BULLETIN of the M ALAYSIAN M ATHEMATICAL S CIENCES S OCIETY
Bull. Malays. Math. Sci. Soc. (2) 35(3) (2012), 601–610
http://math.usm.my/bulletin
Multivalent Harmonic Functions Defined by Dziok-Srivastava Operator 1 R ASHIDAH
O MAR AND 2 S UZEINI A BDUL H ALIM
1,2 Institute
of Mathematical Sciences, Faculty of Science, Universiti Malaya, Malaysia 1 Faculty of Computer and Mathematical Sciences, Universiti Teknologi Mara, Malaysia 1
[email protected], 2
[email protected]
Abstract. Necessary and sufficient coefficient bounds and convolution condition for certain multivalent harmonic functions whose convolution with generalized hypergeometric functions is starlike of order γ are investigated. Results on extreme points, convex combination and distortion bounds using the coefficient condition are also obtained. 2010 Mathematics Subject Classification: 30C45, 30C50 Keywords and phrases: Multivalent harmonic functions, starlike functions, generalized hypergeometric functions, Dziok-Srivastava operator.
1. Introduction A complex-valued continuous function f = u + iv in a complex domain E ⊂ C is said to be harmonic if both u and v are real harmonic in E. There is an interrelation between harmonic functions and analytic functions. If E is a simply connected domain, then f = h + g¯ where h and g are analytic in E; the functions h and g are respectively called the analytic part and co-analytic part of f . The function f = h + g¯ is said to be harmonic univalent in E if the mapping z → f (z) is orientation preserving, harmonic and univalent in E. This mapping is orientation preserving and locally univalent in E if and only if the Jacobian J f of f given by J f (z) = |h0 (z)|2 − |g0 (z)|2 is positive in E [16]. From the perspective of geometric functions theory, Clunie and Sheil-Small [10] initiated the study on these functions by introducing the class SH consisting of normalized complex-valued harmonic univalent functions f defined on D = {z ∈ C : |z| < 1}. They gave necessary and sufficient conditions for f to be locally univalent and sense-preserving in D. Coefficient bounds for functions in SH were obtained. Since then, various subclasses of SH were investigated by several authors [1, 5, 8, 9, 15, 19, 20, 21]. Note that the class SH reduces to the class of normalized analytic univalent functions if the co-analytic part of f is identically to zero (g ≡ 0). Multivalent harmonic functions in D were introduced by Duren, Hengartner and Laugesen [11] via the argument principle. In [2], the class of multivalent harmonic functions ∗ (p, γ) of multivalent harmonic starlike functions of order γ, where p ≥ 1, and the class SH Communicated by V. Ravichandran. Received: May 5, 2011; Revised: October 4, 2011.
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0 ≤ γ < 1 were discussed and studied. Motivated by [4], we introduce a class of multivalent harmonic functions starlike of order γ using the Dziok-Srivastava operator. Several related work using other linear operators can also be found in [3, 14, 26, 24]. n Recall that the convolution of two analytic functions ϕ(z) = ∑∞ n=0 an z and ψ(z) = ∞ n n ∑n=0 bn z defined on D is the analytic function given by ϕ(z) ∗ ψ(z) = ∑∞ n=0 an bn z = ψ(z) ∗ ϕ(z). Let SH (p) denote the class of multivalent harmonic functions f = h + g¯ where ∞
(1.1)
h(z) = z p + ∑ an+p−1 zn+p−1 ,
∞
g(z) =
n=2
∑ bn+p−1 zn+p−1 .
n=1
For αi ∈ C (i = 1, 2, . . . , l) and β j ∈ C\{0, −1, −2, . . .} ( j = 1, 2, . . . , m), the generalized hypergeometric function l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) is given by ∞
l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) =
(α1 )n . . . (αl )n
∑ (β1 )n . . . (βm )n n! zn
n=0
(l ≤ m + 1; l, m ∈ N0 := N ∪ {0}; z ∈ D) where (λ )n is the Pochhammer symbol defined, in terms of gamma function, by Γ(λ + n) (λ )n := = Γ(λ )
( 1, n = 0, λ 6= 0 λ (λ + 1)(λ + 2) · · · (λ + n − 1), n = 1, 2, 3, . . .
