DOI: 10.1515/auom-2016-0021
An. Şt. Univ. Ovidius Constanţa
Vol. 24(1),2016, 353–369
Some characteristic properties of analytic
functions
R. K. Raina, Poonam Sharma and G. S. Sălăgean
Abstract
In this paper, we consider a class L (λ, µ; φ) of analytic functions f
defined in the open unit disk U satisfying the subordination condition
that
Dλ+1 f (z)
≺ φ(z) (λ ∈ N0 , µ ≥ 0; z ∈ U),
q(z) λ
D f (z)
µ−2
, Dλ is the Sălăgean operator and φ(z) is a
where q(z) = Dλzf (z)
convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination
results, and an inclusion result for the above class when the function
φ(z) is a bilinear mapping in the open unit disk. For these functions
f (z) , sharp bounds for the initial coefficient and for the Fekete-Szegö
functional are determined, and also some integral representations are
given.
1
Introduction
Let A denote a class of functions f analytic in the open unit disk
U = {z : |z| < 1} with the normalization that f (0) = 0 = f 0 (0) − 1, that is the
function f has the series expansion
f (z) = z +
∞
X
ak+1 z k+1 , z ∈ U.
(1.1)
k=1
Key Words: Analytic functions, subordination, starlike function, Fekete-Szegö functional, Sălăgean operator.
2010 Mathematics Subject Classification: Primary 30C45, 30C50.
Received: October, 2014.
Revised: November, 2014.
Accepted: January, 2015.
353
354
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
For f ∈ A of the form (1.1), we define the operator denoted Dλ , λ ∈ Z =
N ∪ {0} ∪ −N = {· · · , −2, −1, 0, 1, 2, · · · } by
Dλ f (z) = z +
∞
X
(k + 1)λ ak+1 z k+1 , z ∈ U.
k=1
The operator Dλ was considered in [16] and for λ ∈ N0 = N ∪ {0} it is known
as Sălăgean operator of order λ. In this case, it can be defined equivalently by
D0 f (z) = f (z), D1 f (z) = Df (z) = zf 0 (z), Dλ f (z) = D Dλ−1 f (z) , λ ∈ N.
We note that Dλ D−λ f (z) = f (z), for all λ ∈ Z.
Classes of analytic functions f ∈ A involving the quotient
zf 0 (z)
f (z)
f (z)
z
=
z 2 f 0 (z)
f 2 (z)
have been studied in [2, 10, 11, 17, 19]. Also, the classes involving the quotient
1+µ
zf 0 (z)
f (z)
z
0
=
f
(z)
have been studied for µ > −1 in [13] (for −1 < µ <
µ
f (z)
( f (z)
z )
0 in [9] and for 0 < µ < 1 in [22]). Moreover, for µ 6= 0, a class involving a
certain linear operator under a subordination condition is investigated in [4].
1+µ
µ
z
z
and f (z)
for 0 < µ < 1
Interestingly, a combination of both f 0 (z) f (z)
was studied in [23] (see also [18, Definition 1.1, p. 5]).
It may be observed that the operator Dλ preserves the class A and hence
λ
D f (z) = 0 at z = 0. Let λ ∈ N0 and let f ∈ A be such that Dλ f (z) 6= 0 for
z ∈ U \ {0}. We define a function q(z) by
µ−2
z
q(z) =
(µ ≥ 0, µ 6= 2, z ∈ U \ {0}) and q(0) = 1, (1.2)
Dλ f (z)
µ−2
are taken into
where we assume that only principal values of Dλzf (z)
consideration. Clearly, the function q(z) is analytic in the open unit disk U.
