ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 4(2010), Pages 97-105.
AN APPLICATION OF HYPERGEOMETRIC FUNCTIONS ON HARMONIC UNIVALENT FUNCTIONS
(DEDICATED IN OCCASION OF THE 70-YEARS OF PROFESSOR HARI M. SRIVASTAVA)
SAURABH PORWAL AND KAUSHAL KISHORE DIXIT
Abstract. The purpose of the present paper is to establish connections
be-tween various subclasses of harmonic functions by applying certain convolu-tion operator involving hypergeometric funcconvolu-tions. To be more precise, we investigate such connections with Goodman-Rønning-type harmonic univalent functions in the open unit disc𝑈.
1. Introduction
Let𝐴denote the class of functions of the form
𝑓(𝑧) =𝑧+
∞
∑
𝑛=2
𝑎𝑛𝑧𝑛 (1.1)
which are analytic in the open unit disc𝑈. Hohlov [8] introduced the convolution operator𝐻(𝑎, 𝑏;𝑐) :𝐴→𝐴defined by
𝐻(𝑎, 𝑏;𝑐)𝑓(𝑧) =𝑧𝐹(𝑎, 𝑏;𝑐;𝑧)∗𝑓(𝑧),
where the symbol ”∗” stands for the convolution of two power series, which is defined
for two functions𝑓, 𝑔∈𝐴, where𝑓(𝑧) and𝑔(𝑧) are of the form𝑓(𝑧) =𝑧+
∞
∑
𝑛=2 𝑎𝑛𝑧𝑛
and𝑔(𝑧) =𝑧+
∞
∑
𝑛=2
𝑏𝑛𝑧𝑛 as
2000Mathematics Subject Classification. 30C45, 30C50, 33A30.
Key words and phrases. Harmonic, Starlike, Convex, Hypergeometric functions. c
⃝2008 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
The authors are thankful to the referee for his valuable comments and suggestions.
The present investigation was supported by the University grant commission under grant No. F. 11-12/2006(SA-I).
(𝑓∗𝑔)(𝑧) =𝑓(𝑧)∗𝑔(𝑧) =𝑧+
∞
∑
𝑛=2
𝑎𝑛𝑏𝑛𝑧𝑛and𝐹(𝑎, 𝑏;𝑐;𝑧) is a well-known Gaussian
hypergeometric function and given by the series
𝐹(𝑎, 𝑏;𝑐;𝑧) =
∞
∑
𝑛=0
(𝑎)𝑛(𝑏)𝑛
(𝑐)𝑛(1)𝑛
𝑧𝑛,
where 𝑎, 𝑏, 𝑐 are complex numbers such that 𝑐 ∕= 0,−1,−2, ... and (𝑎)𝑛 is the
Pochhammer symbol defined in terms of the Gamma function, by
(𝑎)𝑛=
Γ(𝑎+𝑛) Γ(𝑎)
= {
1 if𝑛= 0
𝑎(𝑎+ 1)...(𝑎+𝑛−1) if𝑛∈𝑁 ={1,2,3...}.
A hypergeometric function𝐹(𝑎, 𝑏;𝑐;𝑧) is analytic in𝑈 and plays an important role in Geometric Function Theory. See, for example, the works by Ahuja [3], Carleson and Shaffer [5], Miller and Mocanu [9], Owa and Srivastava [10], Ponnusamy and Rønning [11], Ruscheweyh and Singh [13] and Swaminathan [14].
Let𝐻 be the family of all harmonic functions of the form𝑓 =ℎ+𝑔, where
ℎ(𝑧) =𝑧+
∞
∑
𝑛=2
𝐴𝑛𝑧𝑛, 𝑔(𝑧) = ∞
∑
𝑛=1
𝐵𝑛𝑧𝑛, ∣𝐵1∣<1, (𝑧∈𝑈), (1.2)
are in the class𝐴. For complex parameters𝑎1, 𝑏1, 𝑐1, 𝑎2, 𝑏2, 𝑐2(𝑐1, 𝑐2∕= 0,−1,−2, ...), we define the functions Φ1(𝑧) =𝑧𝐹(𝑎1, 𝑏1;𝑐1;𝑧) and Φ2(𝑧) =𝑧𝐹(𝑎2, 𝑏2;𝑐2;𝑧) .
