Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2 Issue 1(2010), Pages 56-65.
A CLASS OF MULTIVALENT ANALYTIC FUNCTIONS WITH
FIXED ARGUMENT OF COEFFICIENTS INVOLVING
WRIGHT’S GENERALIZED HYPERGEOMETRIC FUNCTIONS
POONAM SHARMA
Abstract. In this paper, a class of analytic functions with fixed argument
of its coefficients involving Wright’s generalized hypergeometric function is
defined with the help of subordination. The coefficient inequalities have been
derived. Growth, distortion bounds and extreme points for functions belonging
to the defined class have been investigated with consequent results.
1. Introduction and Preliminaries
Let A𝑚 denote a class of functions 𝑓 (𝑧) of the form:
𝑓 (𝑧) = 𝑧
𝑚
+
∞
∑
𝑎𝑚+𝑘 𝑧 𝑚+𝑘 (𝑎𝑚+𝑘 ∈ C, 𝑚 ∈ N = {1, 2, 3...}),
(1.1)
𝑘=1
which are analytic in an open unit disk U := {𝑧 : 𝑧 ∈ C and ∣𝑧∣ < 1} and its
subclass is denoted by A𝜃𝑚 whose members are of the form:
𝑓 (𝑧) = 𝑧 𝑚 + 𝑒𝑖𝜃
∞
∑
∣𝑎𝑚+𝑘 ∣𝑧 𝑚+𝑘 (𝑚 ∈ N = {1, 2, 3...}),
(1.2)
𝑘=1
where 𝜃 is the fixed argument of 𝑎𝑚+𝑘 ∕= 0 (𝑘 ≥ 1). Note that A1 ≡ A.
A function 𝑓 (𝑧) ∈ A𝑚 is said to be in the class S∗𝑚 (𝛼) if it satisfies
{ ′
}
𝑧𝑓 (𝑧)
Re
> 𝛼, (0 ≤ 𝛼 < 1, 𝑧 ∈ U)
𝑓 (𝑧)
and functions therein are called starlike of order 𝛼.
The Wright’s (psi) function 𝑝 Ψ𝑞 (𝑧) is a generalized hypergeometric function [12]
whose series representation is given by
2000 Mathematics Subject Classification. Primary 30C45, 30C55; Secondary 26A33, 33C60.
Key words and phrases. Analytic functions; starlike functions; Wright’s generalized hypergeometric function; subordination.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
56
A CLASS OF MULTIVALENT ANALYTIC FUNCTIONS WITH FIXED ARGUMENT
𝑝
∏
Γ(𝑎𝑖 + 𝐴𝑖 𝑘)𝑧 𝑛
∞
∑
𝑖=1
,
𝑝 Ψ𝑞 ((𝑎1 , 𝐴1 ), ..., (𝑎𝑝 , 𝐴𝑝 ); (𝑏1 , 𝐵1 ), ..., (𝑏𝑞 , 𝐵𝑞 ); 𝑧) =
𝑞
∏
𝑘=0
Γ(𝑏𝑖 + 𝐵𝑖 𝑘)𝑘!
57
(1.3)
𝑖=1
where 𝑎𝑖 (𝑖 = 1, 2, ..., 𝑝), 𝑏𝑖 (𝑖 = 1, 2, ..., 𝑞) are positive real numbers and 𝐴𝑖 (𝑖 =
𝑞
𝑝
∑
∑
1, 2, ..., 𝑝), 𝐵𝑖 (𝑖 = 1, 2, ..., 𝑞) are positive integers such that 1 +
𝐵𝑖 −
𝐴𝑖 ≥ 0.
𝑖=1
𝑖=1
𝑞
∏
Taking 𝐴𝑖 = 1(𝑖 = 1, 2, ..., 𝑝), 𝐵𝑖 = 1(𝑖 = 1, 2, ..., 𝑞), we see that
Γ(𝑏𝑖 )
𝑖=1
𝑝
∏
𝑝 Ψ𝑞 (𝑧)
Γ(𝑎𝑖 )
𝑖=1
reduces to a familiar generalized hypergeometric function:
𝑝 F𝑞 (𝑧)
≡𝑝 F𝑞 (𝑎1 , ..., 𝑎𝑝 ; 𝑏1 , ..., 𝑏𝑞 ; 𝑧) =
∞
∑
(𝑎1 )𝑘 ....(𝑎𝑝 )𝑘 𝑧 𝑘
,
(𝑏1 )𝑘 ...(𝑏𝑞 )𝑘 𝑘!