λ+1
(z)
Recently, by considering the expression q(z) DDλ f f(z)
, µ ≥ 0, Prajapat and
Raina [14] investigated a class B (λ, µ; α) of functions f ∈ A satisfying the
condition that
λ+1
f (z)
< 1 − α, µ ≥ 0, 0 ≤ α < 1, z ∈ U.
q(z) D
−
1
λ
D f (z)
It may be noted that for λ = 0, α = 0, µ = 3, the class B (0, 3; 0) = U was
earlier studied by Ozaki and Nunukawa in [11] (see also Obradovic et al. [10]
and Singh [19]), where it is proved that the functions f ∈ U are univalent.
For two analytic functions p, q such that p(0) = 1 = q(0), we say that p is
subordinate to q in U and write p(z) ≺ q(z), z ∈ U, if there exists a Schwarz
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
355
function w, analytic in U with w(0) = 0, and |w(z)| < 1, z ∈ U such that
p(z) = q(w(z)), z ∈ U. Furthermore, if the function q is univalent in U, then
we have the following equivalence:
p(z) ≺ q(z) ⇔ p(0) = q(0) and p(U) ⊂ q(U).
Janowski [5] defined a class P(A, B) of analytic functions p(z), z ∈ U, with
1+Az
, −1 ≤ B < A ≤ 1, z ∈ U. If p ∈ P(A, B), then it
p(0) = 1, if p(z) ≺ 1+Bz
follows that
p(z) − 1 − AB < A − B for − 1 < B < A ≤ 1, z ∈ U
(1.3)
2
1 − B 1 − B2
and for B = −1,
< (p(z)) >
1−A
, −1 < A ≤ 1, z ∈ U.
2
(1.4)
The class P(1, −1) = P is a Carathéodory class of functions which are analytic
with positive real part in U.
In this paper, we consider a new class L (λ, µ; φ) of analytic functions
(which evidently generalizes the class B (λ, µ; α)) comprising of functions f ∈
A if and only if (for Dλzf (z) 6= 0 in U) :
q(z)
Dλ+1 f (z)
≺ φ(z) (λ ∈ N0 , µ ≥ 0; z ∈ U),
Dλ f (z)
where q(z) is given by (1.2), Dλ is the Sălăgean operator and φ ∈ P is a convex
function in U; see also the works in [20] and [21].
We note that L (0, 2; φ) = S ∗ [φ] and L (1, 2; φ) = K [φ] are the classes
introduced by Ma and Minda [7] which include several well-known starlike
and convex mappings as special cases.
1+Az
For the bilinear transformation φ(z) = 1+Bz
(−1 ≤ B < A ≤ 1, z ∈ U),
1+Az
we denote L λ, µ; 1+Bz by T (λ, µ; A, B) .
We observe that the class T (λ, 2; A, B) = Pλ+1
(A, B) was earlier considλ
ered by Kuroki and Owa [6, Remark 2, p. 4] for any integer λ, and for complex parameters A and B, the class T (0, 3; A, B) = T (A, B) was studied by
Shanmugam and Gangadharan [17]. The class T (0, 2; A, B) = S(A, B) is the
class of Janowski starlike functions [5]. Further, the classes T (λ, µ; 1 − α, 0) =
B (λ, µ; α) and T (0, 3; 1 − α, 0) = B (α) (0 ≤ α < 1) were studied in [2] and
various subordination properties and sufficient conditions were investigated in
these classes of functions.
356
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
For the purpose of this paper, we consider the functions f ∈ A of the form
(1.1) such that the coefficients bk (k ∈ N) defined by
q(z) =
z
Dλ f (z)
µ−2
=1+
∞
X
bk z k , z ∈ U,
(1.5)
k=1
to be non-negative.
Example 1. Let µ ≥ 0, µ 6= 2 and let λ ∈ N0 ; if we consider f ∈ A of the
form f (z) = D−λ zez/(2−µ) , then q(z) = ez has the form (1.5).
