Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 541371, 7 pages http://dx.doi.org/10.1155/2014/541371
Research Article On Certain Subclass of Meromorphic Spirallike Functions Involving the Hypergeometric Function Lei Shi and Zhi-Gang Wang School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China Correspondence should be addressed to Lei Shi;
[email protected] Received 23 March 2014; Revised 11 May 2014; Accepted 12 May 2014; Published 25 May 2014 Academic Editor: Jin-Lin Liu Copyright © 2014 L. Shi and Z.-G. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce and investigate a new subclass M1𝑚 (𝜃, 𝜆, 𝜂) of meromorphic spirallike functions. Such results as integral representations, convolution properties, and coefficient estimates are proved. The results presented here would provide extensions of those given in earlier works. Several other results are also obtained.
For 𝜃 which is real with |𝜃| < 𝜋/2, 0 ≤ 𝛾 < 1, we denote by MS∗ (𝜃, 𝛾) and MK∗ (𝜃, 𝛾) the subclasses of 𝑓 ∈ M which are defined, respectively, by
1. Introduction Let M denote the class of functions 𝑓 of the form 𝑓 (𝑧) =
1 ∞ + ∑ 𝑎 𝑧𝑛 , 𝑧 𝑛=1 𝑛
(1)
R (𝑒𝑖𝜃
which are analytic in the punctured open unit disk: ∗
U := {𝑧 : 𝑧 ∈ C, 0 < |𝑧| < 1} =: U \ {0} .
(2)
(3)
Then the Hadamard product (or convolution) 𝑓 ∗ 𝑔 of 𝑓 and 𝑔 is defined by (𝑓 ∗ 𝑔) (𝑧) =
1 ∞ + ∑ 𝑎 𝑏 𝑧𝑛 := (𝑔 ∗ 𝑓) (𝑧) . 𝑧 𝑛=1 𝑛 𝑛
(4)
Let P denote the class of functions 𝑝 given by ∞
𝑛
𝑝 (𝑧) = 1 + ∑ 𝑝𝑛 𝑧
(𝑧 ∈ U) ,
(5)
which are analytic in U and satisfy the condition (𝑧 ∈ U) .
𝑖𝜃
R (𝑒
(𝑧𝑓 (𝑧)) 𝑓 (𝑧)
(7) ) < −𝛾 cos 𝜃
∗
(𝑧 ∈ U ) .
By setting 𝜃 = 0 in (7), we get the well-known subclasses of 𝑓 ∈ M consisting of meromorphic functions which are starlike and convex of order 𝛾 (0 ≤ 𝛾 < 1), respectively. For some recent investigations on meromorphic spirallike functions and related topics, see, for example, the earlier works [1–4] and the references cited therein. For 𝜂 > 1, Wang et al. [5] and Nehari and Netanyahu [6] introduced and studied the subclass M(𝜂) of M consisting of functions 𝑓 satisfying R(
𝑛=1
R (𝑝 (𝑧)) > 0
(𝑧 ∈ U∗ ) ,
Let 𝑓, 𝑔 ∈ M, where 𝑓 is given by (1) and 𝑔 is defined by 1 ∞ 𝑔 (𝑧) = + ∑ 𝑏𝑛 𝑧𝑛 . 𝑧 𝑛=1
𝑧𝑓 (𝑧) ) < −𝛾 cos 𝜃 𝑓 (𝑧)
𝑧𝑓 (𝑧) ) > −𝜂 𝑓 (𝑧)
(𝑧 ∈ U∗ ) .
