J. Math. Anal. Appl. 332 (2007) 109–122 www.elsevier.com/locate/jmaa
On certain subclasses of multivalent functions associated with an extended fractional differintegral operator J. Patel a,∗ , A.K. Mishra b a Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar-751 004, Orissa, India b Department of Mathematics, Berhampur University, Bhanja Bihar, Berhampur-760 007, Orissa, India
Received 1 December 2005 Available online 7 November 2006 Submitted by H.M. Srivastava
Abstract (λ,p)
(−∞ < λ < p + 1; p ∈ N), In the present paper an extended fractional differintegral operator Ωz suitable for the study of multivalent functions is introduced. Various mapping properties and inclusion relationships between certain subclasses of multivalent functions are investigated by applying the techniques of differential subordination. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out. © 2006 Elsevier Inc. All rights reserved. Keywords: Analytic function; Multivalent function; Differential subordination; Fractional differintegral operator; Gauss hypergeometric function; Hadamard product (or convolution)
1. Introduction and main results Let Ap denote the class of functions normalized by f (z) = zp +
∞
ap+n zp+n
p ∈ N = {1, 2, 3, . . .} ,
n=1
which are analytic and p-valent in the open unit disk U = {z: z ∈ C and |z| < 1}. * Corresponding author.
E-mail addresses:
[email protected] (J. Patel),
[email protected] (A.K. Mishra). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.09.067
(1.1)
110
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A function f (z) ∈ Ap is said to be in the class Sp∗ (α) of p-valently starlike functions of order α in U, if zf (z) > α (0 α < p; z ∈ U). (1.2) f (z) Furthermore, a function f (z) ∈ Ap is said to be in the class Kp (α) of p-valently convex functions of order α in U, if zf (z) > α (0 α < p; z ∈ U). (1.3) 1+ f (z) Indeed, it follows from (1.2) and (1.3) that f (z) ∈ Kp (α)
⇐⇒
zf (z) ∈ Sp∗ (α) p
(0 α < p; z ∈ U).
(1.4)
We note that Sp∗ (α) ⊆ Sp∗ (0) ≡ Sp∗
and Kp (α) ⊆ Kp (0) ≡ Kp
(0 α < p),
Sp∗
and Kp denote the subclasses of Ap consisting of functions which are p-valently where starlike in U and p-valently convex in U, respectively (see, for details, [5]; see also [20] and [1]). If f (z) and g(z) are analytic in U, we say that f (z) is subordinate to g(z), written symbolically as f ≺g
in U
or f (z) ≺ g(z)
(z ∈ U),
if there exists a Schwarz function w(z), which (by definition) is analytic in U with w(0) = 0 and |w(z)| < 1 in U such that f (z) = g(w(z)), z ∈ U. It is known that f (z) ≺ g(z)
(z ∈ U)
⇒
f (0) = g(0)
and f (U) ⊂ g(U).
In particular, if the function g(z) is univalent in U, then we have the following equivalence (cf., e.g., [10], see also [12]): f (z) ≺ g(z)
(z ∈ U)
⇐⇒
f (0) = g(0)
and f (U) ⊂ g(U).
Furthermore, f (z) is said to be subordinate to g(z) in the disk Ur = {z ∈ C: |z| < r} if the function fr (z) = f (rz) is subordinate to the function gr (z) = g(rz) in U. It follows from the Schwarz lemma that if f ≺ g in U, then f ≺ g in Ur for every r (0 < r < 1). The general theory of differential subordination introduced by Miller and Mocanu is given in [10]. Namely, if Ψ : Ω → C (where Ω ⊆ C2 ) is an analytic function, h is analytic and univalent in U, and if φ is analytic in U with (φ(z), zφ (z)) ∈ Ω when z ∈ U, then we say that φ satisfies a first-order differential subordination provided that (1.5) Ψ φ(z), zφ (z) ≺ h(z) (z ∈ U) and Ψ φ(0), 0 = h(0). We say that a univalent function q(z) is a dominant of the differential subordination (1.5), if φ(0) = q(0) and φ(z) ≺ q(z) for all analytic functions φ(z) that satisfy the differential subordination (1.5). A dominant q(z) ˜ is called as the best dominant of (1.5), if q(z) ˜ ≺ q(z) for all dominants q(z) of (1.5) (cf., [10,11]). For functions fj (z) ∈ Ap , given by fj (z) = zp +
∞ n=1
ap+n,j zp+n
(j = 1, 2; p ∈ N),
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111
we define the Hadamard product (or convolution) of f1 (z) and f2 (z) by (f1 f2 )(z) = zp +
∞
ap+n,1 ap+n,2 zp+n = (f2 f1 )(z)
(p ∈ N; z ∈ U).