Example 2. Let 0 < µ < 2, µ = pr , p, r ∈ N and let λ = 1; if
f (z) =
1 (1 + z)r+1 − 1 , then q(z) = (1 + z)2r−p .
r+1
Example 3. Let µ > 2 and let λ ∈ N0 ; if we consider f ∈ A, f (z) = z − an z n ,
where an > 0 and n ≥ 2, then
2−µ
q(z) = 1 − nλ an z n−1
=
1+
∞
X
(−µ + 2)(−µ + 1)(−µ) · · · (−µ − (k − 3))
k!
k=1
1+
∞
X
(µ − 2)(µ − 1)(µ) · · · (µ + k − 3)
k=1
k!
−nλ an z n−1
k
=
k
nkλ (an ) z k(n−1) .
Example 4. Let µ ≥ 0, µ 6= 2 and
let λ ∈ N0 ; if we consider f ∈ A of the
form f (z) = D−λ z(1 + z)1/(2−µ) , then q(z) = 1 + z.
Example 5. Evidently, for f of the form (1.1) with ak+1 ≥ 0 and for µ = 1
λ
and λ ∈ N0 , the coefficients bk are given by bk = (k + 1) ak+1 , k ∈ N.
In this paper, we concentrate ourselves in investigating some basic characteristic properties such as the coefficient inequality, the radius result, subordination and inclusion properties for the functions f ∈ T (λ, µ; A, B) . Sharp
bounds for the initial coefficient, the Fekete-Szegö functional of functions f (z)
and integral representations belonging to this class are also determined.
2
A Coefficient Inequality
We begin to investigate the coefficient inequality of functions f ∈ T (λ, µ; A, B) ,
which is contained in the following:
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
357
Theorem 1. Let −1 ≤ B < A ≤ 1, µ ∈ [1, 3] \ {2} , let f ∈ A of the form
(1.1) and let bk , k ∈ N defined by (1.5) be non-negative. If
∞
X
k−µ+2
k=1
|µ − 2|
bk ≤
A−B
,
1 + |B|
(2.1)
then f ∈ T (λ, µ; A, B) . The condition (2.1) is necessary for f ∈ T (λ, µ; A, B)
provided that −1 ≤ B ≤ 0 < A ≤ 1, µ ∈ (2, 3] .
Proof. Let
p(z) = q(z)
Dλ+1 f (z)
, z ∈ U,
Dλ f (z)
(2.2)
where q(z) is given by (1.2) then, we get
p(z) = q(z) −
zq 0 (z)
.
µ−2
Since f ∈ T (λ, µ; A, B) , if and only if
p(z) − 1 A − Bp(z) < 1, z ∈ U,
(2.3)
(2.4)
therefore, if we consider
P = |p(z) − 1| − |A − Bp(z)| ,
then in view of (1.5) and (2.3), we get
∞
∞
Xk − µ + 2
X
k
−
µ
+
2
P = −
bk z k − A − B +
Bbk z k µ−2
µ−2
k=1
k=1
"
#
∞
∞
X
X
k−µ+2
k−µ+2
<
bk − A − B −
|B| bk
|µ − 2|
|µ − 2|
=
k=1
∞
X
k=1
k=1
k−µ+2
(1 + |B|) bk − (A − B) ≤ 0,
|µ − 2|
on using (2.1). For the necessary part, we consider for −1 ≤ B ≤ 0 < A ≤ 1,
µ ∈ (2, 3] that f ∈ T (λ, µ; A, B) , then from (2.4), in view of (1.5) and (2.3),
we have
∞
P
k−µ+2
k
b
z
−
k
µ−2
k=1
< 1, z ∈ U.
(2.5)
∞
A − B − P k−µ+2 |B| bk z k µ−2
k=1
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
358
Since p(z) in (2.3) is real for real z, letting z → 1− along real axis, we get
from the condition that
1−A
< (p(z)) >
1−B
which ensures that the denominator under the mod sign in the inequality (2.5)
remains positive and then we have
∞
P
k=1
A−B−
k−µ+2
µ−2 bk
∞
P
k=1
≤ 1,
k−µ+2
µ−2
|B| bk
which proves (2.1). This completes the proof of Theorem 1.