(8)
Let A be the class of functions of the form ∞
(6)
𝑓 (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑧𝑛 , 𝑛=1
(9)
2
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which are analytic in U. A function 𝑓 ∈ A is said to be in the class S∗ (𝜃, 𝛿, 𝛾) if it satisfies the condition
L (𝑎, 𝑐) 𝑓 (𝑧) := H [𝛼1 , 1; 𝛽1 ] 𝑓 (𝑧)
𝑒𝑖𝜃 𝑧𝑓 (𝑧) R( ) > 𝛾 cos 𝜃 (1 − 𝛿) 𝑓 (𝑧) + 𝛿𝑧𝑓 (𝑧) 𝜋 (𝑧 ∈ U; |𝜃| < ; 0 ≤ 𝛿 < 1; 0 ≤ 𝛾 < 1) . 2
(10)
The function class S∗ (𝜃, 𝛿, 𝛾) is introduced and studied recently by Orhan et al. [7]. An analogous of the class S∗ (𝜃, 𝛿, 𝛾) has been studied by Murugusundaramoorthy [8]. For complex parameters 𝛼1 , . . . , 𝛼𝑙 and 𝛽1 , . . . , 𝛽𝑚 (𝛽𝑗 ≠ 0, −1, −2, . . . ; 𝑗 = 1, 2, . . . , 𝑚) the generalized hypergeometric function 𝑙 𝐹𝑚 (𝑧) is defined by 𝑙 𝐹𝑚
we obtain the following linear operator:
(𝑧) ≡ 𝑙 𝐹𝑚 (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ; 𝑧) ∞
(𝛼1 )𝑛 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛 𝑧𝑛 𝑛=0 (𝛽1 )𝑛 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛 𝑛!
:= ∑
Definition 1. For |𝜃| < 𝜋/2, 0 ≤ 𝜆 < 1/2, and 𝜂 > 1, let M𝑙𝑚 (𝜃, 𝜆, 𝜂) denote a subclass of M consisting of functions satisfying the condition that
R(
where N denotes the set of all positive integers and (𝑎)𝑘 is the Pochhammer symbol defined by (𝑎)𝑛 = {
1, 𝑛 = 0, 𝑎 (𝑎 + 1) (𝑎 + 2) ⋅ ⋅ ⋅ (𝑎 + 𝑛 − 1) , 𝑛 ∈ N; 𝑎 ∈ C. (12)
For a function 𝑓 ∈ M, we consider a linear operator (which is a meromorphically modified version of the familiar Dziok-Srivastava linear operator [9, 10]: H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) = H (𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ) 𝑓 (𝑧) −1
= [𝑧
𝑙 𝐹𝑚
(𝛼1 , . . . , 𝛼𝑙 ; 𝛽1 , . . . , 𝛽𝑚 ; 𝑧)] (13)
∗ 𝑓 (𝑧) ,
𝑒𝑖𝜃 (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) > −𝜂 cos 𝜃,
1 ∞ H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) = + ∑ 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎𝑛 𝑧𝑛 , 𝑧 𝑛=1
)
(18)
(𝑧 ∈ U∗ ) ,
where H𝑙𝑚 [𝛼, 𝛽]𝑓 is given by (13). We note that, for 𝑙 = 2, 𝑚 = 1, 𝛼1 = 𝛼2 = 1, 𝛽1 = 1, and 𝜃 = 𝜆 = 0, the class M21 (0, 0, 𝜂) becomes the class M(𝜂). In the present paper, we aim at proving some interesting properties such as integral representations, convolution properties, and coefficient estimates for the class M𝑙𝑚 (𝜃, 𝜆, 𝜂). The following lemma will be required in our investigation. Lemma 2. Suppose that the sequence {𝐴 𝑛 }∞ 𝑛=1 is defined by 𝐴1 =
where 𝛼𝑠 > 0, 𝛽𝑡 > 0 (𝑠 = 1, . . . , 𝑙; 𝑡 = 1, . . . , 𝑚; 𝑙 ≤ 𝑚 + 1; 𝑙, 𝑚 ∈ N0 ). From the definition of the operator H𝑙𝑚 [𝛼, 𝛽]𝑓, it is easy to observe that
(17)
which was introduced and investigated earlier by Liu and Srivastava [14] and was further studied in a subsequent investigation by Srivastava et al. [15]. It should also be remarked that the linear operator H𝑙𝑚 [𝛼, 𝛽] is a generalization of other linear operators considered in many earlier investigations (see, e.g., [16–18]). Using the operator H𝑙𝑚 [𝛼, 𝛽]𝑓, we introduce the following class of meromorphic functions.