For an analytic function h of the form (1.1), Dziok and Srivastava [12] introduced the linear operator H pl,m [α1 ] h(z) = z p l Fm (α1 , . . . , αl ; β1 , . . . , βm ; z) ∗ h(z) which includes well known operators such as the Hohlov operator [13], Carlson-Shaffer operator [7], Ruscheweyh derivative operator [22], the generalized Bernardi-Libera-Livington integral operator [6], [17], [18] and the Srivastava-Owa fractional derivative operator [25]. For a harmonic function f = h + g, ¯ with h and g given by (1.1), the Dziok-Srivastava operator is defined by H pl,m [α1 ] f (z) = H pl,m [α1 ] h(z) + H pl,m [α1 ] g(z), n+p−1 , H l,m [α ] g(z) = ∞ φ b where H pl,m [α1 ] h(z) = z p + ∑∞ ∑n=1 n n+p−1 zn+p−1 p 1 n=2 φn an+p−1 z and
(1.2)
φn =
(α1 )n−1 · · · (αl )n−1 , (β1 )n−1 · · · (βm )n−1 (n − 1)!
α1 , . . . , αl , β1 , . . . , βm are positive real numbers such that l ≤ m + 1. ∗ (p, α , γ), the class of multivalent harmonic functions satisfying Denote by SH 1 0 0 l,m l,m z H [α ] h(z) − z H [α ] g(z) p p 1 1 (1.3) Re ≥ pγ H pl,m [α1 ] h(z) + H pl,m [α1 ] g(z) ∗ (1, α , γ) ≡ S∗ (α , γ) is the class defined for p ≥ 1, 0 ≤ γ < 1, |z| = r < 1. Note that SH 1 1 H ∗ (p, 1, γ) ≡ S∗ (p, γ) is invesin [4]. In the case of l = m + 1 and α2 = β1 , . . . , αl = βm , SH H ∗ (1, 1, γ) ≡ S∗ (γ) is the class introduced by Jahangiri [15]. Further tigated in [2] and SH H
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∗ (p, α , γ) where h and g TH∗ (p, α1 , γ), p ≥ 1 denotes the class of functions f = h + g¯ ∈ SH 1 are functions of the form ∞
h(z) = z p − ∑ |an+p−1 |zn+p−1 , g(z) =
(1.4)
n=2
∞
∑ |bn+p−1 |zn+p−1 .
n=1
2. Main results Necessary coefficient conditions for the harmonic starlike functions and harmonic convex functions can be found in [10] and [23]. Now we derive sufficient coefficient bound for the ∗ (p, α , γ). class SH 1 Theorem 2.1. Let f = h + g¯ be given by (1.1) and ∏li=1 (αi )n−1 ≥ ∏mj=1 (β j )n−1 (n − 1)!. If ∞ n + p (1 + γ) − 1 1+γ n + p (1 − γ) − 1 |an+p−1 | + |bn+p−1 | |φn | ≤ 1 − |b p | (2.1) ∑ p (1 − γ) p (1 − γ) 1 −γ n=2 where |b p | < (1 − γ)/(1 + γ) , 0 ≤ γ < 1 and φn is given by (1.2), then the harmonic function ∗ (p, α , γ). f is orientation preserving in D and f ∈ SH 1 Proof. The inequality |h0 (z)| ≥ |g0 (z)| is enough to show that f is orientation preserving. Note that ∞
|h0 (z)| ≥ p |z| p−1 − ∑ (n + p − 1)|an+p−1 ||z|n+p−2 n=2
= p|z|
p−1
≥ p|z| p−1 ≥ |z|
p−1
∞
(n + p − 1) 1− ∑ |an+p−1 ||z|n−1 p n=2 ! ∞ (n + p − 1) 1− ∑ |an+p−1 | p n=2
!