𝑘=0
(𝑎)𝑘 = Γ(𝑎+𝑘)
Γ(𝑎) for non-negative
(1.4)
where the Pochhammer symbol is defined by
integrs
𝑘.
Similar to the operator defined earlier by Dziok and Raina [4] (see also [1, 2],
[6] and [5]), involving psi function 𝑞+1 Ψ𝑞 (𝑧) for positive real numbers 𝑎𝑖 , 𝑏𝑖 (𝑖 =
𝑞
∑
1, 2, ..., 𝑞) and for positive integers 𝐴𝑖 , 𝐵𝑖 (𝑖 = 1, 2, ..., 𝑞) such that
(𝐵𝑖 − 𝐴𝑖 ) ≥ 0,
𝑖=1
𝑞
𝑞
((𝑎𝑖 , 𝐴𝑖 )1,𝑞 , (𝑏𝑖 , 𝐵𝑖 )1,𝑞 ) 𝑓 : A𝑚 → A𝑚 is defined as:
([𝑎1 ]) 𝑓 ≡ 𝐼𝑚
an operator 𝐼𝑚
𝑞
𝐼𝑚
([𝑎1 ]) 𝑓 = 𝑧
𝑚
𝑞
∏
Γ(𝑏𝑖 )
𝑖=1
Γ(𝑎𝑖 )
𝑞+1 Ψ𝑞 (𝑧)
∗ 𝑓,
(1.5)
where
𝑞+1 Ψ𝑞 (𝑧)
≡ 𝑞+1 Ψ𝑞 ((𝑎1 , 𝐴1 ), ..., (𝑎𝑞 , 𝐴𝑞 ), (1, 1); (𝑏1 , 𝐵1 ), ..., (𝑏𝑞 , 𝐵𝑞 ); 𝑧) .
𝑞
([𝑎1 ]) 𝑓 is
Let 𝑓 (𝑧) ∈ A𝑚 be of the form (1.1), then the series expansion of 𝐼𝑚
given by
𝑞
𝐼𝑚
([𝑎1 ]) 𝑓 = 𝑧 𝑚 +
∞
∑
𝜃𝑚+𝑘 𝑎𝑚+𝑘 𝑧 𝑚+𝑘 ,
(1.6)
𝑘=1
where
𝜃𝑚+𝑘 =
𝑞
∏
Γ(𝑎𝑖 + 𝐴𝑖 𝑘)Γ(𝑏𝑖 )
𝑖=1
Γ(𝑏𝑖 + 𝐵𝑖 𝑘)Γ(𝑎𝑖 )
, 𝑘 ≥ 1.
(1.7)
𝑞
The series expansion of 𝐼𝑚
([𝑎1 + 1]) 𝑓 ≡ 𝐼𝑚 [(𝑎1 + 1, 𝐴1 ), (𝑎𝑖 , 𝐴𝑖 )2,𝑞 , (𝑏𝑖 , 𝐵𝑖 )1,𝑞 ] 𝑓 :
A𝑚 → A𝑚 is given by
𝑞
𝐼𝑚
([𝑎1 + 1]) 𝑓 = 𝑧 𝑚 +
∞
∑
+
𝜃𝑚+𝑘
𝑎𝑚+𝑘 𝑧 𝑚+𝑘
𝑘=1
with
(𝑎1 + 𝐴1 𝑘)
𝜃𝑚+𝑘 .
𝑎1
From (1.6) and (1.8), we get an identity:
+
𝜃𝑚+𝑘
=
(1.8)
58
P. SHARMA
′
𝑞
𝑞
𝑞
𝐴1 𝑧 (𝐼𝑚
([𝑎1 ]) 𝑓 ) = 𝑎1 𝐼𝑚
([𝑎1 + 1]) 𝑓 − (𝑎1 − 𝑚𝐴1 )𝐼𝑚
([𝑎1 ]) 𝑓
𝑞
𝑚𝐴𝑖 , 𝐼𝑚
which shows that for 𝑎𝑖 =
([𝑎1 + 1]) 𝑓 =
if 𝐴𝑖 = 𝐵𝑖 , 𝑎𝑖 = 𝑏𝑖 (𝑖 = 1, 2, ..., 𝑞).