(11)
(𝑙 ≤ 𝑚 + 1; 𝑙, 𝑚 ∈ N0 := N ∪ {0} ; 𝑧 ∈ U) ,
(𝑧 ∈ U∗ ) ,
𝐴 𝑛+1 =
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 , (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚) 2 (𝜂 − 1) cos 𝜃 (𝑛 + 2) (1 − 𝜆) 𝜙𝑛+1 (𝛼, 𝛽; 𝑙; 𝑚) 𝑛
× [1 − 2𝜆 + ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝐴 𝑘 ] .
(14)
𝑘=1
(19)
where (𝛼1 )𝑛+1 ⋅ ⋅ ⋅ (𝛼𝑙 )𝑛+1
1 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) = (𝛽1 )𝑛+1 ⋅ ⋅ ⋅ (𝛽𝑚 )𝑛+1 (𝑛 + 1)!
Then (15)
is a positive number for all 𝑛 ∈ N. Recently, Aouf [11], Liu and Srivastava [12], and Raina and Srivastava [13] obtained many interesting results involving the linear operator H𝑙𝑚 [𝛼, 𝛽]. In particular, for 𝑙 = 2,
𝑚 = 1,
𝛼1 = 𝑎,
𝛽1 = 𝑐,
𝛼2 = 1, (16)
𝐴𝑛 =
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 (1 − 𝜆)𝑛 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑛−1
(𝑘 + 1) (1 − 𝜆) + 2 (1 − 𝜆 + 𝑘𝜆) (𝜂 − 1) cos 𝜃 , 𝑘+2 𝑘=1
×∏
(𝑛 ≥ 2) . (20)
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3 Theorem 3. Let 𝑓 ∈ M𝑙𝑚 (𝜃, 𝜆, 𝜂). Then
Proof. From (19), we have (𝑛 + 2) (1 − 𝜆) 𝜙𝑛+1 (𝛼, 𝛽; 𝑙; 𝑚) 𝐴 𝑛+1
∞
𝑓 (𝑧) = (𝑧−1 + ∑ 𝜙𝑛−1 (𝛼, 𝛽; 𝑙, 𝑚) 𝑧𝑛 ) 𝑛=1
= 2 (𝜂 − 1) cos 𝜃 [1 − 2𝜆
𝑧
∗ [𝑧−1 ⋅ exp (∫ ( ( (1 − 2𝜆) (𝜂 − 1) 0
𝑛
+ ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝐴 𝑘 ] ,
× (1 + 𝑒−2𝑖𝜃 ) 𝜔 (𝑡))
𝑘=1
(𝑛 + 1) (1 − 𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝐴 𝑛
(24)
× ( [(1 − 𝜆) (1 − 𝜔 (𝑡)) − 𝜆 (𝜂 − 1) (1 + 𝑒−2𝑖𝜃 )
= 2 (𝜂 − 1) cos 𝜃 [1 − 2𝜆
−1
𝑛−1
× 𝜔 (𝑡)] 𝑡) ) 𝑑𝑡)] ,
𝑘=1
(𝑧 ∈ U∗ ) ,
+ ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝐴 𝑘 ] . (21)
where 𝜔 is analytic in U with 𝜔(0) = 0 and |𝜔(𝑧)| < 1.