! (n + p (1 − γ) − 1) 1− ∑ |φn ||an+p−1 | p (1 − γ) n=2 ∞
By hypothesis, since |φn | ≥ 1 and by (2.1), 0
|h (z)| ≥ |z|
p−1
= |z| p−1 ≥ |z| p−1
! ∞ (n + p (1 + γ) − 1) 1+γ |b p | + ∑ |φn ||bn+p−1 | 1−γ p (1 − γ) n=2 ! ∞ (n + p (1 + γ) − 1) |φn ||bn+p−1 | ∑ p (1 − γ) n=1 ! ∞
∑ (n + p − 1)|bn+p−1 |
n=1
≥ |z| p−1
∞
!
∑ (n + p − 1)|bn+p−1 ||z|n−1
n=1 ∞
=
∑ (n + p − 1)|bn+p−1 ||z|n+p−2 = |g0 (z)|
n=1
Thus, f is orientation preserving in D.
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∗ (p, α , γ) by establishing condition (1.3). First, let Next, we prove f ∈ SH 1 0 0 z H pl,m [α1 ] h(z) − z H pl,m [α1 ] g(z) A(z) , w(z) = = B(z) H pl,m [α1 ] h(z) + H pl,m [α1 ] g(z)
where 0 0 A(z) = z H pl,m [α1 ] h(z) − z H pl,m [α1 ] g(z) ,
B(z) = H pl,m [α1 ] h(z) + H pl,m [α1 ] g(z) .
Now, |A(z) + p (1 − γ)B(z)| − |A(z) − p (1 + γ)B(z)| ∞
≥ (2p − pγ)|z p | − ∑ (n + 2p − pγ − 1)|φn an+p−1 zn+p−1 | n=2
∞
− ∑ (n + pγ − 1)|φn bn+p−1 zn+p−1 | − pγ|z p | n=1 ∞
∞
− ∑ (n − pγ − 1)|φn an+p−1 zn+p−1 | − ∑ (n + 2p + pγ − 1)|φn bn+p−1 zn+p−1 | n=2
n=1
∞
= 2p (1 − γ)|z p | − ∑ (2n + 2p − 2pγ − 2)|φn ||an+p−1 ||zn+p−1 | n=2
∞
− ∑ (2n + 2p + 2pγ − 2)|φ n ||bn+p−1 ||zn+p−1 | n=1
∞
(n + p − pγ − 1) |φn ||an+p−1 ||zn−1 | p (1 − γ) n=2 ! ∞ (n + p + pγ − 1) −∑ |φn ||bn+p−1 ||zn−1 | p (1 − γ) n=1
= 2p (1 − γ)|z p | 1 − ∑
∞
(n + p − pγ − 1) |φn ||an+p−1 | p (1 − γ) n=2 ! ∞ (n + p + pγ − 1) −∑ |φn ||bn+p−1 | p (1 − γ) n=1 ∞ (n + p − pγ − 1) 1+γ p = 2p (1 − γ)|z | 1 − |b p | − ∑ |an+p−1 | 1−γ p (1 − γ) n=2 (n + p + pγ − 1) + |bn+p−1 | |φn | p (1 − γ) ≥ 2p (1 − γ)|z p | 1 − ∑
The last expression is non-negative by (2.1). Since Re w ≥ pγ if and only if |A(z) + p (1 − ∗ (p, α , γ). γ)B(z)| ≥ |A(z) − p (1 + γ)B(z)|, f ∈ SH 1 ∞ For ∑n=1 (|xn+p−1 | + |y¯n+p−1 |) = 1 and x p = 0, the function (2.2) ∞ ∞ p (1 − γ) p(1 − γ) f1 (z) = z p + ∑ xn+p−1 zn+p−1 + ∑ y¯n+p−1 z¯n+p−1 [n + p(1 − γ) − 1]|φ | [n + p(1 + γ) − 1]|φ | n n n=2 n=1
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shows equality in the coefficient bound given by (2.1). For the function f1 defined in (2.2), the coefficients are an+p−1 =
p (1 − γ) xn+p−1 [n + p(1 − γ) − 1]|φn |
and bn+p−1 =
p(1 − γ) y¯n+p−1 , [n + p(1 + γ) − 1]|φn |