n=1
In our present investigation, we shall also make use of the Gauss hypergeometric function 2 F1 defined by 2 F1 (a, b; c; z) =
∞ (a)n (b)n zn
(c)n
n=0
n!
a, b, c ∈ C; c ∈ / Z− 0 = {0, −1, −2, . . .} ,
(1.6)
where (κ)n denotes the Pochhammer symbol (or the shifted factorial) given, in terms of the Gamma function , by (κ + n) 1 (n = 0), = (κ)n = κ(κ + 1) · · · (κ + n − 1) (n ∈ N). (κ) We note that the series defined by (1.6) converges absolutely for z ∈ U and hence 2 F1 represents an analytic function in the open unit disk U (see, for details, [23, Chapter 14]). With a view to introducing an extended fractional differintegral operator, we begin by recalling here the following definitions of fractional calculus (that is, fractional integral and fractional derivative of an arbitrary order) considered by Owa [13] (see also [14] and [19]). Definition 1.1. The fractional integral of order λ (λ > 0) is defined, for a function f (z), analytic in a simply-connected region of the complex plane containing the origin by Dz−λ f (z) =
1 (λ)
z 0
f (ζ ) dζ, (z − ζ )1−λ
(1.7)
where the multiplicity of (z − ζ )λ−1 is removed by requiring log(z − ζ ) to be real when z − ζ > 0. Definition 1.2. Under the hypothesis of Definition 1.1, the fractional derivative of f (z) of order λ (λ 0) is defined by ⎧ z ⎪ 1 d f (ζ ) ⎪ ⎪ ⎪ dζ (0 λ < 1), ⎨ (1 − λ) dz (z − ζ )λ Dzλ f (z) = (1.8) 0 ⎪ ⎪ n ⎪ ⎪ ⎩ d D λ−n f (z) n λ < n + 1; n ∈ N0 = N ∪ {0} , dzn z where the multiplicity of (z − ζ )−λ is removed as in Definition 1.1. (λ,p)
We now define the extended fractional differintegral operator Ωz : Ap → Ap for a function f (z) of the form (1.1) and for a real number λ (−∞ < λ < p + 1) by (λ,p)
Ωz
f (z) = zp +
∞ (n + p + 1) (p + 1 − λ) n=1
=z
p
(p + 1) (n + p + 1 − λ)
ap+n zp+n
2 F1 (1, p + 1; p + 1 − λ; z) f (z)
(−∞ < λ < p + 1; z ∈ U).
(1.9)
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It is easily seen from (1.9) that (λ,p) (1+λ,p) (λ,p) z Ωz f (z) = (p − λ)Ωz f (z) + λΩz f (z)
(−∞ < λ < p; z ∈ U). (1.10)
We also note that (0,p)
Ωz
(1,p)
f (z) = f (z),
Ωz
f (z) =
zf (z) , p
and, in general (λ,p)
Ωz
f (z) =
(p + 1 − λ) λ λ z Dz f (z) (p + 1)
(−∞ < λ < p + 1; z ∈ U),
(1.11)
where Dzλ f (z) is, respectively, the fractional integral of f (z) of order −λ when −∞ < λ < 0 and the fractional derivative of f (z) of order λ when 0 λ < p + 1. For integral values of λ, (1.11) further simplifies to (k,p)
Ωz
f (z) =
(p − k)!zk f (k) (z) p!
