(a + 1, c)f (z) − (a − 1)L(a, c)f (z)
(a > 0).
(1.4)
Earlier in 1975, Ruscheweyh [16] introduced another linear operator Dλ : A → A defined by the Hadamard product (or convolution) as follows: z Dλ f (z) := ∗ f (z) (λ > −1; z ∈ U), (1 − z)λ+1 which implies that Dn f (z) =
z(zn−1 f (z))(n) n!
n ∈ N0 := N ∪ {0} .
Clearly, we have D0 f (z) = f (z),
D1 f (z) = zf (z),
and Dλ f (z) = L(λ + 1, 1)f (z) (f ∈ A). Remark 1. The Carlson–Shaffer operator L(a, c) not only contains, as its special cases, the Ruscheweyh operator Dλ defined above, but also the familiar fractional derivative operator Dzλ (see, for details, [14] and [17]). Motivated essentially by the Ruscheweyh operator Dn (n ∈ N0 ), by setting z fn (z) = (n ∈ N0 ) (1 − z)n+1 (†)
and defining fn (z) in terms of the Hadamard product (or convolution): z (z ∈ U), fn (z) ∗ fn(†) (z) = (1 − z)2 Noor [12] considered an integral operator In : A → A defined by (see also [8], [9], and [13]) (†) z (†) ∗ f (z) In f (z) = fn (z) ∗ f (z) = (1 − z)n+1 (f ∈ A; n ∈ N0 ; z ∈ U),
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so that, obviously, I0 f (z) = zf (z)
and I1 f (z) = f (z).
Furthermore, it is easily observed that In f (z) = L(2, n + 1)f (z), which exhibits the (hitherto unnoticed) fact that the integral operator In is a very specialized case of the Carlson–Shaffer operator L(a, c) when (cf. Eq. (1.3)) a = 2 and c = n + 1
(n ∈ N0 ).
Remark 2. Noor and Noor [13] investigated several properties of a certain subclass of close-to-convex functions, which was defined by means of the integral operator In , but many of their notations and assertions seem to be in error. By analogy with the general Ruscheweyh operator Dλ (λ > −1), we now set z (λ > −1) fλ (z) = (1 − z)λ+1 and define fλ,µ by means of the Hadamard product (or convolution): z (fλ ∗ fλ,µ )(z) = (z ∈ U). (1.5) (1 − z)µ Then a natural generalization of In is provided by the integral operator Iλ,µ : A → A, which we define here by Iλ,µ f (z) = (fλ,µ ∗ f )(z) (f ∈ A; λ > −1; µ > 0).
(1.6)
In particular, by taking λ = n (n ∈ N0 )
and µ = 2
in (1.6), we have the following relationship In,2 f (z) = In f (z)
(n ∈ N0 )
with the integral operator In (n ∈ N0 ) considered by Noor [12] (see also [8], [9], and [13]). Also, in view of the definitions (1.3) and (1.6), we have Iλ,µ f (z) = L(µ, λ + 1)f (z) (λ > −1; µ > 0).
(1.7)
By applying (1.4), (1.5), (1.7), and certain well-known identities involving Dλ f , we obtain z Iλ+1,µ f (z) = (λ + 1)Iλ,µ f (z) − λIλ+1,µ f (z) (f ∈ A; λ > −1; µ > 0) and
(1.8)
z Iλ,µ f (z) = µIλ,µ+1 f (z) − (µ − 1)Iλ,µ f (z) (f ∈ A; λ > −1; µ > 0).
(1.9)
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Next, by using the general operator Iλ,µ , we introduce the following (presumably new) classes of analytic functions for φ, ψ ∈ N , λ > −1, and µ > 0: S∗λ,µ (φ) := f : f ∈ A and Iλ,µ f (z) ∈ S∗ (φ) , Kλ,µ (φ) := f : f ∈ A and Iλ,µ f (z) ∈ K(φ) , and Cλ,µ (φ, ψ) = f : f ∈ A and Iλ,µ f (z) ∈ C(φ, ψ) . We also note that f (z) ∈ Kλ,µ (φ)
⇔
zf (z) ∈ S∗λ,µ (φ).