Corresponding to these functions, we consider the following convolution operator
Ω≡Ω (
𝑎1, 𝑏1, 𝑐1 𝑎2, 𝑏2, 𝑐2 )
:𝐻→𝐻
defined by
Ω (
𝑎1, 𝑏1, 𝑐1 𝑎2, 𝑏2, 𝑐2 )
𝑓 =𝑓∗(Φ1+ Φ2) =ℎ∗Φ1+𝑔∗𝜙2
for any function𝑓 =ℎ+𝑔 in𝐻. Letting
Ω (
𝑎1, 𝑏1, 𝑐1 𝑎2, 𝑏2, 𝑐2 )
𝐹(𝑧) =𝐻(𝑧) +𝐺(𝑧),
we have
𝐻(𝑧) =𝑧+
∞
∑
𝑛=2
(𝑎1)𝑛−1(𝑏1)𝑛−1 (𝑐1)𝑛−1(1)𝑛−1
𝐴𝑛𝑧𝑛, 𝐺(𝑧) = ∞
∑
𝑛=1
(𝑎2)𝑛−1(𝑏2)𝑛−1 (𝑐2)𝑛−1(1)𝑛−1
𝐵𝑛𝑧𝑛. (1.3)
We observe that
Ω (
𝑎1, 1, 𝑎1 𝑎2, 1, 𝑎2 )
𝑓(𝑧) =𝑓(𝑧) =𝑓(𝑧)∗ (
𝑧 1−𝑧 +
𝑧 1−𝑧
)
is the identity mapping.
This convolution operator Ω were defined and studied by the author in [2]. Denote by𝑆𝐻 the subclass of 𝐻 that are univalent and sense-preserving in 𝑈.
Note that 1𝑓−∣−𝐵1𝑓𝐵1∣2 ∈𝑆𝐻 whenever𝑓 ∈𝑆𝐻. We also let the subclass𝑆𝐻0 of 𝑆𝐻
The classes 𝑆𝐻0 and 𝑆𝐻 were first studied in [6]. Also, we let 𝐾𝐻0, 𝑆 ∗,0
𝐻 and 𝐶
0
𝐻
denote the subclasses of𝑆𝐻0 of harmonic functions which are, respectively, convex, starlike and close-to-convex in 𝑈. For definitions and properties of these classes, one may refer to ([1], [6]) or [7].
For 0≤𝛾 <1, let
𝑁𝐻(𝛾) = {
𝑓 ∈𝐻 : Re𝑓
′(𝑧)
𝑧′ ≥𝛾, 𝑧=𝑟𝑒 𝑖𝜃∈𝑈
} ,
𝐺𝐻(𝛾) = {
𝑓 ∈𝐻: Re {
(
1 +𝑒𝑖𝛼)𝑧𝑓
′(𝑧)
𝑓(𝑧) −𝑒
𝑖𝛼 }
≥𝛾, 𝛼∈𝑅, 𝑧∈𝑈 }
,
where
𝑧′= ∂ ∂𝜃
(
𝑧=𝑟𝑒𝑖𝜃)
, 𝑓′(𝑧) = ∂ ∂𝜃𝑓
( 𝑟𝑒𝑖𝜃)
.
Define
𝑇 𝑁𝐻(𝛾) =𝑁𝐻(𝛾)∩𝑇 and𝑇 𝐺𝐻(𝛾) =𝐺𝐻(𝛾)∩𝑇,
where𝑇 consists of the functions𝑓 =ℎ+𝑔 in𝑆𝐻 so thatℎand𝑔 are of the form
ℎ(𝑧) =𝑧−
∞
∑
𝑛=2
∣𝐴𝑛∣𝑧𝑛, 𝑔(𝑧) = ∞
∑
𝑛=1
∣𝐵𝑛∣𝑧𝑛. (1.4)
The classes𝑁𝐻(𝛾) and 𝐺𝐻(𝛾) were initially introduced and studied, respectively,
in ([4], [12]). A function in 𝐺𝐻(𝛾) is called Goodman-Rønning-type harmonic
univalent function in𝑈.
Throughout this paper, we will frequently use the notations
Ω (𝑓) = Ω (
𝑎1, 𝑏1, 𝑐1 𝑎2, 𝑏2, 𝑐2
) 𝑓,
𝐷𝑛−1=
(∣𝑎1∣)𝑛−1(∣𝑏1∣)𝑛−1
(𝑐1)𝑛−1(1)𝑛−1 , 𝐸𝑛−1=
(∣𝑎2∣)𝑛−1(∣𝑏2∣)𝑛−1 (𝑐2)𝑛−1(1)𝑛−1 ,
and a well-known formula
𝐹(𝑎, 𝑏;𝑐; 1) =Γ (𝑐−𝑎−𝑏) Γ (𝑐)
Γ (𝑐−𝑎) Γ (𝑐−𝑏), Re (𝑐−𝑎−𝑏)>0.