From Theorem 1, for the cases when B = 0 and B = −1 (µ ∈ (2, 3]),
respectively, and applying the well-known assertions (1.3) and (1.4), we get
the following results.
Corollary 1. Let 0 < A ≤ 1, µ ∈ (2, 3] and let f ∈ A of the form (1.1) and
let bk , k ∈ N defined by (1.5) be non-negative. Then
λ+1
f (z)
q(z) D
− 1 < A, z ∈ U,
Dλ f (z)
if and only if
∞
X
k−µ+2
k=1
µ−2
bk ≤ A.
Corollary 2. Let −1 < A ≤ 1, µ ∈ (2, 3] and let f ∈ A of the form (1.1) and
let bk , k ∈ N defined by (1.5) be non-negative. Then
Dλ+1 f (z)
1−A
, z ∈ U,
>
< q(z) λ
D f (z)
2
if and only if
∞
X
k−µ+2
k=1
µ−2
bk ≤
1+A
.
2
Remark 1. For λ = 0, µ = 3, A = 1, Corollary 1 corresponds to the known
result of Ponnusamy and Sahoo [13, Theorem 7, p. 400].
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
3
359
Radius Result
Theorem 2. Let −1 ≤ B < A ≤ 1, µ ∈ [1, 3] \ {2} and let f ∈ A of the
form (1.1) and let bk , k ∈ N defined by (1.5) be non-negative and satisfy the
condition that
∞
X
k−µ+2
2
(bk ) ≤ 1.
(3.1)
|µ − 2|
k=1
Then
1
f (rz) ∈ T (λ, µ; A, B)
r
for 0 < r ≤ r0 , where r0 = r0 (µ, A, B) is given by
p
η 2 |µ − 2|
r0 =
(3.2)
1/2
[3 − µ + 2η 2 |µ − 2| + E]
q
2
where E = {3 − µ + 2η 2 |µ − 2|} + 4η 2 |µ − 2| (µ − 2 − η 2 |µ − 2|) and η =
A−B
.
1 + |B|
Proof. Let f ∈ A be of the form (1.1) with µ ∈ [1, 3] \ {2} . Then for
0 < r ≤ 1, we have
µ−2
∞
X
z
q(rz) = 1 λ
=1+
bk rk z k , bk ≥ 0, z ∈ U,
D
f
(rz)
r
k=1
where q(z) is given by (1.2). Thus, by Theorem 1, 1r f (rz) ∈ T (λ, µ; A, B) if
R :=
∞
X
k−µ+2
k=1
|µ − 2|
bk r k ≤
A−B
.
1 + |B|
By Cauchy-Schwarz inequality and the condition (3.1), we obtain that
!1/2 ∞
!1/2
∞
X
Xk − µ + 2
k−µ+2
2
2k
R ≤
(bk )
r
|µ − 2|
|µ − 2|
k=1
k=1
!1/2
∞
X
1
2k
≤ p
(k − µ + 2) r
|µ − 2| k=1
!1/2
r2
1
r4
+ (3 − µ)
= p
1 − r2
|µ − 2| (1 − r2 )2
=
1/2
1
r A−B
p
3 − µ + (µ − 2) r2
≤
,
2
1 + |B|
|µ − 2| 1 − r
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
360
provided that the inequality
p
1/2
r ≤ η |µ − 2|
3 − µ + (µ − 2) r2
2
1−r
holds, where
A−B
1+|B|
= η, or equivalently
1 2
1 η |µ − 2| − 2 3 − µ + 2η 2 |µ − 2| − µ − 2 − η 2 |µ − 2| ≥ 0
4
r
r
holds, which provides the value of r0 given by (3.2). This proves Theorem 2.
Remark 2. By setting λ = 0, µ = 3 − α (0 ≤ α < 1) and B = 0, A = η,
Theorem 2 coincides with the result of Ponnusamy and Sahoo [13, Theorem
5, p. 398] for univalent functions f (z).