Combining (21), we find that 𝐴 𝑛+1 (𝑛 + 1) (1 − 𝜆) + 2 (1 − 𝜆 + 𝑛𝜆) (𝜂 − 1) cos 𝜃 = 𝐴𝑛 (𝑛 + 2) (1 − 𝜆) 𝜙 (𝛼, 𝛽; 𝑙; 𝑚) ⋅ 𝑛 . 𝜙𝑛+1 (𝛼, 𝛽; 𝑙; 𝑚)
Proof. Suppose that 𝑓 ∈ M𝑙𝑚 (𝜃, 𝜆, 𝜂) and (22)
𝜏 (𝑧) := (( (𝑒𝑖𝜃 (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) ) × ( (1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)
Thus, for 𝑛 ≥ 2, we deduce from (22) that 𝐴𝑛 = =
𝐴𝑛 𝐴 𝐴 ⋅ ⋅ ⋅ 3 ⋅ 2 ⋅ 𝐴1 𝐴 𝑛−1 𝐴2 𝐴1 𝑛 (1 − 𝜆) + 2 (1 − 𝑛𝜆) (𝜂 − 1) cos 𝜃 ⋅⋅⋅ (𝑛 + 1) (1 − 𝜆) ×
3 (1 − 𝜆) + 2 (1 + 𝜆) (𝜂 − 1) cos 𝜃 4 (1 − 𝜆)
2 (1 − 𝜆) + 2 (𝜂 − 1) cos 𝜃 ⋅ 3 (1 − 𝜆)
=
−1
+𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) ) )
(25)
+𝜂 cos 𝜃 + 𝑖 sin 𝜃) × ((𝜂 − 1) cos 𝜃)
−1
,
(𝑧 ∈ U) . We know that 𝜏 ∈ P, which implies
(( (𝑒𝑖𝜃 (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) )
⋅
𝜙𝑛−1 (𝛼, 𝛽; 𝑙; 𝑚) 𝜙 (𝛼, 𝛽; 𝑙; 𝑚) ⋅⋅⋅ 2 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝜙3 (𝛼, 𝛽; 𝑙; 𝑚)
×((1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) ) )
⋅
𝜙1 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 2𝜆) (𝜂 − 1) cos 𝜃 ⋅ 𝜙2 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚)
+ 𝜂 cos 𝜃 + 𝑖 sin 𝜃) × ((𝜂 − 1) cos 𝜃)
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 (1 − 𝜆)𝑛 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚)
−1
−1
=
1 + 𝜔 (𝑧) , 1 − 𝜔 (𝑧)
(26)
𝑛−1
(𝑘 + 1) (1 − 𝜆) + 2 (1 − 𝜆 + 𝑘𝜆) (𝜂 − 1) cos 𝜃 ×∏ . 𝑘+2 𝑘=1 (23)
where 𝜔 is analytic in U with 𝜔(0) = 0 and |𝜔(𝑧)| < 1. We find from (26) that (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
This completes the proof of Lemma 2.
2. Main Results We begin by proving the following integral representation for the class M𝑙𝑚 (𝜃, 𝜆, 𝜂).
(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) =
−1 + [1 + (𝜂 − 1) (1 + 𝑒−2𝑖𝜃 )] 𝜔 (𝑧) 1 − 𝜔 (𝑧)
,
(27)
4
The Scientific World Journal Proof. From the definition (18), we know that 𝑓 M𝑙𝑚 (𝜃, 𝜆, 𝜂) if and only if
which follows
(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)
+
(( (𝑒𝑖𝜃 (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) )
1 𝑧 −2𝑖𝜃
(1 − 2𝜆) (𝜂 − 1) (1 + 𝑒
=
−1
×((1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) ) )
) 𝜔 (𝑧)
. 𝑧 [(1 − 𝜆) (1 − 𝜔 (𝑧)) − 𝜆 (𝜂 − 1) (1 + 𝑒−2𝑖𝜃 ) 𝜔 (𝑧)] (28)
1+𝜉 , 1−𝜉 (𝑧 ∈ U∗ ; 𝜉 = 1) , −1
+𝜂 cos 𝜃 + 𝑖 sin 𝜃) × ((𝜂 − 1) cos 𝜃)
Integrating both sides of (28) yields
0
(32)
− (1 − 𝜉) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) + [𝜉𝜂 + 𝜉 (𝜂 − 1) 𝑒−2𝑖𝜃 − 1]
(1 − 2𝜆) (𝜂 − 1) (1 + 𝑒−2𝑖𝜃 ) 𝜔 (𝑡)
𝑧
≠
which is equivalent to
log (𝑧H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) =∫
∈
[(1 − 𝜆) (1 − 𝜔 (𝑡)) − 𝜆 (𝜂 − 1) (1 +
𝑒−2𝑖𝜃 ) 𝜔 (𝑡)] 𝑡
𝑑𝑡.