𝑞
𝑧(𝐼𝑚
([𝑎1 ])𝑓 )′
.
𝑚
Note that
(1.9)
𝑞
𝐼𝑚
𝑓
≡𝑓
With the use of subordination, classes in which the linear operator 𝐼1𝑞 ([𝑎1 ]) 𝑓
𝑞
is involved, are defined and studied in [2] and [6]. Involving 𝐼𝑚
([𝑎1 ]) 𝑓 in [9], a
𝜃
class of functions 𝑓 ∈ A𝑚 with the help of subordination is defined and studied.
𝑞
𝑞
Motivated with the work of Raina [9], involving 𝐼𝑚
([𝑎1 ]) 𝑓 and 𝐼𝑚
([𝑎1 + 1]) 𝑓 ,
we define a class for functions 𝑓 ∈ A𝑚 as follows:
Definition 1.1. Let A𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) denote a class of functions 𝑓 ∈ A𝑚 satisfying
𝑞
𝐼 𝑞 ([𝑎1 + 1]) 𝑓
1 + 𝐴𝑧
([𝑎1 ]) 𝑓
𝐼𝑚
+𝜆 𝑚
≺
,
(1.10)
𝑚
𝑧
𝑧𝑚
1 + 𝐵𝑧
where 0 ≤ 𝜆 ≤ 1, and −1 ≤ 𝐴 < 𝐵 ≤ 1, 0 ≤ 𝐵. Denote A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) ≡
A𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) ∩ A𝜃𝑚 .
(1 − 𝜆)
For positive real 𝑎 and for positive integer 𝐴, we have [[11], 240, Eq. (1.26)]:
(
)
( 𝑎 ) (𝑎 + 1)
𝑎+𝐴−1
𝑘𝐴
Γ(𝑎 + 𝑘𝐴) = Γ(𝑎)
...
(𝐴) , 𝑘 = 0, 1, 2, ...
𝐴 𝑘
𝐴
𝐴
𝑘
𝑘
when it is used together with the result [[3], p.57]:
[
( )]
Γ(𝑎 + 𝑘)Γ(𝑏 + 𝑘)
1
𝑎+𝑏−𝑐−𝑑
=𝑘
1+𝑂
, 𝑘 = 1, 2, 3, ...
Γ(𝑐 + 𝑘)Γ(𝑑 + 𝑘)
𝑘
we obtain that for positive real numbers 𝑎𝑖 , 𝑏𝑖 (𝑖 = 1, 2, ..., 𝑞) and positive integers
𝑞
∞
∑
∑
𝐴𝑖 , 𝐵𝑖 (𝑖 = 1, 2, ..., 𝑞) such that
(𝐵𝑖 − 𝐴𝑖 ) ≥ 0, the series
𝜃𝑚+𝑘 , where 𝜃𝑚+𝑘
𝑖=1
𝑘=1
is given by (1.7), converges absolutly if
𝑞
𝑞
∑
1∑
(𝐴𝑖 − 𝐵𝑖 ).
(𝑏𝑖 − 𝑎𝑖 ) > 1 +
2 𝑖=1
𝑖=1
(1.11)
For details one may refer to [8] and [10].
The purpose of this paper is to investigate the coefficient inequalities, growth
and distortion bounds with certain conditions on parameters and extreme points
for functions belonging to the class A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆). Some consequent results
are also mentioned.
2. Coefficient Inequalities
Theorem 2.1. Let 𝑓 (𝑧) ∈ A𝜃𝑚 of the form (1.2) belong to A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆),
then
∞
∑
𝑎1 (𝐵 − 𝐴)
],
(2.1)
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ < [√
1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
𝑘=1
where 𝜃𝑚+𝑘 is given by (1.7) and 0 ≤ 𝜆 ≤ 1, −1 ≤ 𝐴 < 𝐵 ≤ 1, 0 ≤ 𝐵 , 𝜃 is the
argument of 𝑎𝑚+𝑘 ∕= 0 (𝑘 ≥ 1).