(1.10)
In particular, we set
1+z ∗ Sn,2 = Sn∗ , 1−z
1 + Az ∗ ∗ Sλ,µ [A, B] (−1 B < A 1), = Sλ,µ 1 + Bz and
Kλ,µ
1 + Az 1 + Bz
= Kλ,µ [A, B]
(−1 B < A 1).
In this paper, we investigate several inclusion properties of the classes S∗λ,µ (φ), Kλ,µ (φ), and Cλ,µ (φ, ψ) associated with the general integral operator Iλ,µ . Some applications involving these and other families of integral operators are also considered.
2. Inclusion properties involving Iλ,µ The following results will be required in our investigation. Lemma 1 (Eenigenburg et al. [3]). Let h be convex univalent in U with h(0) = 1 and βh(z) + γ > 0 (β, γ ∈ C). If p(z) is analytic in U with p(0) = 1, then p(z) +
zp (z) ≺ h(z) βp(z) + γ
implies that p(z) ≺ h(z) in U.
in U
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Lemma 2 (Miller and Mocanu [11]). Let h be convex in U with h(0) = 1. Suppose also that Q(z) is analytic in U with {Q(z)} 0 (z ∈ U). If p(z) is analytic in U with p(0) = 1, then p(z) + Q(z)zp (z) ≺ h(z)
in U
implies that p(z) ≺ h(z) in U. We begin by proving our first inclusion relation given by Theorem 1. Let λ 0 and µ 1. Then S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) ⊂ S∗λ+1,µ (φ)
(φ ∈ N ).
Proof. First of all, we show that S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) Let
f (z) ∈ S∗λ,µ+1 (φ)
(φ ∈ N ; λ 0; µ 1).
and set
z(Iλ,µ f (z)) = p(z), Iλ,µ f (z)
(2.1)
where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. Applying (1.9) and (2.1), we obtain µ
Iλ,µ+1 f (z) = p(z) + µ − 1. Iλ,µ f (z)
(2.2)
By using the logarithmic differentiation on both side of (2.2), we have zp (z) z(Iλ,µ+1 f (z)) z(Iλ,µ f (z)) = + Iλ,µ+1 f (z) Iλ,µ f (z) p(z) + µ − 1
zp (z) . = p(z) + p(z) + µ − 1 Since µ 1, φ(z) ∈ N , and f (z) ∈ S∗λ,µ+1 (φ), from (2.3) we see that φ(z) + µ − 1 > 0 (z ∈ U) and p(z) +
zp (z) ≺ φ(z) p(z) + µ − 1
in U.
Thus, by using Lemma 1 and (2.1), we observe that p(z) ≺ φ(z)
in U,
so that f (z) ∈ S∗λ,µ (φ).
(2.3)
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This implies that S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ). To prove the second part, let f (z) ∈ S∗λ,µ (φ) (λ 0; µ 1) and put z(Iλ+1,µ f (z)) = q(z), Iλ+1,µ f (z) where q(z) = 1 + d1 z + d2 z2 + · · · is analytic in U and q(z) = 0 for all z ∈ U. Then, by using arguments similar to those detailed above with (1.8), it follows that q(z) ≺ φ(z)
in U,
which implies that f (z) ∈ S∗λ+1,µ (φ). Hence we conclude that S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) ⊂ S∗λ+1,µ (φ), which completes the proof of Theorem 1.
✷
Remark 3. By putting λ=n
(n ∈ N0 ),
µ = 2,
1+z (z ∈ U) 1−z (n ∈ N0 ), which was asserted earlier by
and φ(z) =
∗ in Theorem 1, we obtain Sn∗ ⊂ Sn+1 Noor [12].
