ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 2(2010), Pages 69-76.
HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH GENERALIZED HYPERGEOMETRIC FUNCTIONS
GANGADHARAN. MURUGUSUNDARAMOORTHY AND KALIEPPAN.UMA
Abstract. The generalized hypergeometric function is used here to introduce a new class of complex valued harmonic functions which are orientation pre-serving and univalent in the open unit disc. Among the results presented in this paper include the coefficient bounds, distortion inequality and covering property, extreme points and certain inclusion results for this generalized class of functions.
1. Introduction
A continuous function 𝑓 = 𝑢+𝑖𝑣 is a complex- valued harmonic function in a complex domain Ω if both𝑢 and 𝑣 are real and harmonic in Ω. In any simply-connected domain 𝐷⊂Ω,we can write𝑓 =ℎ+𝑔,where ℎand 𝑔 are analytic in 𝐷. We call ℎ the analytic part and 𝑔 the co-analytic part of 𝑓. A necessary and sufficient condition for𝑓 to be locally univalent and orientation preserving in𝐷 is that∣ℎ′(𝑧)∣>∣𝑔′(𝑧)∣in𝐷 (see [2]).
Denote byℋthe family of functions
𝑓 =ℎ+𝑔 (1.1)
which are harmonic, univalent and orientation preserving in the open unit disc
𝒰 ={𝑧:∣𝑧∣<1}so that𝑓 is normalized by𝑓(0) =ℎ(0) =𝑓𝑧(0)−1 = 0.Thus, for
𝑓 =ℎ+𝑔∈ ℋ,the functionsℎand 𝑔analytic𝒰 can be expressed in the following forms:
ℎ(𝑧) =𝑧+
∞ ∑ 𝑛=2
𝑎𝑛𝑧𝑛, 𝑔(𝑧) = ∞ ∑ 𝑛=1
𝑏𝑛𝑧𝑛 (∣𝑏1∣<1),
and𝑓(𝑧) is then given by
𝑓(𝑧) =𝑧+
∞ ∑ 𝑛=2
𝑎𝑛𝑧𝑛+ ∞ ∑ 𝑛=1
𝑏𝑛𝑧𝑛 (∣𝑏1∣<1). (1.2)
We note that the familyℋof orientation preserving, normalized harmonic univalent functions reduces to the well known class𝒮of normalized univalent functions if the
2000Mathematics Subject Classification. 30C45, 30C50.
Key words and phrases. Harmonic univalent functions, hypergeometric function , distortion bounds, extreme points, convolution. inclusion property.
c
⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted November, 2009. Published May, 2010.
co-analytic part of 𝑓 is identically zero, i.e. 𝑔 ≡ 0. Also, we denote by ℋ the subfamily ofℋconsisting of harmonic functions𝑓 =ℎ+𝑔of the form
𝑓(𝑧) =𝑧− ∞ ∑ 𝑛=2
∣𝑎𝑛∣𝑧𝑛+
∞ ∑ 𝑛=1
∣𝑏𝑛∣𝑧𝑛, (∣𝑏1∣<1) (1.3)
introduced and studied by Silverman[12].