A CLASS OF MULTIVALENT ANALYTIC FUNCTIONS WITH FIXED ARGUMENT
59
Proof. Let a function 𝑓 belong to the class A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆), then from Definition
1.1, we have
𝑞
𝑞
𝐼𝑚
([𝑎1 ]) 𝑓
𝐼𝑚
([𝑎1 + 1]) 𝑓
1 + 𝐴𝑤(𝑧)
+
𝜆
=
,
𝑧𝑚
𝑧𝑚
1 + 𝐵𝑤(𝑧)
where 𝑤(0) = 0 and ∣𝑤(𝑧)∣ < 1 for 𝑧 ∈ U. Hence we have
(1 − 𝜆)
𝑞
𝑞
𝑧 𝑚 − (1 − 𝜆)𝐼𝑚
([𝑎1 ]) 𝑓 − 𝜆𝐼𝑚
([𝑎1 + 1]) 𝑓 < 1, 𝑧 ∈ U.
∣𝑤(𝑧)∣ = 𝑞
𝑞
𝐵 [(1 − 𝜆)𝐼𝑚
([𝑎1 ]) 𝑓 + 𝜆𝐼𝑚
([𝑎1 + 1]) 𝑓 ] − 𝐴𝑧 𝑚 (2.2)
On using (1.6) and (1.8), inequality (2.2) yields
∞
∞
∑
∑
𝑖𝜃
𝑘
𝑘
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ 𝑧 .
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ 𝑧 < 𝑎1 (𝐵 − 𝐴) + 𝐵𝑒
𝑘=1
𝑘=1
(2.3)
∞
∑
For 0 < 𝑧 = 𝑟 < 1,
𝑘
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ 𝑟 =: 𝜂 is real, we get
𝑘=1
∣𝜂 ∣< 𝑎1 (𝐵 − 𝐴) + 𝐵𝑒𝑖𝜃 𝜂 .
On solving (2.4) for 𝜂, we get
(2.4)
𝑎1 (𝐵 − 𝐴)
],
1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
which proves the inequality (2.1).
𝜂 < [√
□
The inequality (2.1) is sharp and the extremal function is given by
𝑓𝑘 (𝑧) = 𝑧 𝑚 + 𝑒𝑖𝜃
𝑎1 (𝐵 − 𝐴)
[√
] 𝑧 𝑚+𝑘 , 𝑘 ≥ 1.
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
(2.5)
Corollary 2.2. Let 𝑓 (𝑧) ∈ A𝜃𝑚 of the form (1.2) belong to A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) then
𝑞
𝑞
∑
∑
for
(𝑏𝑖 − 𝑎𝑖 ) > 1 + 12 (𝐴𝑖 − 𝐵𝑖 ),
𝑖=1
𝑖=1
∞
∑
∣𝑎𝑚+𝑘 ∣ <
𝑘=1
𝑎1 (𝐵 − 𝐴)
[√
],
(𝑎1 + 𝜆𝐴1 )𝜙 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
where 𝜙 = min {𝜃𝑚+𝑘 } , 𝜃𝑚+𝑘 is given by (1.7).
𝑘≥1
Proof. Under the given hypothesis, convergence of the series
∞
∑
𝜃𝑚+𝑘 implies that
𝑘=1
𝜃𝑚+𝑘 , given in (1.7) is bounded for 𝑘 ≥ 1. Let 𝜙 := min {𝜃𝑚+𝑘 }, then by Theorem
𝑘≥1
2.1 we get the result.
□
A𝜃𝑚
Corollary 2.3. Let 𝑓 (𝑧) ∈
of the form (1.2) belong to
𝑞
𝑞
∑
∑
for
(𝑏𝑖 − 𝑎𝑖 ) > 12 (𝐴𝑖 − 𝐵𝑖 ),
A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆)
𝑖=1
𝑖=1
∞
∑
(𝑚 + 𝑘)∣𝑎𝑚+𝑘 ∣ <
𝑘=1
where 𝜑 = min
𝑘≥1
{
𝜃𝑚+𝑘
𝑚+𝑘
}
𝑎1 (𝐵 − 𝐴)
[√
],
(𝑎1 + 𝜆𝐴1 )𝜑 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
, 𝜃𝑚+𝑘 is given by (1.7).
then
60
P. SHARMA
Proof. Under the given hypothesis, convergence of the series
∞
∑
𝑘=1
𝜃𝑚+𝑘
for
𝑚+𝑘{ is bounded
}
𝜃𝑚+𝑘
min 𝑚+𝑘 , then by
𝑘≥1
𝜃𝑚+𝑘
𝑚+𝑘
implies that
𝑘 ≥ 1, where 𝜃𝑚+𝑘 is given by (1.7) and hence let 𝜑 :=
Theorem 2.1 we get the result.