Theorem 2. Let λ 0 and µ 1. Then Kλ,µ+1 (φ) ⊂ Kλ,µ (φ) ⊂ Kλ+1,µ (φ)
(φ ∈ N ).
Proof. Applying (1.10) and Theorem 1, we observe that f (z) ∈ Kλ,µ+1 (φ) ⇔ Iλ,µ+1 f (z) ∈ K(φ) ⇔ z Iλ,µ+1 f (z) ∈ S∗ (φ) ⇔ Iλ,µ+1 zf (z) ∈ S∗ (φ) ⇔ zf (z) ∈ S∗λ,µ+1 (φ) ⇒ zf (z) ∈ S∗λ,µ (φ) ⇔ Iλ,µ zf (z) ∈ S∗ (φ) ⇔ z Iλ,µ f (z) ∈ S∗ (φ) ⇔ Iλ,µ f (z) ∈ K(φ) ⇔ f (z) ∈ Kλ,µ (φ) and f (z) ∈ Kλ,µ (φ) ⇔ zf (z) ∈ S∗λ,µ (φ)
⇒ zf (z) ∈ S∗λ+1,µ (φ) ⇔ z Iλ+1,µ f (z) ∈ S∗ (φ) ⇔ Iλ+1,µ f (z) ∈ K(φ) ⇔ f (z) ∈ Kλ+1,µ (φ),
which evidently proves Theorem 2.
✷
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Taking 1 + Az (−1 B < A 1; z ∈ U) 1 + Bz in Theorems 1 and 2, we have φ(z) =
Corollary 1. Let λ 0, µ 1, and −1 B < A 1. Then ∗ ∗ ∗ [A, B] ⊂ Sλ,µ [A, B] ⊂ Sλ+1,µ [A, B] Sλ,µ+1
and Kλ,µ+1 [A, B] ⊂ Kλ,µ [A, B] ⊂ Kλ+1,µ [A, B]. Next, by using Lemma 2, we obtain the following inclusion theorem for the class Cλ,µ (φ, ψ). Theorem 3. Let λ 0 and µ 1. Then Cλ,µ+1 (φ, ψ) ⊂ Cλ,µ (φ, ψ) ⊂ Cλ+1,µ (φ, ψ)
(φ, ψ ∈ N ).
Proof. We begin by proving that Cλ,µ+1 (φ, ψ) ⊂ Cλ,µ (φ, ψ)
(λ 0; µ 1; φ, ψ ∈ N ).
Let f (z) ∈ Cλ,µ+1 (φ, ψ). Then, in view of (1.2), there exists a function q(z) ∈ S∗ (φ) such that z(Iλ,µ+1 f (z)) ≺ ψ(z) q(z)
in U.
Choose the function g(z) such that Iλ,µ+1 g(z) = q(z). Then g(z) ∈ S∗λ,µ+1 (φ) and z(Iλ,µ+1 f (z)) ≺ ψ(z) in U. (2.4) Iλ,µ+1 g(z) Now let z(Iλ,µ f (z)) = p(z), Iλ,µ g(z)
(2.5)
where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. Thus, by using (1.9), we obtain z(Iλ,µ+1 f (z)) Iλ,µ+1 (zf (z)) = Iλ,µ+1 g(z) Iλ,µ+1 g(z) z(Iλ,µ (zf (z))) + (µ − 1)Iλ,µ (zf (z)) = z(Iλ,µ g(z)) + (µ − 1)Iλ,µ g(z)
J.H. Choi et al. / J. Math. Anal. Appl. 276 (2002) 432–445
=
z(Iλ,µ (zf (z))) I (zf (z)) + (µ − 1) λ,µ Iλ,µ g(z) Iλ,µ g(z) . z(Iλ,µ g(z)) Iλ,µ g(z) + µ − 1
441
(2.6)
Since g(z) ∈ S∗λ,µ+1 (φ) ⊂ S∗λ,µ (φ) (φ ∈ N ), by Theorem 1, we set z(Iλ,µ g(z)) = G(z), Iλ,µ g(z) where G(z) ≺ φ(z) in U for φ ∈ N . Then, by virtue of (2.5) and (2.6), we observe that Iλ,µ zf (z) = p(z)Iλ,µ g(z) (2.7) and z(Iλ,µ+1 f (z)) = Iλ,µ+1 g(z)
z(Iλ,µ (zf (z))) Iλ,µ g(z)
+ (µ − 1)p(z)
G(z) + µ − 1
.