(k ∈ N; k < p + 1)
and (−m,p) Ωz f (z) =
p+m zm
z
(−m+1,p)
t m−1 Ωz
f (t) dt
(m ∈ N)
0
= F1,p ◦ F2,p ◦ · · · ◦ Fm,p (f )(z) p p p z z z F2,p · · · Fm,p f (z), = F1,p 1−z 1−z 1−z where Fμ,p is the familiar integral operator defined by (1.17) and ◦ stands for the usual composition of functions. (λ,p) with 0 λ < 1 was investigated by Srivastava and The fractional differential operator Ωz Aouf [16]. More recently, Srivastava and Mishra [18] obtained several interesting properties and characteristics for certain subclasses of p-valent analytic functions involving the differintegral (λ,p) (λ,1) when −∞ < λ < 1. We, further observe that Ωz = Ωzλ is the operator introoperator Ωz duced by Owa and Srivastava [14]. (λ,p) (−∞ < λ < p + 1), we Now, by using the extended fractional differintegral operator Ωz introduce the following subclass of functions in Ap . Definition 1.3. For fixed parameters A, B (−1 B < A 1) and 0 α < p, we say that a function f (z) ∈ Ap is in the class Vpλ (α; A, B) if it satisfies the following subordination condition: (λ,p) z(Ωz 1 + Az f (z)) 1 − α ≺ (λ,p) p−α 1 + Bz Ωz f (z)
(−∞ < λ < p; z ∈ U).
(1.12)
For convenience, we write
(λ,p) f (z)) z(Ωz Vpλ (α; 1, −1) = Vpλ (α) = f (z) ∈ Ap : > α, 0 α < p, z ∈ U . (λ,p) Ωz f (z) We, further observe that
J. Patel, A.K. Mishra / J. Math. Anal. Appl. 332 (2007) 109–122
Vpλ (α; A, B) = Vpλ
113
α 0; A + (B − A), B , p
Vp0 (α; 1, −1) = Sp∗ (α)
and Vp1 (α; 1, −1) = Kp (α).
Srivastava et al. [17] have studied some interesting properties of the class V1λ (α; 1, −1) := Sλ (α) (0 λ < 1; 0 α < 1) by using the techniques of Hadamard product. In the present paper several sharp inclusion relationships and other interesting properties of the class Vpλ (α; A, B) are found for all admissible nonnegative values of λ and also for negative values of λ by using the techniques of differential subordination. Mapping properties of a variety (λ,p) are also investigated. Consequences of the main results of operators involving the operator Ωz and their relevance to known results are pointed out. We now state our main results. Unless otherwise mentioned, we assume throughout the sequel that −1 B < A 1,
0 α < p,
−∞ < λ < p,
and p ∈ N.
Theorem 1.4. Let f (z) ∈ Vpλ+1 (α; A, B), (p − α)(1 − A) + (α − λ)(1 − B) 0, and the function Q(z) be defined on U by ⎧ (p−α)(A−B)/B 1 ⎪ ⎪ ⎪ ⎪ t p−λ−1 1 + Btz ⎪ dt ⎪ ⎪ 1 + Bz ⎪ ⎨ 0 Q(z) = ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ t p−λ−1 exp (p − α)A(t − 1)z dt ⎪ ⎪ ⎩
(1.13)
(B = 0), (1.14) (B = 0).
0
Then
(λ,p) 1 z(Ωz 1 1 f (z)) − α + λ = q(z) − α ≺ (λ,p) p−α p − α Q(z) Ωz f (z) 1 + Az (z ∈ U), ≺ 1 + Bz
(1.15)
Vpλ+1 (α; A, B) ⊂ Vpλ (α; A, B), and q(z) is the best dominant of (1.15). If, in addition to (1.13) one has A−
(α − λ + 1)B p−α
with −1 B < 0,
then Vpλ+1 (α; A, B) ⊂ Vpλ (α; 1 − 2ρ, −1), where ρ=
−1 (p − α)(B − A) B 1 (p − λ) 2 F1 1, ; p − λ + 1; −α+λ . p−α B B −1
The bound ρ in (1.16) is the best possible.
(1.16)
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Taking A = 1, B = −1 and p = 1 in Theorem 1.4, we get the following result which both extends and sharpens the work of Srivastava et al. [17, Theorem 1]. Corollary 1.5. If −∞ < max{λ, λ/2} α < 1, then Sλ+1 (α) ⊆ Sλ (γ ) ⊆ Sλ (α), where
1 −1 + λ. γ = (1 − λ) 2 F1 1, 2(1 − α); 2 − λ; 2
The value of γ is the best possible. For a function f (z) ∈ Ap , the generalized Bernardi–Libera–Livingston integral operator Fμ,p : Ap → Ap is defined by (cf., e.g., [2]) μ+p Fμ,p (f )(z) = zμ = z + p
z t μ−1 f (t) dt 0 ∞ k=1
μ + p p+k f (z) z μ+p+k
(μ > −p; z ∈ U).