□
Taking 𝜃 = (2𝑛 − 1)𝜋, 𝑛 = 1, 2, 3, .. in Corollary 2.2 we get following result.
(2𝑛−1)𝜋
(2𝑛−1)𝜋
Corollary 2.4. Let 𝑓 (𝑧) ∈ A𝑚
of the form (1.2) belong to A𝑚
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆),
𝑞
𝑞
∑
∑
then for
(𝑏𝑖 − 𝑎𝑖 ) > 1 + 12 (𝐴𝑖 − 𝐵𝑖 ),
𝑖=1
𝑖=1
∞
∑
∣𝑎𝑚+𝑘 ∣ <
𝑘=1
𝑎1 (𝐵 − 𝐴)
,
(𝑎1 + 𝜆𝐴1 )𝜙(1 + 𝐵)
where 𝜙 = min {𝜃𝑚+𝑘 } , 𝜃𝑚+𝑘 is given by (1.7).
𝑘≥1
Remark. Raina [9] studied the class of functions 𝑓 (𝑧) ∈ A𝜃𝑚 and obtained results
𝑚+𝑘
assuming 𝜃𝑚+𝑘 and 𝜃𝑚+𝑘
to be increasing functions.
(2𝑛−1)𝜋
(2𝑛−1)𝜋
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆)
of the form (1.2) belong to A𝑚
Theorem 2.5. A function 𝑓 (𝑧) ∈ A𝑚
if and only if
∞
∑
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ <
𝑘=1
𝑎1 (𝐵 − 𝐴)
,
(1 + 𝐵)
(2.6)
where 𝜃𝑚+𝑘 is given by (1.7) and 0 ≤ 𝜆 ≤ 1, −1 ≤ 𝐴 < 𝐵 ≤ 1, 0 ≤ 𝐵.
(2𝑛−1)𝜋
Proof. We need to show only sufficient condition for 𝑓 ∈ A𝑚
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆).
Consider
∞
∞
∑
∑
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ 𝑧 𝑘 (𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ 𝑧 𝑘 − 𝑎1 (𝐵 − 𝐴) − 𝐵
𝑘=1
≤
𝑘=1
∞
∑
[
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ − 𝑎1 (𝐵 − 𝐴) − 𝐵
𝑘=1
≤
∞
∑
∞
∑
]
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣
𝑘=1
(1+𝐵)(𝑎1 +𝜆𝐴1 𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣−𝑎1 (𝐵 − 𝐴) < 0,
𝑘=1
if (2.6) holds, which proves (2.3) for 𝜃 = (2𝑛 − 1)𝜋 and hence the result.
□
The result is sharp for the function:
𝑓𝑘 (𝑧) = 𝑧 𝑚 −
𝑎1 (𝐵 − 𝐴)
𝑧 𝑚+𝑘 , 𝑘 ≥ 1.
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
(2.7)
A CLASS OF MULTIVALENT ANALYTIC FUNCTIONS WITH FIXED ARGUMENT
61
Corollary 2.6. Let a function 𝑓 (𝑧) ∈ A(2𝑛−1)𝜋
of the form (1.2) belong to
𝑚
A(2𝑛−1)𝜋
(𝑝,
[𝑎
],
𝐴,
𝐵,
𝜆)
then
1
𝑚
𝑎1 (𝐵 − 𝐴)
, 𝑘 ≥ 1.
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
∣𝑎𝑚+𝑘 ∣ <
(2.8)
3. Growth and Distortion Bounds
In this section we find growth and distortion bounds for functions belonging to
the class A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) with the use of Corollaries 2.2 and 2.3.