The main object of this paper is to establish some important connections between the classes𝐾0
𝐻,𝑆 ∗,0
𝐻 ,𝐶𝐻0,𝑁𝐻(𝛾) and𝐺𝐻(𝛾) by applying the convolution operator
Ω.
2. Connections with Goodman-Rønning-type Harmonic Univalent
Functions
In order to establish connections between harmonic convex functions, we need following results in Lemma 2.1 [6], Lemma 2.2 [12] and Lemma 2.3 [2].
Lemma 2.1. If 𝑓 =ℎ+𝑔∈𝐾𝐻0 whereℎand𝑔 are given by (1.2), then
∣𝐴𝑛∣ ≤
𝑛+ 1
2 , ∣𝐵𝑛∣ ≤ 𝑛−1
Lemma 2.2. Let 𝑓 =ℎ+𝑔 be given by (1.2). If
∞
∑
𝑛=2
(2𝑛−1−𝛾)∣𝐴𝑛∣+ ∞
∑
𝑛=1
(2𝑛+ 1 +𝛾)∣𝐵𝑛∣ ≤1−𝛾, (2.1)
then 𝑓 is sense-preserving, Goodman-Rønning-type harmonic univalent functions in𝑈 and𝑓 ∈𝐺𝐻(𝛾).
Remark 1. In [12], it is also shown that 𝑓 = ℎ+𝑔 given by (1.4) is in the family 𝑇 𝐺𝐻(𝛾), if and only if the coefficient condition (2.1) holds. Moreover, if
𝑓 ∈𝑇 𝐺𝐻(𝛾), then
∣𝐴𝑛∣ ≤
1−𝛾
2𝑛−1−𝛾, 𝑛≥2,
∣𝐵𝑛∣ ≤
1−𝛾
2𝑛+ 1 +𝛾, 𝑛≥1.
Lemma 2.3. If 𝑎, 𝑏, 𝑐 >0, then
(𝑖)
∞
∑
𝑛=2
(𝑛−1)(𝑎)𝑛−1(𝑏)𝑛−1 (𝑐)𝑛−1(1)𝑛−1 =
𝑎𝑏
𝑐−𝑎−𝑏−1𝐹(𝑎, 𝑏;𝑐; 1), 𝑖𝑓 𝑐 > 𝑎+𝑏+ 1.
(𝑖𝑖)
∞
∑
𝑛=2
(𝑛−1)2(𝑎)𝑛−1(𝑏)𝑛−1 (𝑐)𝑛−1(1)𝑛−1 =
[ (𝑎) 2(𝑏)2 (𝑐−𝑎−𝑏−2)2+
𝑎𝑏 𝑐−𝑎−𝑏−1
]
𝐹(𝑎, 𝑏;𝑐; 1), 𝑖𝑓 𝑐 > 𝑎+𝑏+2.
(𝑖𝑖𝑖)
∞
∑
𝑛=2
(𝑛−1)3(𝑎)𝑛−1(𝑏)𝑛−1 (𝑐)𝑛−1(1)𝑛−1 =
[ (𝑎) 3(𝑏)3 (𝑐−𝑎−𝑏−3)3+
3 (𝑎)2(𝑏)2 (𝑐−𝑎−𝑏−2)2 +
𝑎𝑏 𝑐−𝑎−𝑏−1
]
𝐹(𝑎, 𝑏;𝑐; 1), 𝑖𝑓 𝑐 > 𝑎+𝑏+ 3.