It follows from (1.9) and (1.17) that (λ,p) (λ,p) (λ,p) Fμ,p (f )(z) = (μ + p)Ωz f (z) − μΩz Fμ,p (f )(z) z Ωz
(z ∈ U).
(1.17)
(1.18)
Theorem 1.6. Let μ be a real number satisfying (p − α)(1 − A) + (μ + α)(1 − B) 0. (i) If f (z) ∈ Vpλ (α; A, B), then (λ,p) Fμ,p (f )(z)) z(Ωz 1 1 1 − μ − α = q(z) ˜ −α ≺ (λ,p) p−α p − α Q(z) Ωz Fμ,p (f )(z) 1 + Az (z ∈ U), ≺ 1 + Bz where
Q(z) =
⎧ (p−α)(A−B)/B 1 ⎪ ⎪ ⎪ ⎪ μ+p−1 1 + Btz ⎪ t dt ⎪ ⎪ 1 + Bz ⎪ ⎨
(1.19)
(B = 0),
0
⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ t μ+p−1 exp (p − α)A(t − 1)z dt ⎪ ⎪ ⎩
(B = 0),
0
and q(z) ˜ is the best dominant of (1.19). Consequently, the operator Fμ,p maps the class Vpλ (α; A, B) into itself.
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115
(ii) If −1 B < 0 and
(p − α)(1 − A) (p − α)(B − A) − p − 1, − −α , μ max B 1−B
(1.20)
then the operator Fμ,p maps the class Vpλ (α; A, B) into the class Vpλ (α; 1 − 2 , −1), where −1 1 (p − α)(B − A) B
= −μ−α . (μ + p) 2 F1 1, ; μ + p + 1; p−α B B −1 The bound is the best possible. Putting A = 1 and B = −1 in Theorem 1.6, we get the following result which is an improvement of a result due to Srivastava et al. [17, Theorem 5] for p = 1. Corollary 1.7. If μ is a real number satisfying μ max{p − 2α − 1, −α}, then Fμ,p Vpλ (α) ⊂ Vpλ (σ ), where
1 −1 − μ. σ = (μ + p) 2 F1 1, 2(p − α); μ + p + 1; 2
The result is the best possible. (λ,p)
We now obtain some properties of the operator Ωz . By employing the same method as in the proof of Theorem 4 and Theorem 5 in [15], we may obtain the following theorems (Theorem 1.8 and Theorem 1.9) stated below. Theorem 1.8. Let δ > 0 and the function f (z) ∈ Ap satisfy (λ,p)
(1 − δ) Then
Ωz
f (z)
zp (λ,p)
Ωz
f (z)
zp
(1+λ,p)
+δ
Ωz
f (z)
zp
≺
1 + Az 1 + Bz
(z ∈ U).
1/m > χ 1/m
(m ∈ N; z ∈ U),
where
⎧ A p−λ A B −1 ⎪ ⎪ (B = 0), ⎨ B + 1 − B (1 − B) 2 F1 1, 1; δ + 1; B − 1 χ= ⎪ (p − λ)A ⎪ ⎩1 − (B = 0). p−λ+δ The result is the best possible. Theorem 1.9. Let δ > 0, −∞ < λ < p + 1 and f (z) ∈ Ap . If the function Fμ,p (f )(z) defined by (1.17) satisfies (λ,p)
(1 − δ)
Ωz
(λ,p)
Fμ,p (f )(z) f (z) 1 + Az Ωz +δ ≺ zp zp 1 + Bz
(z ∈ U),
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then
(λ,p)
Ωz
Fμ,p (f )(z) zp
1/m (m ∈ N; z ∈ U),
> ξ 1/m
where
⎧ A p+μ A B −1 ⎪ ⎪ (B = 0), ⎨ B + 1 − B (1 − B) 2 F1 1; 1; δ + 1; B − 1 ξ= ⎪ (p + μ)A ⎪ ⎩1 − (B = 0). p+μ+δ
The result is the best possible. Theorem 1.10. Let μ be a real number satisfying (1.20). If f (z) ∈ Vpλ (α; A, B), then the function Fμ,p (f )(z) defined by (1.17) satisfies −1 (λ,p) (p − α)(B − A) B f (z) Ωz 1, ; μ + p + 1; > F (z ∈ U). 2 1 (λ,p) B B −1 Ωz Fμ,p (f )(z) The result is the best possible. The proof of the following theorem is much akin to that of [15, Theorem 3] (see also [8]) and we omit its proof. Theorem 1.11. Let δ > 0 and −1 Bj < Aj 1 (j = 1, 2). If each of the function fj (z) ∈ Ap (j = 1, 2) satisfies (λ,p)