The Hadamard product (or convolution) of two power series
𝜙(𝑧) =𝑧+
∞ ∑ 𝑛=2
𝜙𝑛𝑧𝑛 (1.4)
and
𝜓(𝑧) =𝑧+
∞ ∑ 𝑛=2
𝜓𝑛𝑧𝑛 (1.5)
in𝑆 is defined (as usual) by
(𝜙∗𝜓)(𝑧) =𝜙(𝑧)∗𝜓(𝑧) =𝑧+
∞ ∑ 𝑛=2
𝜙𝑛𝜓𝑛𝑧𝑛. (1.6)
For positive real values of𝛼1, . . . , 𝛼𝑙and𝛽1, . . . , 𝛽𝑚(𝛽𝑗 ∕= 0,−1, . . .;𝑗= 1,2, . . . , 𝑚)
thegeneralized hypergeometric function𝑙𝐹𝑚(𝑧) is defined by
𝑙𝐹𝑚(𝑧)≡𝑙𝐹𝑚(𝛼1, . . . 𝛼𝑙;𝛽1, . . . , 𝛽𝑚;𝑧) := ∞ ∑ 𝑛=0
(𝛼1)𝑛. . .(𝛼𝑙)𝑛
(𝛽1)𝑛. . .(𝛽𝑚)𝑛
𝑧𝑛
𝑛! (1.7)
(𝑙≤𝑚+ 1; 𝑙, 𝑚∈𝑁0 := 𝑁∪ {0};𝑧∈𝑈),
where𝑁 denotes the set of all positive integers and (𝑎)𝑛is the Pochhammer symbol
defined by
(𝑎)𝑛= {
1, 𝑛= 0
𝑎(𝑎+ 1)(𝑎+ 2). . .(𝑎+𝑛−1), 𝑛∈𝑁. (1.8)
The notation 𝑙𝐹𝑚 is quite useful for representing many well-known functions
such as the exponential, the Binomial, the Bessel and Laguerre polynomial. Let
𝐻[𝛼1, . . . 𝛼𝑙;𝛽1, . . . , 𝛽𝑚] :𝒮 → 𝒮
be a linear operator defined by
𝐻[𝛼1, . . . 𝛼𝑙;𝛽1, . . . , 𝛽𝑚]𝜙(𝑧) := 𝑧 𝑙𝐹𝑚(𝛼1, 𝛼2, . . . 𝛼𝑙;𝛽1, 𝛽2. . . , 𝛽𝑚;𝑧)∗𝜙(𝑧)
= 𝑧+
∞ ∑ 𝑛=2
𝜔𝑛(𝛼1;𝑙;𝑚)𝜙𝑛𝑧𝑛, (1.9)
where
𝜔𝑛(𝛼1;𝑙;𝑚) =
(𝛼1)𝑛−1. . .(𝛼𝑙)𝑛−1
(𝛽1)𝑛−1. . .(𝛽𝑚)𝑛−1
1
(𝑛−1)!. (1.10)
For notational simplicity, we use a shorter notation𝐻𝑙
𝑚[𝛼1] for𝐻[𝛼1, . . . 𝛼𝑙;𝛽1, . . . , 𝛽𝑚]
in the sequel.It follows from (1.9) that
𝐻01[1]𝜙(𝑧) =𝜙(𝑧), 𝐻01[2]𝜙(𝑧) =𝑧𝜙′(𝑧).
The linear operator𝐻𝑙
𝑚[𝛼1] is called Dziok-Srivastava operator (see [4])introduced
by using the generalized hypergeometric function , recently Srivastava etal.([11]) defined the linear operatorℒ𝜏,𝛼1
𝜆,𝑙,𝑚 as follows:
ℒ0
𝜆,𝛼1𝜙(𝑧) =𝜙(𝑧),
ℒ1,𝛼1
𝜆,𝑙,𝑚𝜙(𝑧) = (1−𝜆)𝐻 𝑙
𝑚[𝛼1]𝜙(𝑧) +𝜆(𝐻𝑚𝑙 [𝛼1]𝜙(𝑧))′ =ℒ𝛼𝜆,𝑙,𝑚1 𝜙(𝑧),(𝜆≥0), (1.11)
ℒ2,𝛼1
𝜆,𝑙,𝑚𝜙(𝑧) =ℒ 𝛼1
𝜆,𝑙,𝑚(ℒ 1,𝛼1
𝜆,𝑙,𝑚𝜙(𝑧)) (1.12)
and in general ,
ℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝜙(𝑧) =ℒ 𝛼1
𝜆,𝑙,𝑚(ℒ 𝜏−1,𝛼1
𝜆,𝑙,𝑚 𝜙(𝑧)),(𝑙≤𝑚+1; 𝑙, 𝑚∈𝑁0=𝑁∪{0};𝑧∈𝑈). (1.13)
If the function 𝜙(𝑧) is given by (1.4), then we see from (1.9), (1.10),(1.11) and(1.13) that
ℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝜙(𝑧) := 𝑧+ ∞ ∑ 𝑛=2
𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)𝜙𝑛 𝑧𝑛, (1.14)
where
𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚) = ((𝛼
1)𝑛−1. . .(𝛼𝑙)𝑛−1
(𝛽1)𝑛−1. . .(𝛽𝑚)𝑛−1
[1 +𝜆(𝑛−1)] (𝑛−1)!