Theorem 3.1. Let 𝑓 (𝑧) ∈ A𝜃𝑚 of the form (1.2) belong to A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆), then
𝑞
𝑞
∑
∑
for 0 < ∣𝑧∣ = 𝑟 < 1 and for
(𝑏𝑖 − 𝑎𝑖 ) > 1 + 21 (𝐴𝑖 − 𝐵𝑖 )
𝑖=1
𝑖=1
𝑟𝑚 − 𝑟𝑚+1
𝑎1 (𝐵 − 𝐴)
[√
] ≤ ∣𝑓 (𝑧)∣
(𝑎1 + 𝜆𝐴1 )𝜙 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
≤ 𝑟𝑚 + 𝑟𝑚+1
𝑎1 (𝐵 − 𝐴)
[√
],
(𝑎1 + 𝜆𝐴1 )𝜙 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
(3.1)
where 𝜙 = min {𝜃𝑚+𝑘 } , 𝜃𝑚+𝑘 is given by (1.7).
𝑘≥1
Proof. From (1.2), we have
∣𝑓 (𝑧)∣ ≤ 𝑟𝑚 +
∞
∑
∣𝑎𝑚+𝑘 ∣ 𝑟𝑚+𝑘 , 0 < ∣𝑧∣ = 𝑟 < 1.
𝑘=1
Using Corollary 2.2, we get
∣𝑓 (𝑧)∣ ≤ 𝑟𝑚 + 𝑟𝑚+1
𝑎1 (𝐵 − 𝐴)
[√
] , 0 < ∣𝑧∣ = 𝑟 < 1.
(𝑎1 + 𝜆𝐴1 )𝜙 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
Similary we get
∣𝑓 (𝑧)∣ ≥ 𝑟𝑚 − 𝑟𝑚+1
𝑎1 (𝐵 − 𝐴)
[√
],
(𝑎1 + 𝜆𝐴1 )𝜙 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
which completes the proof.
□
Let 𝑓 (𝑧) ∈ A𝜃𝑚 of the form (1.2) belong to A𝜃𝑚 (𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆),
𝑞
𝑞
∑
∑
then for 0 < ∣𝑧∣ = 𝑟 < 1 and for
(𝑏𝑖 − 𝑎𝑖 ) > 21 (𝐴𝑖 − 𝐵𝑖 ),
Theorem 3.2.
𝑖=1
𝑚𝑟𝑚−1 − 𝑟𝑚
𝑎1 (𝐵 − 𝐴)
[√
] ≤ ∣𝑓 ′ (𝑧)∣
(𝑎1 + 𝜆𝐴1 )𝜑 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
≤ 𝑚𝑟𝑚−1 + 𝑟𝑚
where 𝜑 = min
𝑘≥1
𝑖=1
{
𝜃𝑚+𝑘
𝑚+𝑘
}
𝑎1 (𝐵 − 𝐴)
[√
],
(𝑎1 + 𝜆𝐴1 )𝜑 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
, 𝜃𝑚+𝑘 is given by (1.7).
(3.2)
62
P. SHARMA
Proof. From (1.2), we have
∣𝑓 ′ (𝑧)∣ ≤ 𝑚𝑟𝑚−1 +
∞
∑
(𝑚 + 𝑘) ∣𝑎𝑚+𝑘 ∣ 𝑟𝑚+𝑘−1 , 0 < ∣𝑧∣ = 𝑟 < 1.
𝑘=1
Using Corollary 2.3, we get
∣𝑓 ′ (𝑧)∣ ≤ 𝑚𝑟𝑚−1 + 𝑟𝑚
𝑎1 (𝐵 − 𝐴)
[√
].
(𝑎1 + 𝜆𝐴1 )𝜑 1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
Similary we get
∣𝑓 ′ (𝑧)∣ ≥ 𝑚𝑟𝑚−1 − 𝑟𝑚
(𝑎1 + 𝜆𝐴1 )𝜑
𝑎1 (𝐵 − 𝐴)
],
[√
1 − 𝐵 2 𝑠𝑖𝑛2 𝜃 − 𝐵𝑐𝑜𝑠𝜃
which completes the proof.
□
4. Extreme Points
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆).
In this section we find extreme points for the class A(2𝑛−1)𝜋
𝑚
(2𝑛−1)𝜋
and
Let 𝑓 (𝑧) ∈ A𝑚
Theorem 4.1.
𝑓0 (𝑧) = 𝑧 𝑚 , 𝑓𝑘 (𝑧) = 𝑧 𝑚 −
𝑎1 (𝐵 − 𝐴)
𝑧 𝑚+𝑘 , 𝑘 ≥ 1.