∗ (p, α , γ). and since condition (2.1) holds, this implies f1 ∈ SH 1 To show that the converse need not be true, consider the function
f (z) = z p +
γ −1 p p(1 − γ) z p+1 + z¯ . [1 + p(1 − γ)]φ2 2(1 + γ)
It can be shown that h h i0 i0 p + p(1−γ) z p+1 − z¯ (γ−1) z¯ p z z [1+p(1−γ)] 2(1+γ) Re ≥ pγ, (γ−1) p p(1−γ) p+1 p + 2(1+γ) z¯ z + [1+p(1−γ)] z
(p ≥ 1, 0 ≤ γ < 1)
thus f ∈ S∗p (p, α1 , γ) but ∞
∞ n + p(1 − γ) − 1 n + p(1 + γ) − 1 |an+p−1 ||φn | + ∑ |bn+p−1 ||φn | p(1 − γ) p(1 − γ) n=2 n=1 1 + p(1 − γ) p(1 − γ) |φ2 | + 1 + γ γ − 1 > 1. = p(1 − γ) [1 + p(1 − γ)]φ2 1 − γ 2(1 + γ)
∑
∗ (p, α , γ). The next result provide a convolution condition for f to be in the class SH 1 ∗ (p, α , γ) if and only if Theorem 2.2. f ∈ SH 1 2p(1 − γ)z p + (ξ − 2p + 2pγ + 1)z p+1 H pl,m [α1 ] h(z) ∗ (1 − z)2 2p(ξ + γ)¯z p + (ξ − 2pξ − 2pγ + 1)¯z p+1 − H pl,m [α1 ] g(z) ∗ 6= 0, (1 − z¯)2
|ξ | = 1, z ∈ D.
∗ (p, α , γ) is given by (1.3) and we Proof. A necessary and sufficient condition for f ∈ SH 1 have 0 0 l,m l,m z H [α ] g(z) z H [α ] h(z) − p p 1 1 1 Re − pγ ≥ 0. p(1 − γ) l,m l,m H p [α1 ] h(z) + H p [α1 ] g(z)
Since 0 0 l,m l,m z H [α ] h(z) − z H [α ] g(z) p p 1 1 1 − pγ p(1 − γ) l,m l,m H p [α1 ] h(z) + H p [α1 ] g(z) " # n−1 − z¯ p ∞ (n + p − 1)φ b n−1 p + ∑∞ 1 n n+p−1 z n=2 (n + p − 1)φn an+p−1 z z p ∑n=1 − pγ = n−1 + z¯ p ∑∞ φ b n−1 p(1 − γ) 1 + ∑∞ p n=2 φn an+p−1 z n=1 n n+p−1 z z
=1
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at z = 0, the above required condition is equivalent to 0 0 l,m l,m z H [α ] h(z) − z H [α ] g(z) p p 1 1 1 ξ −1 (2.3) , − pγ 6= p(1 − γ) ξ +1 l,m l,m H p [α1 ] h(z) + H p [α1 ] g(z) |ξ | = 1, ξ 6= −1, 0 < |z| < 1. Simple algebraic manipulation in (2.3) yields 0 0 l,m l,m l,m l,m 0 6= (ξ + 1) z H p [α1 ] h(z) − z H p [α1 ] g(z) − pγH p [α1 ] h(z) − pγH p [α1 ] g(z) − (ξ − 1)p(1 − γ)H pl,m [α1 ] h(z) − (ξ − 1)p(1 − γ)H pl,m [α1 ] g(z) (1 − p)z p (2pγ + pξ − p)z p zp l,m − − = H p [α1 ] h(z) ∗ (ξ + 1) (1 − z)2 (1 − z) (1 − z) ! ¯ − p)z p p p z (2pγ + p ξ (1 − p)z l,m − H p [α1 ] g(z) ∗ (ξ¯ + 1) − + (1 − z)2 (1 − z) (1 − z) 2p(1 − γ)z p + (ξ − 2p + 2pγ + 1)z p+1 = H pl,m [α1 ] h(z) ∗ (1 − z)2 2p(ξ + γ)¯z p + (ξ − 2pξ − 2pγ + 1)¯z p+1 − H pl,m [α1 ] g(z) ∗ . (1 − z¯)2 The coefficient bound