)𝜏
,(𝑛∈𝑁∖ {1}, 𝜏 ∈𝑁0(1.15))
unless otherwise stated. We note that when𝜏 = 1 and𝜆= 0 the linear operatorℒ𝜏,𝛼1
𝜆,𝑙,𝑚
would reduce to the familiar Dziok-Srivastava linear operator given by (see [4]), in-cludes (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer [1], Owa [9]and Ruscheweyh [10] .
In view of the relationship (1.15) and the linear operator (1.14) for the harmonic function𝑓 =ℎ+𝑔 given by (1.1), we define the operator
ℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝑓(𝑧) =ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧) +ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧), (1.16)
and introduce below a new subclassℒ𝐻(𝜏;𝜆;𝛾) ofℋin terms of the operator given
by (1.16).
Letℒ𝐻(𝜏;𝜆;𝛾) denote a subclass ofℋ consisting of functions of the form𝑓 = ℎ+𝑔 given by (1.2) satisfying the condition that
∂ ∂𝜃
(
𝑎𝑟𝑔ℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝑓(𝑧) )
> 𝛾= Re
{
𝑧(ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧))′−𝑧(ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧))′ ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧) + (ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧)) }
≥𝛾, (1.17)
(
𝑧=𝑟𝑒𝑖𝜃; 0≤𝜃 <2𝜋; 0≤𝑟 <1; 0≤𝛾 <1;𝑧∈𝕌)
whereℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝑓(𝑧) is given by (1.16). We also letℒ𝐻(𝜏;𝜆;𝛾) =ℒ𝐻(𝜏;𝜆;𝛾)∩ ℋ.
In this paper, we obtain coefficient conditions for the classes ℒ𝐻(𝜏;𝜆;𝛾) and
ℒ𝐻(𝜏;𝜆;𝛾). A representation theorem, inclusion properties and distortion bounds for the classℒ𝐻(𝜏;𝜆;𝛾) are also established.
2. Coefficient bounds
Due to Jahangiri [7], we state the following sufficient coefficient bound for the harmonic functions 𝑓 ∈ ℋ(𝛾) the class of harmonic starlike functions of order 𝛾,(0≤𝛾 <1) a subclass ofℋconsisting of functions of the form𝑓 =ℎ+𝑔 given by (1.2) satisfying the condition that ∂
Theorem 2.1. [7]. Let𝑓 =ℎ+𝑔 be given by (1.2). If
∞ ∑ 𝑛=1 [
𝑛−𝛾 1−𝛾∣𝑎𝑛∣+
𝑛+𝛾 1−𝛾∣𝑏𝑛∣
]
≤2, (2.1)
where𝑎1= 1 and0≤𝛾 <1,then 𝑓 ∈ ℋ(𝛾).
The following result gives a sufficient coefficient condition for the harmonic func-tions𝑓 ∈ ℒ𝐻(𝜏;𝜆;𝛾).
Theorem 2.2. Let 𝑓 =ℎ+𝑔 be given by (1.2). If
∞ ∑ 𝑛=1
[𝑛 −𝛾 1−𝛾∣𝑎𝑛∣+
𝑛+𝛾 1−𝛾∣𝑏𝑛∣
]
𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)≤2, (2.2)
where𝑎1= 1 and0≤𝛾 <1,then 𝑓 ∈ ℒ𝐻(𝜏;𝜆;𝛾).