Theorem 2.1. Let 𝑎𝑗, 𝑏𝑗 ∈𝐶∖ {0},𝑐𝑗 ∈𝑅 and𝑐𝑗 >∣𝑎𝑗∣+∣𝑏𝑗∣+ 2 for𝑗= 1,2. If
for some𝛾(0≤𝛾 <1), the inequality
𝑄1𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) +𝑅1𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)≤4 (1−𝛾)
is satisfied, then
Ω( 𝐾𝐻0)
⊂𝐺𝐻(𝛾),
where
𝑄1=
2 (∣𝑎1∣)2(∣𝑏1∣)2
(𝑐1− ∣𝑎1∣ − ∣𝑏1∣ −2)2 + (7−𝛾)
∣𝑎1𝑏1∣
(𝑐1− ∣𝑎1∣ − ∣𝑏1∣ −1) + 2 (1−𝛾),
𝑅1=
2 (∣𝑎2∣)2(∣𝑏2∣)2
(𝑐2− ∣𝑎2∣ − ∣𝑏2∣ −2)2 + (5 +𝛾)
∣𝑎2𝑏2∣
(𝑐2− ∣𝑎2∣ − ∣𝑏2∣ −1).
Proof. Let𝑓 =ℎ+𝑔∈𝐾0
𝐻 where ℎand𝑔 are of the form (1.2) with𝐵1= 0. We need to show that Ω (𝑓) = 𝐻+𝐺∈𝐺𝐻(𝛾), where𝐻 and𝐺 defined by (1.3) are
analytic functions in𝑈.
In view of Lemma 2.2, we need to prove that
𝑃1≤1−𝛾,
where
𝑃1=
∞
∑
𝑛=2
(2𝑛−1−𝛾)
(𝑎1)𝑛−1(𝑏1)𝑛−1
(𝑐1)𝑛−1(1)𝑛−1 𝐴𝑛 +
∞
∑
𝑛=2
(2𝑛+ 1 +𝛾)
(𝑎2)𝑛−1(𝑏2)𝑛−1
In view of Lemma 2.1 and 2.3, it follows that
𝑃1≤ 1 2
∞
∑
𝑛=2
(𝑛+ 1) (2𝑛−1−𝛾)𝐷𝑛−1+ 1 2
∞
∑
𝑛=2
(𝑛−1) (2𝑛+ 1 +𝛾)𝐸𝑛−1
= 1 2
∞
∑
𝑛=2 [
2 (𝑛−1)2+ (5−𝛾) (𝑛−1) + 2 (1−𝛾)]𝐷𝑛−1+ 1 2
∞
∑
𝑛=2 [
2 (𝑛−1)2+ (3 +𝛾) (𝑛−1)]𝐸𝑛−1
=1
2𝑄1𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) + 1
2𝑅1𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)−(1−𝛾).
Now𝑃1≤1−𝛾 follows from the given condition. □
Analogous to Theorem 2.1, we next find connections of the classes𝑆𝐻∗,0,𝐶𝐻0 with 𝐺𝐻(𝛾). However, we first need the following result which may be found in ([1], [6])
or [15].
Lemma 2.4. If𝑓 =ℎ+𝑔∈𝑆𝐻∗,0or𝐶𝐻0 withℎand𝑔as given by (1.2)with𝐵1= 0, then
∣𝐴𝑛∣ ≤
(2𝑛+ 1) (𝑛+ 1)
6 , ∣𝐵𝑛∣ ≤
(2𝑛−1) (𝑛−1)
6 .
Theorem 2.2. Let 𝑎𝑗, 𝑏𝑗 ∈𝐶∖ {0},𝑐𝑗 ∈𝑅 and𝑐𝑗 >∣𝑎𝑗∣+∣𝑏𝑗∣+ 3 for𝑗= 1,2. If
for some𝛾(0≤𝛾 <1), the inequality
𝑄2𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) +𝑅2𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)≤12 (1−𝛾)
is satisfied, then
Ω(𝑆𝐻∗,0)⊂𝐺𝐻(𝛾)andΩ(𝐶𝐻0 )
⊂𝐺𝐻(𝛾),
where
𝑄2= 4 (∣𝑎1∣)3(∣𝑏1∣)3 (𝑐1− ∣𝑎1∣ − ∣𝑏1∣ −3)3
+ 2 (14−𝛾) (∣𝑎1∣)2(∣𝑏1∣)2 (𝑐1− ∣𝑎1∣ − ∣𝑏1∣ −1)2
+ 3 (13−3𝛾) ∣𝑎1𝑏1∣ (𝑐1− ∣𝑎1∣ − ∣𝑏1∣ −1)
+ 6 (1−𝛾),
𝑅2= 4 (∣𝑎2∣)3(∣𝑏2∣)3
(𝑐2− ∣𝑎2∣ − ∣𝑏2∣ −3)3+2 (10 +𝛾)
(∣𝑎2∣)2(∣𝑏2∣)2
(𝑐2− ∣𝑎2∣ − ∣𝑏2∣ −1)2+3 (5 +𝛾)
∣𝑎2𝑏2∣
(𝑐2− ∣𝑎2∣ − ∣𝑏2∣ −1).