(1 − δ)
Ωz
(1 − δ)
Ωz
fj (z)
zp
(1+λ,p)
+δ
Ωz
+δ
Ωz
fj (z)
≺
1 + Aj z 1 + Bj z
H (z)
≺
1 + (1 − 2η0 )z 1−z
zp
(j = 1, 2; z ∈ U),
then (λ,p)
H (z)
zp
(1+λ,p)
zp
(z ∈ U),
where (λ,p)
H (z) = Ωz and η0 = 1 −
(f1 f2 )(z)
(z ∈ U)
(1.21)
p−λ 1 1 4(A1 − B1 )(A2 − B2 ) 1 − 2 F1 1, 1; + 1; . (1 − B1 )(1 − B2 ) 2 δ 2
The result is the best possible when B1 = B2 = −1. With a view to stating a well-known result, we denote by P(γ ) the class of functions ϕ(z) of the form ϕ(z) = 1 + c1 z + c2 z2 + · · · which are analytic in U and satisfy the following inequality: ϕ(z) > γ (0 γ < 1; z ∈ U).
(1.22)
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It is known [22] that if ϕj (z) ∈ P(γj ) (0 γj < 1; j = 1, 2), then (ϕ1 ϕ2 )(z) ∈ P(γ3 ) γ3 = 1 − 2(1 − γ1 )(1 − γ2 ) ,
117
(1.23)
and the bound γ3 is the best possible. We now state (1+λ,p)
Theorem 1.12. Let fj (z) ∈ Ap (j = 1, 2). If the functions Ωz ηj < 1; j = 1, 2), then the function H (z), given by (1.21) satisfies (1+λ,p) H (z) Ωz > 0 (z ∈ U), (λ,p) Ωz H (z)
fj (z)/zp ∈ P(ηj ) (0
provided (1 − η1 )(1 − η2 )
0
Gλ (z)
(z ∈ U),
where z Gλ (z) = (p − λ)z
λ
H (t) dt t 1+λ
(z ∈ U).
0
For λ = 0 in Theorem 1.14, we obtain the following result which yields the corresponding work of Lashin [7, Theorem 3] for p = 1. Corollary 1.15. Let fj (z) ∈ Ap (j = 1, 2). If 2p + 1 (f1 f2 ) (z) >1− pzp−1 [{2 F1 (1, 1; p + 1; 12 ) − 2}2 + 2p]
(z ∈ U),
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J. Patel, A.K. Mishra / J. Math. Anal. Appl. 332 (2007) 109–122
then z G0 (z) = p
(f1 f2 )(t) dt ∈ Sp . t
0
For the proof of our Theorem 1.12, we need the following lemmas. Lemma 1.16. (Hallenbeck and Ruscheweyh [6], also see [12, p. 71].) Let the function h(z) be analytic and convex (univalent) in U with h(0) = 1. Suppose also that the function φ(z) of the form (1.22) is analytic in U. If φ(z) +
zφ (z) ≺ h(z) γ
γ = 0, (γ ) 0; z ∈ U ,
then γ φ(z) ≺ ψ(z) = γ z
z
t γ −1 h(t) dt ≺ h(z)
(z ∈ U),
(1.26)
0
and ψ(z) is the best dominant of (1.26). Lemma 1.17. (Miller and Mocanu [12, p. 35]) Suppose that the function Ψ : C2 ×U → C satisfies the condition Ψ (ix, y; z) ε for ε > 0, real x, y −(1 + x 2 )/2 and for all z ∈ U. If φ(z), given by (1.22) is analytic in U and Ψ φ(z), zφ (z); z > ε, then (φ(z)) > 0 in U. 2. Proofs of Theorems 1.4, 1.6, 1.10, 1.12 and 1.14 Proof of Theorem 1.4. Let f (z) ∈ Vpλ+1 (α; A, B), and suppose that the function g(z) is defined by g(z) = z
(λ,p)
Ωz
f (z)
zp
1 p−α
(z ∈ U).