(29)
⋅
(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) 1 − 2𝜆
≠ 0. (33)
On the other hand, we find from (14) that
From (29), we obtain
− 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)
=
𝑧
= 𝑧−1 ⋅ exp (∫ ( ((1 − 2𝜆) (1 + 𝑒−2𝑖𝜃 ) (𝜂 − 1) 𝜔 (𝑡)) 0
× ([ (1 − 𝜆) (1 − 𝜔 (𝑡)) − 𝜆 (𝜂 − 1)
1 ∞ − ∑ 𝑛𝜙 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎𝑛 𝑧𝑛 𝑧 𝑛=1 𝑛
1 ∞ = 𝑓 (𝑧) ∗ [ − ∑ 𝑛𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑧𝑛 ] , 𝑧 𝑛=1
−1
× (1 + 𝑒−2𝑖𝜃 ) 𝜔 (𝑡)] 𝑡) ) 𝑑𝑡) .
(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) 1 − 2𝜆
(30) = Thus, the assertion (24) of Theorem 3 follows directly from (30).
Next, we derive a convolution property for the class M𝑙𝑚 (𝜃, 𝜆, 𝜂). Theorem 4. Let 𝜉 ∈ C and |𝜉| = 1. Then 𝑓 ∈ M𝑙𝑚 (𝜃, 𝜆, 𝜂) if and only if
(31)
(𝑧 ∈ U∗ ) .
1 ∞ 1 − 𝜆 + 𝑛𝜆 = 𝑓 (𝑧) ∗ [ + ∑ 𝜙 (𝛼, 𝛽; 𝑙; 𝑚) 𝑧𝑛 ] . 𝑧 𝑛=1 1 − 2𝜆 𝑛 Combining (33) and (34), we get assertion (31) of Theorem 4. Now, we discuss the coefficient estimates for functions in the class M𝑙𝑚 (𝜃, 𝜆, 𝜂). Theorem 5. Suppose that 𝑓 ∈ M𝑙𝑚 (𝜃, 𝜆, 𝜂). Then
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 𝑎𝑛 ≤ (1 − 𝜆)𝑛 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚)
+ [𝜉𝜂 + 𝜉 (𝜂 − 1) 𝑒−2𝑖𝜃 − 1] 1 − 𝜆 + 𝑛𝜆 1 ⋅[ + ∑ 𝜙 (𝛼, 𝛽; 𝑙; 𝑚) 𝑧𝑛 ]} ≠ 0, 𝑧 𝑛=1 1 − 2𝜆 𝑛
1 ∞ 1 − 𝜆 + 𝑛𝜆 +∑ 𝜙 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎𝑛 𝑧𝑛 𝑧 𝑛=1 1 − 2𝜆 𝑛
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 , 𝑎1 ≤ (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚)
1 ∞ 𝑓 ∗ {(1 − 𝜉) ⋅ [ − ∑ 𝑛𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑧𝑛 ] 𝑧 𝑛=1
∞
(34)
𝑛−1
(𝑘 + 1) (1 − 𝜆) + 2 (1 − 𝜆 + 𝑘𝜆) (𝜂 − 1) cos 𝜃 , 𝑘+2 𝑘=1
×∏
(𝑛 ∈ N \ {1}) . (35)
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Proof. Let 𝑓 ∈ M𝑙𝑚 (𝜃, 𝜆, 𝜂). Then there exists 𝜏 ∈ P such that
Now we define the sequence {𝐴 𝑛 }∞ 𝑛=1 as follows:
𝑒𝑖𝜃 (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
𝐴1 =
(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) = (𝜂 − 1) cos 𝜃𝜏 (𝑧) − 𝜂 cos 𝜃 − 𝑖 sin 𝜃,
(36) 𝐴 𝑛+1 =
(𝑧 ∈ U∗ ) .
𝑒 (1 −
2𝜆) 𝑧(H𝑙𝑚
2 (𝜂 − 1) cos 𝜃 (𝑛 + 2) (1 − 𝜆) 𝜙𝑛+1 (𝛼, 𝛽; 𝑙; 𝑚) 𝑛
× [1 − 2𝜆 + ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝐴 𝑘 ] .
It follows from (36) that 𝑖𝜃
𝑘=1
(42)
[𝛼, 𝛽] 𝑓 (𝑧))
= [(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) ] (37)
In order to prove that 𝑎𝑛 ≤ 𝐴 𝑛
× [(𝜂 − 1) cos 𝜃𝜏 (𝑧) − 𝜂 cos 𝜃 − 𝑖 sin 𝜃] .