Proof. Let𝑓 =ℎ+𝑔∈𝑆∗𝐻,0(𝐶0
𝐻) whereℎand𝑔of the form (1.2) with𝐵1= 0. We
need to show that Ω (𝑓) = 𝐻+𝐺∈𝐺𝐻(𝛾), where𝐻 and𝐺 defined by (1.3) are
analytic functions in𝑈.
In view of Lemma 2.2, we need to prove that
𝑃1≤1−𝛾,
where
𝑃1=
∞
∑
𝑛=2
(2𝑛−1−𝛾)
(𝑎1)𝑛−1(𝑏1)𝑛−1
(𝑐1)𝑛−1(1)𝑛−1 𝐴𝑛 +
∞
∑
𝑛=2
(2𝑛+ 1 +𝛾)
(𝑎2)𝑛−1(𝑏2)𝑛−1
(𝑐2)𝑛−1(1)𝑛−1 𝐵𝑛 .
In view of Lemma 2.3 and 2.4, it follows that
𝑃1≤ 1 6
∞
∑
𝑛=2
[(2𝑛+ 1) (𝑛+ 1) (2𝑛−1−𝛾)]𝐷𝑛−1+ 1 6
∞
∑
𝑛=2
= 1 6
∞
∑
𝑛=2 [
4 (𝑛−1)3+ 2 (8−𝛾) (𝑛−1)2+ (19−7𝛾) (𝑛−1) + 6 (1−𝛾)]𝐷𝑛−1
+1 6
∞
∑
𝑛=2 [
4 (𝑛−1)3+ 2 (4 +𝛾) (𝑛−1)2+ (3 +𝛾) (𝑛−1)]𝐸𝑛−1
=1
6𝑄2𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) + 1
6𝑅2𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)−(1−𝛾)
Now𝑃1≤1−𝛾 follows from the given condition. □
In order to determine connection between 𝑇 𝑁𝐻(𝛽) and 𝐺𝐻(𝛾), we need the
following results in Lemma 2.5 [4] and Lemma 2.6 [3].
Lemma 2.5. Let 𝑓 = ℎ+𝑔 where ℎ and 𝑔 as given by (1.4) with 𝐵1 = 0, and suppose that0≤𝛽 <1. Then
𝑓 ∈𝑇 𝑁𝐻(𝛽)⇔ ∞
∑
𝑛=2
𝑛∣𝐴𝑛∣+ ∞
∑
𝑛=2
𝑛∣𝐵𝑛∣ ≤1−𝛽.
Remark 2. If 𝑓 ∈𝑇 𝑁𝐻(𝛽), then
∣𝐴𝑛∣ ≤
1−𝛽
𝑛 , 𝑛≥2,
∣𝐵𝑛∣ ≤
1−𝛽
𝑛 , 𝑛≥1.
Lemma 2.6. Let 𝑎, 𝑏 ∈ 𝐶∖ {0}, 𝑎 ∕= 1, 𝑏 ∕= 1, 𝑐 ∈ (0,1)∪(1,∞) and 𝑐 > max{0,∣𝑎∣+∣𝑏∣ −1}. Then
∞
∑
𝑛=1 1 𝑛
(∣𝑎∣)𝑛−1(∣𝑏∣)𝑛−1 (𝑐)𝑛−1(1)𝑛−1 =
(𝑐− ∣𝑎∣ − ∣𝑏∣)
(∣𝑎∣ −1) (∣𝑏∣ −1)𝐹(∣𝑎∣,∣𝑏∣;𝑐; 1)−
(𝑐−1) (∣𝑎∣ −1) (∣𝑏∣ −1).