(2.1)
Write r1 = sup r: g(z) = 0, 0 < |z| < r < 1 . Then g(z) is single-valued and analytic in Ur1 . Taking logarithmic differentiation in (2.1), it follows that the function (λ,p) zg (z) 1 z(Ωz f (z)) φ(z) = = −α (2.2) (λ,p) g(z) p−α Ωz f (z)
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119
is of the form (1.22) and is analytic in Ur1 . Using the identity (1.10) in (2.2) and again carrying out logarithmic differentiation in the resulting equation, we get (λ+1,p) z(Ωz 1 + Az zφ (z) f (z)) 1 ≺ − α = φ(z) + (λ+1,p) p−α (p − α)φ(z) + α − λ 1 + Bz Ωz f (z)
(z ∈ Ur1 ).
Hence by applying a result [11, Corollary 3.2], we deduce that 1 1 + Az 1 − α + λ = q(z) ≺ φ(z) ≺ p − α Q(z) 1 + Bz
(z ∈ Ur1 ),
where q(z) is the best dominant of (1.15) and Q(z) is given by (1.14). The remaining part of the proof can now be deduced on the same lines as in [15, Theorem 1]. This evidently completes the proof of Theorem 1.4. 2
Proof of Theorem 1.6. Upon replacing g(z)
by z
(λ,p)
Ωz
Fμ,p (f )(z) zp
1 p−α
(z ∈ U)
in (2.1) and carrying out logarithmic differentiation, it follows that the function φ(z) given by
φ(z) =
(λ,p) Fμ,p (f )(z)) z(Ωz zg (z) 1 = − α (λ,p) g(z) p−α Ωz Fμ,p (f )(z)
(2.3)
is of the form (1.22) and is analytic in Ur1 . Using the identity (1.18) in (2.3) and the fact that (λ,p) Ωz f (z) = 0 in 0 < |z| < 1, we get (λ,p)
Ωz
Fμ,p (f )(z) (λ,p) Ωz f (z)
=
μ+p (p − α)φ(z) + μ + α
(z ∈ Ur1 ).
(2.4)
Again, by taking logarithmic differentiation in (2.4) and using (2.3) in the resulting equation, we deduce that (λ,p) 1 z(Ωz 1 + Az zφ (z) f (z)) ≺ − α = φ(z) + (λ,p) p−α (p − α)φ(z) + μ + α 1 + Bz Ωz f (z)
(z ∈ Ur1 ).
The remaining part of the proof of Theorem 1.6 is similar to that of [15, Theorem 1] and we choose to omit the details. 2
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Proof of Theorem 1.10. Since f (z) ∈ Vpλ (α; A, B), it follows from (1.19) that (λ,p)
Ωz
Fμ,p (f )(z) = 0 for 0 < |z| < 1. Thus, by letting (λ,p)
φ(z) =
Ωz
f (z)
(λ,p) Ωz Fμ,p (f )(z)
(z ∈ U),
and following the arguments used in the proof of Theorem 1.6, we can complete the proof of Theorem 1.10. 2 Proof of Theorem 1.12. By the hypothesis on fj (z), it follows from (1.23) that (1+λ,p) Ωz H (z) z zp p−λ zp (1+λ,p) (1+λ,p) Ωz f1 (z) Ωz f2 (z) > 1 − 2(1 − η1 )(1 − η2 ) = zp zp
(1+λ,p)
Ωz
H (z)
+
(z ∈ U),
(2.5)
which in view of Lemma 1.16 for γ = p − λ,
A = −1 + 4(1 − η1 )(1 − η2 ),
and B = −1
yields
(1+λ,p)
Ωz
H (z)
zp
1 > 1 + 2(1 − η1 )(1 − η2 ) 2 F1 1, 1; p − λ + 1; −2 (z ∈ U). 2 (2.6)
Again, from (2.6) and Theorem 1.8 for
1 −2 , δ = 1, A = −1 − 4(1 − η1 )(1 − η2 ) 2 F1 1, 1; p − λ + 1; 2 B = −1 and m = 1,
we deduce that
2 1 ϑ(z) > 1 − 2(1 − η1 )(1 − η2 ) 2 F1 1, 1; p − λ + 1; −2 2 (λ,p)
where ϑ(z) = Ωz Ωz
(2.7)
H (z)/zp . Now, if we let
(1+λ,p)
φ(z) =
(z ∈ U),
H (z)
(λ,p) Ωz H (z)
(z ∈ U),
then φ(z) is of the form (1.22), is analytic in U and a simple computation shows that (1+λ,p) (1+λ,p) Ωz Ωz H (z) H (z) z zφ (z) 2 + = ϑ(z) φ (z) + zp p−λ zp p−λ = Ψ φ(z), zφ (z); z , where Ψ (u, v; z) = ϑ(z)(u2 + (v/(p − λ))). Thus by using (2.5) in (2.8), we get Ψ φ(z), zφ (z); z > 1 − 2(1 − η1 )(1 − η2 ) (z ∈ U).