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 . 𝑎1 ≤ 𝐴 1 = (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚)
1 ∞ 𝑒𝑖𝜃 (1 − 2𝜆) [− + ∑ 𝑛𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎𝑛 𝑧𝑛 ] 𝑧 𝑛=1 1 ∞ + ∑ (1 − 𝜆 + 𝑛𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎𝑛 𝑧𝑛 ] 𝑧 𝑛=1
𝑖𝜃
(43)
⋅ [−𝑒 + (𝜂 − 1) cos 𝜃𝜏1 𝑧 + (𝜂 − 1) cos 𝜃𝜏2 𝑧 + ⋅ ⋅ ⋅ ] . (38) Evaluating the coefficient of 𝑧𝑛 in both sides of (38) yields 2𝑒𝑖𝜃 (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎1 = (1 − 2𝜆) (𝜂 − 1) cos 𝜃𝜏2 ,
(44)
Therefore, assume that 𝑎𝑘 ≤ 𝐴 𝑘
2
(𝑘 = 1, 2, . . . , 𝑛; 𝑛 ∈ N) .
(45)
Combining (41) and (42), we get 2 (𝜂 − 1) cos 𝜃 𝑎𝑛+1 ≤ (𝑛 + 2) (1 − 𝜆) 𝜙𝑛+1 (𝛼, 𝛽; 𝑙; 𝑚) 𝑛
× [1 − 2𝜆 + ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝑎𝑘 ]
𝑖𝜃
𝑒 (𝑛 + 1) (1 − 𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑎𝑛
𝑘=1
= (𝜂 − 1) cos 𝜃 [ (1 − 2𝜆) 𝜏𝑛+1
≤
2 (𝜂 − 1) cos 𝜃 (𝑛 + 2) (1 − 𝜆) 𝜙𝑛+1 (𝛼, 𝛽; 𝑙; 𝑚)
𝑛−1
+ ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝑎𝑘 𝜏𝑛−𝑘 ] , 𝑘=1
𝑛
× [1 − 2𝜆 + ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝐴 𝑘 ] 𝑘=1
(𝑛 ≥ 2) . (39) By observing the fact that |𝜏𝑛 | ≤ 2 for 𝑛 ∈ N, we find from (39) that (1 − 2𝜆) (𝜂 − 1) cos 𝜃 , 𝑎1 ≤ (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚)
(𝑛 ∈ N) ,
we use the principle of mathematical induction. Note that
Combining (1) and (37), we have
= [(1 − 2𝜆)
(1 − 2𝜆) (𝜂 − 1) cos 𝜃 , (1 − 𝜆) 𝜙1 (𝛼, 𝛽; 𝑙; 𝑚)
(40)
= 𝐴 𝑛+1 . (46) Hence, by the principle of mathematical induction, we have 𝑎𝑛 ≤ 𝐴 𝑛
(𝑛 ∈ N) ,
(47)
as desired. By means of Lemma 2 and (42), we know that (20) holds. Combining (47) and (20), we readily get the coefficient estimates asserted by Theorem 5.
2 (𝜂 − 1) cos 𝜃 𝑎𝑛 ≤ (𝑛 + 1) (1 − 𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙; 𝑚) 𝑛−1
× [1 − 2𝜆 + ∑ 𝜙𝑘 (𝛼, 𝛽; 𝑙; 𝑚) (1 − 𝜆 + 𝑘𝜆) 𝑎𝑘 ] , 𝑘=1
(𝑛 ≥ 2) . (41)
Remark 6. By setting 𝜃 = 0, 𝑙 = 2, 𝑚 = 1, 𝛼1 = 𝛼2 = 𝛽1 = 1, and 𝜆 = 0 in Theorem 5, we get the corresponding result due to Wang et al. [5]. In what follows, we present some sufficient conditions for functions belonging to the class M𝑙𝑚 (𝜃, 𝜆, 𝜂).