Proof. Since𝑛≤𝑚𝑖𝑛{𝑛1−−𝛾𝛾𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚);𝑛1−+𝛾𝛾𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)},it follows from
The-orem2.1 that𝑓 ∈ ℋ(𝛾) and hence𝑓 is harmonic, orientation preserving and univa-lent in𝕌. Suppose the condition (2.2) holds true. To show that𝑓 ∈ ℒ𝐻(𝜏;𝜆;𝛾),
we show (in view of (1.17)) that
ℜ ⎧ ⎨ ⎩
𝑧(ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧) )′
−𝑧(ℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧) )′
ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧) +ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧) ⎫ ⎬ ⎭
=ℜ {𝐴(𝑧)
𝐵(𝑧)
}
≥𝛾 (𝑧∈𝕌),
where
𝐴(𝑧) =𝑧(ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧))
′−𝑧(ℒ𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧))′=𝑧+ ∞ ∑ 𝑛=2
𝑛𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)𝑎𝑛𝑧𝑛− ∞ ∑ 𝑛=1
𝑛𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)𝑏𝑛𝑧𝑛
and
𝐵(𝑧) =𝑧+
∞ ∑ 𝑛=2
𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)𝑎𝑛𝑧𝑛+ ∞ ∑ 𝑛=1
𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)𝑏𝑛𝑧𝑛.
Using the fact that Re{𝑤} ≥𝛾 if and only if∣1−𝛾+𝑤∣ ≥ ∣1 +𝛾−𝑤∣,it suffices to show that
∣𝐴(𝑧) + (1−𝛾)𝐵(𝑧)∣ − ∣𝐴(𝑧)−(1 +𝛾)𝐵(𝑧)∣ ≥0. (2.3)
Substituting for 𝐴(𝑧) and𝐵(𝑧) in (2.3), and performing elementary calculations, we find that
∣𝐴(𝑧) + (1−𝛾)𝐵(𝑧)∣ − ∣𝐴(𝑧)−(1 +𝛾)𝐵(𝑧)∣
≥2(1−𝛾)∣𝑧∣ {
2− ∞ ∑ 𝑛=1
[𝑛−𝛾
1−𝛾∣𝑎𝑛∣+ 𝑛+𝛾 1−𝛾∣𝑏𝑛∣
]
𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)∣𝑧∣𝑛−1 }
>2(1−𝛾)
{
2− ∞ ∑ 𝑛=1
[𝑛−𝛾
1−𝛾∣𝑎𝑛∣+ 𝑛+𝛾 1−𝛾∣𝑏𝑛∣
]
𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚) }
≥0,
The harmonic function
𝑓(𝑧) =𝑧+
∞ ∑ 𝑛=2
1−𝛾 (𝑛−𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
𝑥𝑛𝑧𝑛+ ∞ ∑ 𝑛=1
1−𝛾 (𝑛+𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
𝑦𝑛(𝑧)𝑛,
(2.4)
where
∞ ∑ 𝑛=2
∣𝑥𝑛∣+ ∞ ∑ 𝑛=1
∣𝑦𝑛∣ = 1 shows that the coefficient bound given by (2.2) is
sharp.
The functions of the form (2.4) are inℒ𝐻(𝜏;𝜆;𝛾) because
∞ ∑ 𝑛=1
((𝑛−𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾 ∣𝑎𝑛∣+
(𝑛+𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾 ∣𝑏𝑛∣
)
=1 +
∞ ∑ 𝑛=2
∣𝑥𝑛∣+
∞ ∑ 𝑛=1
∣𝑦𝑛∣= 2.
Our next theorem gives a necessary and sufficient condition for functions of the form (1.3) to be in the classℒ𝐻(𝜏;𝜆;𝛾).
Theorem 2.3. For𝑎1= 1and0≤𝛾 <1, 𝑓=ℎ+𝑔∈ ℒ𝐻(𝜏;𝜆;𝛾)if and only if
∞ ∑ 𝑛=1
[𝑛−𝛾
1−𝛾∣𝑎𝑛∣+ 𝑛+𝛾 1−𝛾∣𝑏𝑛∣
]
𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)≤2. (2.5)
Proof. Since ℒ𝐻(𝜏;𝜆;𝛾) ⊂ ℒ𝐻(𝜏;𝜆;𝛾), we only need to prove the ”only if” part of the theorem. To this end, for functions 𝑓 of the form (1.3), we notice that the condition
ℜ {
𝑧(ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ′(𝑧))−𝑧(ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚𝑔′(𝑧)) ℒ𝜏,𝛼1
𝜆,𝑙,𝑚ℎ(𝑧) + (ℒ 𝜏,𝛼1
𝜆,𝑙,𝑚𝑔(𝑧)) }
≥𝛾
implies that
ℜ ⎧ ⎨ ⎩
(1−𝛾)𝑧− ∞ ∑ 𝑛=2
(𝑛−𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)𝑎𝑛𝑧𝑛− ∞ ∑ 𝑛=1
(𝑛+𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)𝑏𝑛𝑧𝑛
𝑧− ∞ ∑ 𝑛=2
𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)𝑎𝑛𝑧𝑛+ ∞ ∑ 𝑛=1
𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)𝑏𝑛𝑧𝑛
⎫ ⎬ ⎭
≥0.