for class TH∗ (p, α1 , γ) is determined in the following theorem. Furthermore, we use the coefficient condition to obtain extreme points, convex combination and distortion bounds. Theorem 2.3. Let f = h + g¯ be given by (1.4). Then f ∈ TH∗ (p, α1 , γ) if and only if ∞ n + p (1 − γ) − 1 n + p (1 + γ) − 1 1+γ (2.4) ∑ |an+p−1 | + |bn+p−1 | |φn | ≤ 1 − |b p | p (1 − γ) p (1 − γ) 1 −γ n=2 where |b p | < (1 − γ)/(1 + γ) , 0 ≤ γ < 1 and φn is given by (1.2). ∗ (p, α , γ), sufficiency part follows from Theorem 2.1. To Proof. Since TH∗ (p, α1 , γ) ⊂ SH 1 prove the necessity part, suppose that f ∈ TH∗ (p, α1 , γ). Then we obtain p n+p−1 − ∞ (n + p − 1)|b ¯ n+p−1 |φ¯n z¯n+p−1 pz − ∑∞ ∑n=1 n=2 (n + p − 1)|an+p−1 |φn z Re ≥ pγ, n+p−1 + ∑∞ |b ¯ ¯ n+p−1 z p − ∑∞ n=2 |an+p−1 |φn z n=1 n+p−1 |φn z¯
and the result follows by letting r → 1− along real axis. Let clco TH∗ (p, α1 , γ) denote the closed convex hull of TH∗ (p, α1 , γ). Now we determine the extreme points of clco TH∗ (p, α1 , γ). Theorem 2.4. Let f be given by (1.4). Then f ∈ clco TH∗ (p, α1 , γ) if and only if f can be expressed in the form ∞
(2.5)
f=
∑ (Xn+p−1 hn+p−1 +Yn+p−1 gn+p−1 ) ,
n=1
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where h p (z) = z p , hn+p−1 (z) = z p −
p(1 − γ) zn+p−1 [n + p(1 − γ) − 1]|φn |
(n = 2, 3, ...),
p(1 − γ) z¯n+p−1 (n = 1, 2, 3, ...), [n + p(1 + γ) − 1]|φn | φn is given by (1.2), and ∑∞ n=1 (Xn+p−1 +Yn+p−1 ) = 1, with Xn+p−1 ≥ 0,Yn+p−1 ≥ 0. In particular, the extreme points of TH∗ (p, α1 , γ) are hn+p−1 and gn+p−1 . gn+p−1 (z) = z p +
Proof. Let f be of the form (2.5), then we have ∞ f (z) = X p h p (z) + ∑ Xn+p−1 z p −
p(1 − γ) zn+p−1 [n + p(1 − γ) − 1] |φn | n=2 ∞ p(1 − γ) + ∑ Yn+p−1 z p + z¯n+p−1 [n + p(1 + γ) − 1] |φn | n=1
∞
p(1 − γ) Xn+p−1 zn+p−1 [n + p(1 − γ) − 1] |φ | n n=2
= zp − ∑ ∞
p(1 − γ) Yn+p−1 z¯n+p−1 . [n + p(1 + γ) − 1] |φ | n n=1
+∑
(2.6) Furthermore, let |an+p−1 | =
p(1 − γ) Xn+p−1 [n + p(1 − γ) − 1] |φn |
and
|bn+p−1 | =
p(1 − γ) Yn+p−1 . [n + p(1 + γ) − 1] |φn |
Then ∞
p(1 − γ) Xn+p−1 [n + p(1 − γ) − 1] |φn | ∞ [n + p (1 + γ) − 1] |φn | p(1 − γ) +∑ Yn+p−1 p (1 − γ) [n + p(1 + γ) − 1] |φn | n=1
[n + p (1 − γ) − 1] |φn | ∑ p (1 − γ) n=2
∞
=
∞
∑ Xn+p−1 + ∑ Yn+p−1 = 1 − Xp ≤ 1.