(2.8)
Upon differentiating both sides of (2.7), we have z(Iλ,µ (zf (z))) = G(z)p(z) + zp (z). Iλ,µ g(z)
(2.9)
Making use of (2.4), (2.8), and (2.9), we get z(Iλ,µ+1 f (z)) zp (z) = p(z) + ≺ ψ(z) Iλ,µ+1 g(z) G(z) + µ − 1
(z ∈ U).
(2.10)
Since µ 1 and G(z) ≺ φ(z) in U, G(z) + µ − 1 > 0 (z ∈ U). Hence, by taking Q(z) =
1 G(z) + µ − 1
in (2.10), and applying Lemma 2, we can show that p(z) ≺ ψ(z)
in U,
so that f (z) ∈ Cλ,µ (φ, ψ)
(φ, ψ ∈ N ).
For the second part, by using arguments similar to those detailed above with (1.8), we obtain Cλ,µ (φ, ψ) ⊂ Cλ+1,µ (φ, ψ)
(φ, ψ ∈ N ).
The proof of Theorem 3 is thus completed. ✷
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Remark 4. In its special case when 1+z (z ∈ U), 1−z Theorem 3 would reduce immediately to a known result due to Noor and Noor [13, p. 343, Theorem 3.1]. µ = 2,
λ = n (n ∈ N0 ),
and φ(z) = ψ(z) =
3. Inclusion properties involving Jσ In this section, we consider the generalized Bernardi–Libera–Livingston integral operator Jσ (σ > −1) defined by (cf. [1], [7], and [18]) σ +1 Jσ (f )(z) := σ z
z
t σ −1 f (t) dt
(f ∈ A; σ > −1).
(3.1)
0
We first prove Theorem 4. Let σ 0, λ > −1, and µ > 0. If f (z) ∈ S∗λ,µ (φ) (φ ∈ N ), then Jσ (f )(z) ∈ S∗λ,µ (φ)
(φ ∈ N ).
Proof. Let f (z) ∈ S∗λ,µ (φ) for φ ∈ N , and set z(Iλ,µ Jσ (f )(z)) = p(z), Iλ,µ Jσ (f )(z)
(3.2)
where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. From (3.1) we obtain z Iλ,µ Jσ (f )(z) = (σ + 1)Iλ,µ f (z) − σ Iλ,µ Jσ (f )(z) (z ∈ U). (3.3) By applying (3.2) and (3.3), we have (σ + 1)
Iλ,µ f (z) = p(z) + σ. Iλ,µ Jσ (f )(z)
(3.4)
Making use of the logarithmic differentiation on both sides in (3.4), we get z(Iλ,µ f (z)) zp (z) = p(z) + . Iλ,µ f (z) p(z) + σ
(3.5)
Since σ 0, φ(z) ∈ N , and f (z) ∈ S∗λ,µ (φ), from (3.5) we obtain φ(z) + σ > 0
and p(z) +
zp (z) ≺ φ(z) p(z) + σ
Hence, by virtue of Lemma 1, we conclude that p(z) ≺ φ(z)
in U,
(z ∈ U).
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443
which implies that Jσ (f )(z) ∈ S∗λ,µ (φ)
(φ ∈ N ).
✷
Next we derive an inclusion property involving Jσ , which is given by Theorem 5. Let σ 0, λ > −1, and µ > 0. If f (z) ∈ Kλ,µ (φ) (φ ∈ N ), then Jσ (f )(z) ∈ Kλ,µ (φ)
(φ ∈ N ).