Theorem 2.3. Let𝑎𝑗, 𝑏𝑗 ∈𝐶∖ {0},𝑎𝑗∕= 1, 𝑏𝑗 ∕= 1,𝑐𝑗∈𝑅and𝑐𝑗>max{0,∣𝑎𝑗∣+∣𝑏𝑗∣ −1}
for𝑗= 1,2. If for some 𝛽(0≤𝛽 <1) and𝛾(0≤𝛾 <1), the inequality
𝑄3𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) +𝑅3𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)≤ (1−𝛾) (2−𝛽) (1−𝛽)
−(1 +𝛾)
[ (𝑐
1−1)
(∣𝑎1∣ −1) (∣𝑏1∣ −1)−
(𝑐2−1) (∣𝑎2∣ −1) (∣𝑏2∣ −1)
]
is satisfied, then
Ω (𝑇 𝑁𝐻(𝛽))⊂𝐺𝐻(𝛾),
where
𝑄3= 2−(1 +𝛾)
(𝑐1− ∣𝑎1∣ − ∣𝑏1∣) (∣𝑎1∣ −1) (∣𝑏1∣ −1)
and
𝑅3= 2 + (1 +𝛾)
Proof. Let 𝑓 = ℎ+𝑔 ∈ 𝑇 𝑁𝐻(𝛽) where ℎ and 𝑔 are given by (1.4). In view of
Lemma 2.2, it is enough to show that𝑃2≤1−𝛾, where
𝑃2=
∞
∑
𝑛=2
(2𝑛−1−𝛾)
(𝑎1)𝑛−1(𝑏1)𝑛−1
(𝑐1)𝑛−1(1)𝑛−1 𝐴𝑛 +
∞
∑
𝑛=1
(2𝑛+ 1 +𝛾)
(𝑎2)𝑛−1(𝑏2)𝑛−1
(𝑐2)𝑛−1(1)𝑛−1 𝐵𝑛 .
Using Remark 2 and Lemma 2.6, it follows that
𝑃2≤(1−𝛽) [∞
∑
𝑛=2 (
2−(1 +𝛾) 𝑛
)
𝐷𝑛−1+
∞
∑
𝑛=1 (
2 + (1 +𝛾) 𝑛
) 𝐸𝑛−1
]
= (1−𝛽) [
𝑄3𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) +𝑅3𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)−(1−𝛾) + (1 +𝛾) (𝑐1−1) (∣𝑎1∣ −1) (∣𝑏1∣ −1) −
(1 +𝛾) (𝑐2−1) (∣𝑎2∣ −1) (∣𝑏2∣ −1)
]
≤1−𝛾,
by the given hypothesis. □
In next theorem, we establish connections between𝑇 𝐺𝐻(𝛾) and𝐺𝐻(𝛾).
Theorem 2.4. Let 𝑎𝑗, 𝑏𝑗∈𝐶∖ {0},𝑐𝑗∈𝑅 and𝑐𝑗 >∣𝑎𝑗∣+∣𝑏𝑗∣ for𝑗= 1,2. If for
some𝛾(0≤𝛾 <1), the inequality
𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1) +𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)≤2
is satisfied, then
Ω (𝑇 𝐺𝐻(𝛾))⊂𝐺𝐻(𝛾).
Proof. Making use of Lemma 2.2 and the definition of𝑃2 in Theorem 2.3, we only need to prove that𝑃2≤1−𝛾.
Using Remark 1, it follows that
𝑃2=
∞
∑
𝑛=2
(2𝑛−1−𝛾)
(𝑎1)𝑛−1(𝑏1)𝑛−1
(𝑐1)𝑛−1(1)𝑛−1 𝐴𝑛
+
∞
∑
𝑛=1
(2𝑛+ 1 +𝛾)
(𝑎2)𝑛−1(𝑏2)𝑛−1
(𝑐2)𝑛−1(1)𝑛−1 𝐵𝑛
≤(1−𝛾) [∞
∑
𝑛=2
𝐷𝑛−1+
∞
∑
𝑛=1 𝐸𝑛−1
]
= (1−𝛾) [𝐹(∣𝑎1∣,∣𝑏1∣;𝑐1; 1)−1 +𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)]
≤1−𝛾,
by the given condition and this completes the proof. □
In the next result, we establish connections between 𝑇 𝐺𝐻(𝛾) and 𝐺𝐻(𝛾) by
diluting the restrictions on the complex coefficients of Theorem 2.4.