n=2
n=1
TH∗ (p, α1 , γ).
Thus f ∈ clco Conversely, suppose that f ∈ clco TH∗ (p, α1 , γ). Set Xn+p−1 =
[n + p (1 − γ) − 1]|φn ||an+p−1 | p (1 − γ)
[n + p (1 + γ) − 1]|φn ||bn+p−1 | p (1 − γ) ∞ and define X p = 1 − ∑∞ X − ∑ n+p−1 n=2 n=1 Yn+p−1 . Then, Yn+p−1 =
∞
∞
n=2 ∞
n=1
(n = 2, 3, ...), (n = 1, 2, ...),
f (z) = z p − ∑ |an+p−1 |zn+p−1 + ∑ |bn+p−1 |¯zn+p−1 ∞ p (1 − γ)Yn+p−1 p (1 − γ)Xn+p−1 zn+p−1 + ∑ z¯n+p−1 =z −∑ [n + p (1 − γ) − 1]|φ | [n + p (1 + γ) − 1]|φ | n n n=2 n=1 p
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∞ p = X p z + ∑ Xn+p−1 z p −
p (1 − γ) zn+p−1 [n + p (1 − γ) − 1]|φn | n=2 ∞ p (1 − γ) + ∑ Yn+p−1 z p + z¯n+p−1 [n + p (1 + γ) − 1]|φn | n=1
∞
=
∑ (Xn+p−1 hn+p−1 +Yn+p−1 gn+p−1 )
n=1
as required. Theorem 2.5. The class TH∗ (p, α1 , γ) is closed under convex combination. Proof. For i = 1, 2, 3, ..., suppose that fi (z) ∈ TH∗ (p, α1 , γ) , where fi is given by ∞
∞
n=2
n=1
fi (z) = z p − ∑ |ai,n+p−1 |zn+p−1 + ∑ |bi,n+p−1 |¯zn+p−1 . By Theorem 2.3, ∞
∞ n + p (1 − γ) − 1 n + p (1 + γ) − 1 |φ ||a | + n i,n+p−1 ∑ p (1 − γ) ∑ p (1 − γ) |φn ||bi,n+p−1 | ≤ 1. n=2 n=1
(2.7)
For ∑∞ i=1 ti = 1, 0 ≤ ti ≤ 1, the convex combination of f i may be written as, ! ! ∞
∞
∞
∑ ti fi (z) = z p − ∑ ∑ ti |ai,n+p−1 |
i=1
n=2
i=1
∞
zn+p−1 + ∑
n=1
∞
∑ ti |bi,n+p−1 |
z¯n+p−1 .
i=1
Then, by (2.7) ∞
! ∞ ∑ ti |ai,n+p−1 | i=1 ! ∞ [n + p (1 + γ) − 1]|φn | ∞ +∑ ∑ ti |bi,n+p−1 | p (1 − γ) n=1 i=1
[n + p (1 − γ) − 1]|φn | ∑ p (1 − γ) n=2
= ∑ ti i=1 ∞
! ∞ [n + p (1 − γ) − 1]|φn | [n + p (1 + γ) − 1]]|φn | |ai,n+p−1 | + ∑ |bi,n+p−1 | ∑ p (1 − γ) p (1 − γ) n=1 n=2 ∞
∞
≤ ∑ ti = 1. i=1
∗ Hence, ∑∞ i=1 ti f i (z) ∈ TH (p, α1 , γ). In the last theorem below we give distortion inequalities for f in the class TH∗ (p, α1 , γ).