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
(4.1)
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) if only if it can be expressed in the form
Then 𝑓 (𝑧) ∈ A(2𝑛−1)𝜋
𝑚
𝑓 (𝑧) =
∞
∑
𝜆𝑘 𝑓𝑘 (𝑧),
(4.2)
𝑘=0
∞
∑
where 𝜆𝑘 ≥ 0 and
𝑘=0
𝜆𝑘 = 1 and 𝑓𝑘′ 𝑠 for 𝑘 ≥ 0 are the extreme points for
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) class.
A(2𝑛−1)𝜋
𝑚
Proof. Let
𝑓 (𝑧)
=
∞
∑
𝜆𝑘 𝑓𝑘 (𝑧)
𝑘=0
=
(1 −
∞
∑
𝜆𝑘 )𝑧 𝑚 +
𝑘=1
∞
∑
= 𝑧𝑚 −
∞
∑
(
𝜆𝑘 𝑧 𝑚 −
𝑘=1
𝜆𝑘
𝑘=1
𝑎1 (𝐵 − 𝐴)
𝑧 𝑚+𝑘
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
)
𝑎1 (𝐵 − 𝐴)
𝑧 𝑚+𝑘 ,
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
(2𝑛−1)𝜋
which proves that 𝑓 (𝑧) ∈ A𝑚
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆). Since by Theorem 2.5
[
] ∑
∞
∞
∑ (1 + 𝐵)(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘
𝑎1 (𝐵 − 𝐴)𝜆𝑘
=
𝜆𝑘 < 1 − 𝜆0 ≤ 1.
𝑎1 (𝐵 − 𝐴)
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
𝑘=1
𝑘=1
(2𝑛−1)𝜋
Conversely, suppose 𝑓 (𝑧) ∈ A𝑚
(𝑝, [𝑎1 ], 𝐴, 𝐵, 𝜆) then using Corollary 2.6,
∞
∑
(𝑎1 +𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1+𝐵)
𝜆𝑘 .
we set 𝜆𝑘 =
𝑎𝑚+𝑘 for 𝑘 ≥ 1 and 𝜆0 = 1 −
𝑎1 (𝐵−𝐴)
𝑘=1
A CLASS OF MULTIVALENT ANALYTIC FUNCTIONS WITH FIXED ARGUMENT
63
Thus
𝑓 (𝑧)
= 𝑧
𝑚
−
∞
∑
∣𝑎𝑚+𝑘 ∣𝑧 𝑚+𝑘
𝑘=1
∞
∑
= 𝑓0 (𝑧) −
= 𝑓0 (𝑧) −
𝑘=1
∞
∑
𝑎1 (𝐵 − 𝐴)𝜆𝑘
𝑧 𝑚+𝑘
(𝑎1 + 𝜆𝐴1 𝑘)𝜃𝑚+𝑘 (1 + 𝐵)
[𝑧 𝑚 − 𝑓𝑘 (𝑧)] 𝜆𝑘
𝑘=1
∞
∑
=
𝜆𝑘 𝑓𝑘 (𝑧).
𝑘=0
This completes the proof.
□
5. Some Consequences of Main Results
Taking 𝑎1 = 𝑚𝐴1 in Theorm 2.1, we get following result.
of the form (1.2) satisfies
Corollary 5.1. A function 𝑓 (𝑧) ∈ A(2𝑛−1)𝜋
𝑚
′
𝑞
𝑞
([𝑎1 ]) 𝑓 )
𝑧 (𝐼𝑚
1 + 𝐴𝑧
([𝑎1 ]) 𝑓
𝐼𝑚
+
𝜆
≺
𝑧𝑚
𝑧𝑚
1 + 𝐵𝑧
(1 − 𝜆)
if and only if
∞
∑
(𝑚 + 𝜆𝑘)𝜃𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ <
𝑘=1
𝑚 (𝐵 − 𝐴)
,
(1 + 𝐵)
where 𝜃𝑚+𝑘 is defined in (1.7) and 0 ≤ 𝜆 ≤ 1, −1 ≤ 𝐴 < 𝐵 ≤ 1, 0 ≤ 𝐵.
In Corolary 5.1, on replacing 𝜃𝑚+𝑘 by 𝑏𝑚+𝑘 , following result can be obtained.