(2.8)
J. Patel, A.K. Mishra / J. Math. Anal. Appl. 332 (2007) 109–122
121
Now for all real x, y − 12 (1 + x 2 ), we have 1 y 2 − x ϑ(z) − 1 + 2(p − λ) + 1 x 2 ϑ(z) p−λ 2(p − λ) 1 ϑ(z) 1 − 2(1 − η1 )(1 − η2 ) (z ∈ U), − 2(p − λ)
Ψ (ix, y; z) =
by (1.24) and (2.7). Hence by making use of Lemma 1.17, we get (φ(z)) > 0 in U; that is, (1+λ,p) H (z) Ωz > 0 (z ∈ U). (λ,p) Ωz H (z) This completes the proof of Theorem 1.12.
2
Proof of Theorem 1.14. From the definition of the function Gλ (z), we see that
(1+λ,p)
Ωz
H (z)
zp
= >1−
(1+λ,p)
Ωz
zp
Gλ (z)
(1+λ,p) Ωz Gλ (z) z p−λ zp 2(p − λ) + 1 +
[{2 F1 (1, 1; p − λ + 1; 12 ) − 2}2 + 2(p − λ)]
and the proof of Theorem 1.14 is completed similar to Theorem 1.12.
(z ∈ U),
2
3. Consequences and observations (i) For A = 1, B = −1 and λ = 0 in Theorem 1.4, we get the result contained in [21, Corollary 7] which, in turn, yields the corresponding work of MacGregor [9] (see also [24]) for p = 1. (ii) Substituting λ = 0 (or λ = 1, respectively) in Corollary 1.7, we obtain Remark 2 [15, p. 330], which also improves the work of Fukui et al. [4, Remark 2.2] for p = 1. (iii) Setting A = 1 − 2η, B = −1, m = 1 and λ = 0 in Theorem 1.8, we get [15, Corollary 6]. (iv) We note that the results obtained by letting A = 1 − 2η, B = −1, δ = m = 1 and λ = 0 (or A = 1 − 2η, B = −1 and δ = m = λ = 1, respectively) in Theorem 1.9 are also contained in [21, Corollary 1]. (v) Taking A = 1, B = −1 and λ = 0 (or A = 1, B = −1 and λ = 1, respectively) in Theorem 1.10, we obtain [21, Corollaries 5 and 6]. (vi) In the special case when Aj = 1 − 2ηj , Bj = −1 (j = 1, 2) and λ = 0, Theorem 1.11 yields [15, Corollary 4] which gives the corresponding work of Ding et al. [3, Theorem 5] for δ = p = 1. Similarly, letting Aj = 1 − 2ηj , Bj = −1 (j = 1, 2), δ = 1 and λ = −1 in Theorem 1.11, we get [15, Corollary 5]. (vii) If the functions fj (z)/pzp−1 ∈ P(ηj ) (fj ∈ Ap ; 0 ηj < 1; j = 1, 2), then it follows from (1.4) and Corollary 1.13 that the function z G0 (z) = p
(f1 f2 )(t) dt t
0
belongs to the class Kp , provided (1.25) is satisfied.
122
J. Patel, A.K. Mishra / J. Math. Anal. Appl. 332 (2007) 109–122
Acknowledgments The present investigation was supported, in part, by the University Grants Commission of India under its DRS Financial Assistance Program and, in part, by the National Board for Higher Mathematics, Department of Atomic Energy, Government of India, Under Grant No. 48/2/2003-R&D II. The authors express their gratitude to the referee(s) for many valuable suggestions and comments which improve the presentation of the paper.
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