Proof. By applying Theorem 4, it follows that f (z) ∈ Kλ,µ (φ) ⇔ zf (z) ∈ S∗λ,µ (φ) ⇒ Jσ zf (z) ∈ S∗λ,µ (φ) ⇔ z Jσ (f )(z) ∈ S∗λ,µ (φ) ⇔ Jσ (f )(z) ∈ Kλ,µ (φ) which proves Theorem 5.
(φ ∈ N ),
✷
Finally, we prove Theorem 6. Let σ 0, λ > −1, and µ > 0. If f (z) ∈ Cλ,µ (φ, ψ) (φ, ψ ∈ N ), then Jσ (f )(z) ∈ Cλ,µ (φ, ψ)
(φ, ψ ∈ N ).
Proof. Let f (z) ∈ Cλ,µ (φ, ψ) for φ, ψ ∈ N . Then, in view of (1.2), there exists a function g(z) ∈ S∗λ,µ (φ) such that z(Iλ,µ f (z)) ≺ ψ(z) Iλ,µ g(z)
in U.
(3.6)
Thus we set z(Iλ,µ Jσ (f )(z)) = p(z), Iλ,µ Jσ (g)(z) where p(z) = 1 + c1 z + c2 z2 + · · · is analytic in U and p(z) = 0 for all z ∈ U. Applying (3.3), we get z(Iλ,µ f (z)) Iλ,µ (zf (z)) z(Iλ,µ Jσ (zf (z))) + σ Iλ,µ Jσ (zf (z)) = = Iλ,µ g(z) Iλ,µ g(z) z(Iλ,µ Jσ (g)(z)) + σ Iλ,µ Jσ (g)(z) =
z(Iλ,µ Jσ (zf (z))) Iλ,µ Jσ (zf (z)) Iλ,µ Jσ (g)(z) + σ Iλ,µ Jσ (g)(z) z(Iλ,µ Jσ (g)(z)) Iλ,µ Jσ (g)(z) + σ
.
(3.7)
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Since g(z) ∈ S∗λ,µ (φ) (φ ∈ N ), by virtue of Theorem 4, we have Jσ (g)(z) ∈ S∗λ,µ (φ). Let us now put z(Iλ,µ Jσ (g)(z)) = H (z), Iλ,µ Jσ (g)(z) where H (z) ≺ φ(z) in U for φ ∈ N . Then, by using the same techniques as in the proof of Theorem 3, we conclude from (3.6) and (3.7) that z(Iλ,µ f (z)) zp (z) = p(z) + ≺ ψ(z) Iλ,µ g(z) H (z) + σ
in U.
(3.8)
Hence, upon setting Q(z) =
1 H (z) + σ
(z ∈ U)
in (3.8), if we apply Lemma 2, we obtain p(z) ≺ ψ(z)
in U,
which yields Jσ (f )(z) ∈ Cλ,µ (φ, ψ)
(φ, ψ ∈ N ).
The proof of Theorem 6 is evidently completed. ✷ Remark 5. By comparing the definitions (1.3) and (3.1) rather closely, and applying (1.7), it is not difficult to deduce the following relationship: Jσ (f )(z) = L(σ + 1, σ + 2)f (z) = Iσ +1,σ +1 f (z)
(f ∈ A; σ > −1)
(3.9)
among the Carlson–Shaffer linear operator L(a, c), the general integral operator Iλ,µ , and the generalized Bernardi–Libera–Livingston integral operator Jσ (σ > −1). This relationship (3.9) can alternatively be invoked in order to derive the results of this section from those of the preceding section. The details involved in these alternative derivations are being omitted here.
Acknowledgments The present investigation was initiated during the third-named author’s visit to Fukuoka University in April 2002. This work was supported, in part, by the Science Research Promotion Fund from the Japan Private School Promotion Foundation and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
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