The above required condition must hold for all values of𝑧 in 𝕌. Upon choosing the values of𝑧 on the positive real axis where 0≤𝑧=𝑟 <1,we must have
(1−𝛾)− ∞ ∑ 𝑛=2
(𝑛−𝛾)𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)𝑎𝑛𝑟𝑛−1− ∞ ∑ 𝑛=1
(𝑛+𝛾)𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)𝑏𝑛𝑟𝑛−1
1− ∞ ∑ 𝑛=2
𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)𝑎𝑛𝑟𝑛−1+ ∞ ∑ 𝑛=1
𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)𝑏𝑛𝑟𝑛−1
≥0. (2.6)
If the condition (2.5) does not hold, then the numerator in (2.6) is negative for𝑟sufficiently close to 1. Hence, there exist𝑧0 =𝑟0 in (0,1) for which the quotient of (2.6) is negative.
This contradicts the required condition for𝑓(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾).This completes the proof
of the theorem. □
3. Distortion bounds and Extreme Points
Theorem 3.1. (Distortion bounds)Let 𝑓 ∈ ℒ𝐻(𝜏;𝜆;𝛾),then (for∣𝑧∣=𝑟 <1)
(1−𝑏1)𝑟−
1 𝜔𝜏
2(𝛼1;𝜆;𝑙;𝑚) (
1−𝛾 2−𝛾 −
1 +𝛾 2−𝛾𝑏1
)
𝑟2≤ ∣𝑓(𝑧)∣
≤(1 +𝑏1)𝑟+
1 𝜔𝜏
2(𝛼1;𝜆;𝑙;𝑚) (1
−𝛾 2−𝛾 −
1 +𝛾 2−𝛾𝑏1
)
𝑟2.
Corollary 3.2. If 𝑓(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾), then
{
𝑤:∣𝑤∣<2𝜔
𝜏
2(𝛼1;𝜆;𝑙;𝑚)−1−[𝜔2𝜏(𝛼1;𝜆;𝑙;𝑚)−1]𝛾
(2−𝛾)𝜔𝜏
2(𝛼1;𝜆;𝑙;𝑚)
−2𝜔 𝜏
2(𝛼1;𝜆;𝑙;𝑚)−1−[𝜔𝜏2(𝛼1;𝜆;𝑙;𝑚) + 1]𝛾
(2−𝛾)𝜔𝜏
2(𝛼1;𝜆;𝑙;𝑚)
𝑏1
}
⊂𝑓(𝑈).
The extreme points of closed convex hulls ofℒ𝐻(𝜏;𝜆;𝛾) denoted by𝑐𝑙𝑐𝑜ℒ𝐻(𝜏;𝜆;𝛾).
Theorem 3.3. (Extreme Points) A function𝑓(𝑧)∈𝑐𝑙𝑐𝑜ℒ𝐻(𝜏;𝜆;𝛾)if and only
if 𝑓(𝑧) =
∞ ∑ 𝑛=1
[𝑋𝑛ℎ𝑛(𝑧) +𝑌𝑛𝑔𝑛(𝑧)],where
ℎ1(𝑧) =𝑧, ℎ𝑛(𝑧) =𝑧−
1−𝛾
(𝑛−𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
𝑧𝑛(𝑛≥2), 𝑔𝑛(𝑧) =𝑧+
1−𝛾
(𝑛+𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
𝑧𝑛(𝑛≥2),
∞ ∑
𝑛=1
(𝑋𝑛+𝑌𝑛) = 1, 𝑋𝑛≥0𝑎𝑛𝑑 𝑌𝑛≥0.