Theorem 2.6. If f ∈ TH∗ (p, α1 , γ) with φn ≥ φ2 , then for |z| = r < 1, | f (z)| ≤ (1 + |b p |) r p + r p+1
p (1 + γ)|b p | p (1 − γ) − [p (1 − γ) + 1]|φ2 | [p (1 − γ) + 1]|φ2 |
and p
| f (z)| ≥ (1 − |b p |) r − r
p+1
p (1 + γ)|b p | p (1 − γ) − . [p (1 − γ) + 1]|φ2 | [p (1 − γ) + 1]|φ2 |
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Proof. Since ∞ p (1 − γ) + 1 |φ2 | ∑ (|an+p−1 | + |bn+p−1 |) p (1 − γ) n=2 ∞
n + p (1 − γ) − 1 (|an+p−1 | + |bn+p−1 |) |φn | p (1 − γ) n=2 ∞ n + p (1 − γ) − 1 n + p (1 + γ) − 1 ≤∑ |an+p−1 | + |bn+p−1 | |φn |, p (1 − γ) p (1 − γ) n=2 ≤
∑
the result of Theorem 2.3 gives ∞
∑ (|an+p−1 | + |bn+p−1 |) ≤
(2.8)
n=2
p (1 − γ) 1+γ 1− |b p | . [p (1 − γ) + 1]|φ2 | 1−γ
Next, again since f ∈ TH∗ (p, α1 , γ), we have from (2.8) and |z| = r that ∞ ∞ p | f (z)| = z − ∑ |an+p−1 |zn+p−1 + ∑ |bn+p−1 |¯zn+p−1 n=2 n=1 ∞
∞
≤ |z p | + ∑ |an+p−1 | |z|n+p−1 + ∑ |bn+p−1 | |¯z|n+p−1 n=2 ∞
n=1
∞
= r p + ∑ |an+p−1 |rn+p−1 + ∑ |bn+p−1 |rn+p−1 n=2
n=1
!
∞
p
≤ (1 + |b p |)r +
∑ (|an+p−1 | + |bn+p−1 |)
r p+1
n=2 p
≤ (1 + |b p |)r + r
p+1
p (1 + γ)|b p | p (1 − γ) − [p (1 − γ) + 1]|φ2 | [p (1 − γ) + 1]|φ2 |
which gives the first result. In a similar manner, we obtain the following lower bound. ∞
∞
| f (z)| ≥ r p − ∑ |an+p−1 |rn+p−1 − ∑ |bn+p−1 |rn+p−1 n=2
n=1
∞
= (1 − |b p |)r p − ∑ (|an+p−1 | + |bn+p−1 |) rn+p−1 n=2
p
≥ (1 − |b p |)r − r
p+1
p (1 + γ)|b p | p (1 − γ) − . [p (1 − γ) + 1]|φ2 | [p (1 − γ) + 1]|φ2 |
Acknowledgement. The authors thank the referees for useful comments and suggestions. This research was supported by a grant from University Malaya (IPPP/UPGP/geran(RU/ PPP)/PS207/2009A). References [1] O. P. Ahuja, Use of theory of conformal mappings in harmonic univalent mappings with directional convexity, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 3, 775–784. [2] O. P. Ahuja and J. M. Jahangiri, Multivalent harmonic starlike functions, Ann. Univ. Mariae CurieSkłodowska Sect. A 55 (2001), 1–13.
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