Theorem 2.5. If 𝑎1, 𝑏1 > −1, 𝑎1𝑏1 <0, 𝑐1 > max{0, 𝑎1+𝑏1}, 𝑎2, 𝑏2 ∈ 𝐶∖ {0} and𝑐2>∣𝑎2∣+∣𝑏2∣, then a sufficient condition for
Ω (𝑇 𝐺𝐻(𝛾))⊂𝐺𝐻(𝛾)
is that
𝐹(𝑎1, 𝑏1;𝑐1; 1)−𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)≥0
Proof. Let𝑓 =ℎ+𝑔∈𝑇 𝐺𝐻(𝛾) withℎand𝑔 in (1.4). Then
Ω(𝑓) =𝑧−
∞
∑
𝑛=2
(𝑎1)𝑛−1(𝑏1)𝑛−1 (𝑐1)𝑛−1(1)𝑛−1
∣𝐴𝑛∣𝑧𝑛+ ∞
∑
𝑛=1
(𝑎2)𝑛−1(𝑏2)𝑛−1 (𝑐2)𝑛−1(1)𝑛−1
∣𝐵𝑛∣𝑧𝑛.
This function can be rewritten as
Ω(𝑓) =𝑧+∣𝑎1𝑏1∣ 𝑐1
∞
∑
𝑛=2
(𝑎1+ 1)𝑛−2(𝑏1+ 1)𝑛−2
(𝑐1+ 1)𝑛−2(1)𝑛−1 ∣𝐴𝑛∣𝑧
𝑛
+
∞
∑
𝑛=1
(𝑎2)𝑛−1(𝑏2)𝑛−1
(𝑐2)𝑛−1(1)𝑛−1 ∣𝐵𝑛∣𝑧
𝑛.
In view of Lemma 2.2, we need to prove that𝑃3≤1−𝛾, where
𝑃3=
∞
∑
𝑛=2
(2𝑛−1−𝛾)∣𝑎1𝑏1∣ 𝑐1
(𝑎1+ 1)𝑛−2(𝑏1+ 1)𝑛−2
(𝑐1+ 1)𝑛−2(1)𝑛−1 𝐴𝑛
+
∞
∑
𝑛=1
(2𝑛+ 1 +𝛾)
(𝑎2)𝑛−1(𝑏2)𝑛−1
(𝑐2)𝑛−1(1)𝑛−1 𝐵𝑛
≤(1−𝛾) [
∣𝑎1𝑏1∣ 𝑎1𝑏1
∞
∑
𝑛=2
𝐷𝑛−1+
∞
∑
𝑛=1 𝐸𝑛−1
]
≤(1−𝛾) [−𝐹(𝑎1, 𝑏1;𝑐1; 1) + 1 +𝐹(∣𝑎2∣,∣𝑏2∣;𝑐2; 1)]
≤1−𝛾,
by the given condition. □
In next theorem, we present conditions on the parameters𝑎1, 𝑎2, 𝑏1, 𝑏2, 𝑐1, 𝑐2and obtain a characterization for operator Ω which maps𝑇 𝐺𝐻(𝛾) onto itself.
Theorem 2.6. Let 𝑎𝑗, 𝑏𝑗 >0, 𝑐𝑗 > 𝑎𝑗+𝑏𝑗 for (𝑗= 1,2) and𝛾(0≤𝛾 <1).Then
Ω (𝑇 𝐺𝐻(𝛾))⊂𝑇 𝐺𝐻(𝛾), if and only if
𝐹(𝑎1, 𝑏1;𝑐1; 1) +𝐹(𝑎2, 𝑏2;𝑐2; 1)≤2.
Proof. Let𝑓 =ℎ+𝑔 ∈𝑇 𝐺𝐻(𝛾) with ℎand 𝑔 in (1.4). In view of Remark 1, we
only need to prove that Ω(𝑓) given by (1.3) is in𝑇 𝐺𝐻(𝛾), if and only if𝑃2≤1−𝛾, where
𝑃2=
∞
∑
𝑛=2
(2𝑛−1−𝛾)
(𝑎1)𝑛−1(𝑏1)𝑛−1
(𝑐1)𝑛−1(1)𝑛−1 𝐴𝑛
+
∞
∑
𝑛=1
(2𝑛+ 1 +𝛾)
(𝑎2)𝑛−1(𝑏2)𝑛−1
(𝑐2)𝑛−1(1)𝑛−1 𝐵𝑛
.
Using the coefficient estimates stated in Remark 1, we obtain
𝑃2≤(1−𝛾) [∞
∑
𝑛=2
𝐷𝑛−1+
∞
∑
𝑛=1 𝐸𝑛−1
]
≤(1−𝛾) [𝐹(𝑎1, 𝑏1;𝑐1; 1)−1 +𝐹(𝑎2, 𝑏2;𝑐2; 1)]
≤(1−𝛾),
by the given condition. □
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Saurabh Porwal and K.K. Dixit