In particular, the extreme points ofℒ𝐻(𝜏;𝜆;𝛾)are{ℎ𝑛}and{𝑔𝑛}.
4. Inclusion Results
The following result gives the convex combinations of the classℒ𝐻(𝜏;𝜆;𝛾).
Theorem 4.1. The familyℒ𝐻(𝜏;𝜆;𝛾) is closed under convex combinations.
Proof. Let𝑓𝑖∈ ℒ𝐻(𝜏;𝜆;𝛾) (𝑖= 1,2, . . .),where
𝑓𝑖(𝑧) =𝑧− ∞ ∑ 𝑛=2
∣𝑎𝑖,𝑛∣𝑧𝑛+
∞ ∑ 𝑛=2
∣𝑏𝑖,𝑛∣𝑧𝑛.
The convex combination of𝑓𝑖 may be written as
∞ ∑ 𝑖=1
𝑡𝑖𝑓𝑖(𝑧) =𝑧− ∞ ∑ 𝑛=2
(∞ ∑ 𝑖=1
𝑡𝑖∣𝑎𝑖,𝑛∣ )
𝑧𝑛+
∞ ∑ 𝑛=1
(∞ ∑ 𝑖=1
𝑡𝑖∣𝑏𝑖,𝑛∣ )
𝑧𝑛,
provided that
∞ ∑ 𝑖=1
𝑡𝑖= 1 (0≤𝑡𝑖 ≤1).
Applying the inequality (2.5) of Theorem 2.3, we obtain
∞ ∑
𝑛=2
(𝑛−𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾
(∞ ∑
𝑖=1 𝑡𝑖∣𝑎𝑖,𝑛∣
) +
∞ ∑
𝑛=1
(𝑛+𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾
(∞ ∑
𝑖=1 𝑡𝑖∣𝑏𝑖,𝑛∣
)
= ∞ ∑
𝑖=1 𝑡𝑖
(∞ ∑
𝑛=2
(𝑛−𝛾)𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)
1−𝛾 ∣𝑎𝑖,𝑛∣+
∞ ∑
𝑛=1
(𝑛+𝛾)𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾 ∣𝑏𝑖,𝑛∣
)
≤ ∞ ∑
𝑖=1 𝑡𝑖= 1,
and therefore, ∞ ∑ 𝑖=1
Theorem 4.2. Let 𝑓(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾)and𝐹(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛿),where 0≤𝛿≤𝛾 < 1, then
𝑓(𝑧)∗𝐹(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾)⊂ ℒ𝐻(𝜏;𝜆;𝛿).
Proof. Let 𝑓(𝑧) = 𝑧 − ∞ ∑ 𝑛=2
∣𝑎𝑛∣𝑧𝑛 + ∞ ∑ 𝑛=1
∣𝑏𝑛∣𝑧𝑛 ∈ ℒ𝐻(𝜏;𝜆;𝛾) and 𝐹(𝑧) = 𝑧−
∞ ∑ 𝑛=2
∣𝐴𝑛∣𝑧𝑛+ ∞ ∑ 𝑛=1
∣𝐵𝑛∣𝑧𝑛∈ ℒ𝐻(𝜏;𝜆;𝛿),then
𝑓(𝑧)∗𝐹(𝑧) =𝑧− ∞ ∑ 𝑛=2
∣𝑎𝑛∣∣𝐴𝑛∣𝑧𝑛+
∞ ∑ 𝑛=1
∣𝑏𝑛∣∣𝐵𝑛∣𝑧𝑛.