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Theorem 7. Let 𝜁 be a real number with 0 ≤ 𝜁 < 1. If 𝑓 ∈ M satisfies the condition (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) + 1 ≤ 1 − 𝜁, (1 − 𝜆) H𝑙 𝑓 (𝑧) + 𝜆𝑧(H𝑙 [𝛼, 𝛽] 𝑓 (𝑧)) 𝑚 𝑚 (𝑧 ∈ U∗ ) , (48) then 𝑓 ∈
M𝑙𝑚 (𝜃, 𝜆, 𝜂)
provided that cos 𝜃 ≥
1−𝜁 . 𝜂−1
(49)
Proof. From (48), it follows that
(1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
(50)
where 𝜔 is analytic in U with 𝜔(0) = 0 and |𝜔(𝑧)| < 1. Thus, we have (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) (1 −
𝜆) H𝑙𝑚
[𝛼, 𝛽] 𝑓 (𝑧) −
𝑛+1 ∑∞ 𝑛=1 (1 + 𝑛) 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 𝑧 × ∞ 1 − 2𝜆 + ∑𝑛=1 (1 − 𝜆 + 𝑛𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 𝑧𝑛+1 ∞ (1 − 𝜆) ∑𝑛=1 (1 + 𝑛) 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 < . 1 − 2𝜆 − ∑∞ 𝑛=1 (1 − 𝜆 + 𝑛𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 (54) ∞
(1 − 𝜆) ∑ (1 + 𝑛) 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 𝑛=1
= −1 + (1 − 𝜁) 𝜔 (𝑧) ,
R (𝑒
= (1 − 𝜆)
The last expression is bounded by (𝜂 − 1) cos 𝜃, if
(1 − 𝜆) H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
𝑖𝜃
Proof. In virtue of Corollary 8, it suffices to show that condition (52) holds. We observe that (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) + 1 𝑙 𝑙 (1 − 𝜆) H𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑚 [𝛼, 𝛽] 𝑓 (𝑧))
𝜆𝑧(H𝑙𝑚
[𝛼, 𝛽] 𝑓 (𝑧))
𝑛=1
(55) which is equivalent to
𝑖𝜃
= − cos 𝜃 + (1 − 𝜁) R (𝑒 𝜔 (𝑧))
∞
∑ [(1 − 𝜆) (1 + 𝑛) sec 𝜃 + (𝜂 − 1) (1 − 𝜆 + 𝑛𝜆)]
≥ − cos 𝜃 − (1 − 𝜁) |𝜔 (𝑧)|
𝑛=1
This completes the proof of Theorem 9.
≥ −𝜂 cos 𝜃, (51) provided that cos 𝜃 ≥ (1 − 𝜁)/(𝜂 − 1). This completes the proof of Theorem 7. If we take 𝜁 = 1 − (𝜂 − 1) cos 𝜃 in Theorem 7, we obtain the following result. Corollary 8. If 𝑓 ∈ M satisfies the inequality (1 − 2𝜆) 𝑧(H𝑙𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) + 1 (52) 𝑙 𝑙 (1 − 𝜆) H𝑚 [𝛼, 𝛽] 𝑓 (𝑧) + 𝜆𝑧(H𝑚 [𝛼, 𝛽] 𝑓 (𝑧)) ≤ (𝜂 − 1) cos 𝜃, then 𝑓 ∈ M𝑙𝑚 (𝜃, 𝜆, 𝜂). Theorem 9. If a function 𝑓 ∈ M given by (1) satisfies the inequality ∞
∑ [(1 − 𝜆) (1 + 𝑛) sec 𝜃 + (𝜂 − 1) (1 − 𝜆 + 𝑛𝜆)] × 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 ≤ (1 − 2𝜆) (𝜂 − 1) ,
(56)
× 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 ≤ (1 − 2𝜆) (𝜂 − 1) .
> − cos 𝜃 − (1 − 𝜁)
𝑛=1
∞
− ∑ (1 − 𝜆 + 𝑛𝜆) 𝜙𝑛 (𝛼, 𝛽; 𝑙, 𝑚) 𝑎𝑛 ] ,
)
= R (𝑒𝑖𝜃 (−1 + (1 − 𝜁) 𝜔 (𝑧)))
then it belongs to the class M𝑙𝑚 (𝜃, 𝜆, 𝜂).
≤ (𝜂 − 1) cos 𝜃 [1 − 2𝜆
(53)
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008 and 11226088, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110012 of China.
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