Corollary 5.2. Let 𝑔(𝑧) ∈ A𝑚 of the form:
𝑔(𝑧) = 𝑧 𝑚 +
∞
∑
𝑏𝑚+𝑘 𝑧 𝑚+𝑘 (𝑏𝑚+𝑘 ≥ 0, 𝑚 ∈ N = {1, 2, 3...})
𝑘=1
(2𝑛−1)𝜋
of the form
and 0 ≤ 𝜆 ≤ 1, −1 ≤ 𝐴 < 𝐵 ≤ 1, 0 ≤ 𝐵. A function 𝑓 (𝑧) ∈ A𝑚
(1.2) satisfies
′
(1 − 𝜆)
(𝑓 ∗ 𝑔)
𝑧 (𝑓 ∗ 𝑔)
1 + 𝐴𝑧
,𝑧∈U
+𝜆
≺
𝑚
𝑚
𝑧
𝑧
1 + 𝐵𝑧
if and only if
∞
∑
(𝑚 + 𝜆𝑘)𝑏𝑚+𝑘 ∣𝑎𝑚+𝑘 ∣ <
𝑘=1
𝑚 (𝐵 − 𝐴)
.
(1 + 𝐵)
Corollary 5.3. Let 0 < 𝜆 ≤ 1, −1 < 𝐴 < 0 , if a function 𝑓 (𝑧) ∈ A𝑚 satisfies
(1 − 𝜆)
𝑞
𝑞
𝐼𝑚
([𝑎1 ]) 𝑓
𝐼𝑚
([𝑎1 + 1]) 𝑓
+
𝜆
≺ 1 + 𝐴𝑧,
𝑚
𝑧 (
𝑧 𝑚)
𝑞
then 𝐼𝑚
([𝑎1 ]) 𝑓 ∈ S∗𝑚 (𝛼), 𝛼 := 𝑚 −
2𝑎1 ∣𝐴∣
𝜆𝐴1 (1−∣𝐴∣)
.
64
Proof. Let 𝑝(𝑧) :=
P. SHARMA
𝑞
𝐼𝑚
([𝑎1 ])𝑓
𝑧𝑚
then with the use of identity (1.9) we have
𝜆𝐴1 ′
𝑧𝑝 (𝑧) ≺ 1 + 𝐴𝑧.
𝑎1
By a well known Lemma of Hallenbeck and Ruscheweyh [7] we get that 𝑝(𝑧) ≺ 1+𝐴𝑧
and hence:
𝑞
𝑞
𝐼𝑚 ([𝑎1 ]) 𝑓 𝐼𝑚 ([𝑎1 ]) 𝑓
> 1 − ∣𝐴∣ ,
− 1 < ∣𝐴∣ and 𝑧𝑚
𝑧𝑚
𝑝(𝑧) +
which with the use of hypothesis and the identity(1.9), derives:
𝑞
′
𝑞
𝑞
𝑧 (𝐼𝑚
𝐼𝑚 ([𝑎1 ]) 𝑓 ([𝑎1 ]) 𝑓 )
2𝑎1 ∣𝐴∣
𝐼𝑚
([𝑎1 ]) 𝑓 .
−𝑚
< 𝜆𝐴1 (1 − ∣𝐴∣) 𝑧𝑚
𝑧𝑚
𝑧𝑚
𝑞
𝑧(𝐼𝑚 ([𝑎1 ])𝑓 )′
1 ∣𝐴∣
That evidently yields: 𝐼𝑚
− 𝑚 < 𝜆𝐴2𝑎
.
𝑞
([𝑎1 ])𝑓
1 (1−∣𝐴∣)
)
{ 𝑞
}
(
𝑧(𝐼𝑚 ([𝑎1 ])𝑓 )′
2𝑎1 ∣𝐴∣
which proves
Therefore Re
>
𝛼
,
where
𝛼
=
𝑚
−
𝑞
𝜆𝐴1 (1−∣𝐴∣)
𝐼𝑚 ([𝑎1 ])𝑓
the result.
□
Acknowledgments. The author would like to thank the anonymous referee for
his/her valuable comments and suggestions to improve this article.
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Poonam Sharma
Department of Mathematics& Astronomy University of Lucknow Lucknow, 226007, UP
INDIA
E-mail address: poonambaba@yahoo.com