From the assertion that 𝑓(𝑧)∗𝐹(𝑧) ∈ ℒ𝐻(𝜏;𝜆;𝛿), we note that ∣𝐴𝑛∣ ≤ 1 and ∣𝐵𝑛∣ ≤1.In view of Theorem 2.3 and the inequality 0≤𝛿≤𝛾 <1,we have
∞ ∑ 𝑛=2
(𝑛−𝛿)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛿 ∣𝑎𝑛∣ ∣𝐴𝑛∣+
∞ ∑ 𝑛=1
(𝑛+𝛿)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛿 ∣𝑏𝑛∣ ∣𝐵𝑛∣
≤ ∞ ∑ 𝑛=2
(𝑛−𝛿)𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛿 ∣𝑎𝑛∣+
∞ ∑ 𝑛=1
(𝑛+𝛿)𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)
1−𝛿 ∣𝑏𝑛∣
≤ ∞ ∑ 𝑛=2
(𝑛−𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾 ∣𝑎𝑛∣+
∞ ∑ 𝑛=1
(𝑛+𝛾)𝜔𝜏
𝑛(𝛼1;𝜆;𝑙;𝑚)
1−𝛾 ∣𝑏𝑛∣ ≤1,
by Theorem 2.3,𝑓(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾).Hence𝑓(𝑧)∗𝐹(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾)⊂ ℒ𝐻(𝜏;𝜆;𝛿). □
Lastly, we consider the closure property of the classℒ𝐻(𝜏;𝜆;𝛾) under the gen-eralized Bernardi-Libera -Livingston integral operatorℒ𝑐[𝑓(𝑧)] which is defined by
ℒ𝑐[𝑓(𝑧)] =
𝑐+ 1 𝑧𝑐
𝑧 ∫
0
𝑡𝑐−1𝑓(𝑡)𝑑𝑡(𝑐 >−1).
We prove the following result.
Theorem 4.3. Let 𝑓(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾), thenℒ𝑐[𝑓(𝑧)]∈ ℒ𝐻(𝜏;𝜆;𝛾)
Proof. Using (1.1) and (1.3), we get
ℒ𝑐[𝑓(𝑧)] = 𝑐+ 1 𝑧𝑐
𝑧 ∫
0
𝑡𝑐−1ℎ(𝑡)𝑑𝑡+𝑐+ 1 𝑧𝑐
𝑧 ∫
0
𝑡𝑐−1𝑔(𝑡)𝑑𝑡.
= 𝑐+ 1 𝑧𝑐
𝑧 ∫
0
𝑡𝑐−1
(
𝑡− ∞ ∑ 𝑛=2
∣𝑎𝑛∣𝑡𝑛 )
𝑑𝑡+𝑐+ 1 𝑧𝑐
𝑧 ∫
0
𝑡𝑐−1 ( ∞
∑ 𝑛=1
∣𝑏𝑛∣𝑡𝑛 )
𝑑𝑡
=𝑧− ∞ ∑ 𝑛=2
𝐴𝑛𝑧𝑛+ ∞ ∑ 𝑛=1
where𝐴𝑛 =𝑐𝑐+1+𝑛 ∣𝑎𝑛∣and𝐵𝑛= 𝑐𝑐++1𝑛 ∣𝑏𝑛∣.Hence
∞ ∑ 𝑛=1
[𝑛 −𝛾 1−𝛾
(𝑐+ 1
𝑐+𝑛∣𝑎𝑛∣
)
+𝑛+𝛾 1−𝛾
(𝑐+ 1
𝑐+𝑛∣𝑏𝑛∣
)]
𝜔𝑛𝜏(𝛼1;𝜆;𝑙;𝑚)
≤ ∞ ∑ 𝑛=1
[𝑛−𝛾
1−𝛾∣𝑎𝑛∣+ 𝑛+𝛾 1−𝛾∣𝑏𝑛∣
]
𝜔𝜏𝑛(𝛼1;𝜆;𝑙;𝑚)
≤ 2,
since𝑓(𝑧)∈ ℒ𝐻(𝜏;𝜆;𝛾),therefore by Theorem 2.3,ℒ𝑐(𝑓(𝑧))∈ ℒ𝐻(𝜏;𝜆;𝛾). □
Concluding Remarks: By choosing 𝜏 = 1 and 𝜆 = 0 the various results presented in this paper would provide interesting extensions and generalizations of those considered earlier for simpler harmonic function classes (see [5, 6, 8, 12]). The details involved in the derivations of such specializations of the results presented in this paper are fairly straight- forward.
Acknowledgements : The authors would like to thank the referee and Prof.R.K.Raina for their insightful suggestions.
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Gangadharan. Murugusundaramoorthy and Kalieppan.Uma
School of Advanced Sciences , VIT University,, Vellore - 632014, India.,