4
Subordination Result
Theorem 3. Let −1 ≤ B < A ≤ 1, µ ∈ [1, 2) and let f ∈ A of the form (1.1)
and let bk , k ∈ N defined by (1.5) be non-negative. If f ∈ T (λ, µ; A, B) , then
z
Dλ f (z)
µ−2
≺
1 + Az
,z ∈ U
1 + Bz
(4.1)
and hence,
bk ≤ A − B, k ∈ N.
(4.2)
Proof. Let q(z) be defined by (1.2), which is analytic in U with q(0) = 1, then
from (2.3), we have
q(z) +
zq 0 (z)
1 + Az
≺
, z ∈ U,
2−µ
1 + Bz
which by the result of Hallenbeck and Ruscheweyh [3] proves (4.1). Further,
on using a well-known result of Rogosinski [15] on subordination, and in view
of (1.5), the subordination (4.1) gives the coefficient inequality (4.2).
5
Inclusion Result
Theorem 4. Let λ ∈ N0 , −1 ≤ B ≤ 0 < A ≤ 1, µ ∈ (2, 3] and let f ∈ A of
the form (1.1) and let bk , k ∈ N defined by (1.5) be non-negative. If
Dλ+1 f (z)
1 + Az
≺
, z ∈ U,
Dλ f (z)
1 + Bz
(5.1)
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
then
∞
X
(
k−
A−B
1+|B|
(µ − 2)
µ−2
k=1
)
bk ≤
A−B
.
1 + |B|
361
(5.2)
Hence, Pλ+1
(A, B) ⊂ T (λ, µ; A, B) .
λ
Proof. From (5.1), we have
λ+1
D f (z)
λ+1
−
1
λ f (z)
1
+
Az
D
f
(z)
D
λ+1
< 1, z ∈ U.
f ∈ Pλ (A, B) ⇔
≺
⇔
λ+1
D
f (z) Dλ f (z)
1 + Bz
A − B Dλ f (z) (5.3)
Let q(z) be defined by (1.2), then on using (2.2) and (2.3), we get
zq 0 (z)
Dλ+1 f (z)
.
= (µ − 2) 1 −
q(z)
Dλ f (z)
Hence, by (1.5), the condition (5.3) can equivalently be expressed as
∞
P
− kbk z k
k=1
< 1, z ∈ U.
∞
∞
P
P
(A − B) (µ − 2) 1 +
bk z k + B
kbk z k k=1
(5.4)
k=1
(z)
Since DDλ f f(z)
is real for real z, letting z → 1− along the real axis, we get
from (5.1) that
λ+1
1−A
Dλ+1 f (z)
1−A
D
f (z)
>
⇔
>
<
λ
λ
D f (z)
1−B
D f (z)
1−B
λ+1
and hence, for being B ≤ 0,
A−B
Dλ+1 f (z)
1−A
> A + |B|
> 0,
Dλ f (z)
1−B
which ensures that the denominator under the mod sign in the inequality (5.4)
is positive. Thus, we have
∞
P
kbk
k=1
(A − B) (µ − 2) −
∞
P
k=1
{|B| k − (A − B) (µ − 2)} bk
≤ 1,
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
which yields the desired inequality (5.2). Further, since
f∈
A−B
1+|B|
362
≤ 1, if
Pλ+1
(A, B),
λ
we have by (5.2) that
)
(
A−B
∞
∞
X
X
(µ − 2)
k − 1+|B|
A−B
k−µ+2
bk ≤
bk ≤
,
µ−2
µ−2
1 + |B|
k=1
k=1
and consequently by Theorem 1, we conclude that f ∈ T (λ, µ; A, B) . This
proves the inclusion result.
6
Fekete-Szegö Problem
Let f (z) of the form (1.1) be in the class T (λ, µ; A, B), then for some Schwarz
function w(z), we get
q(z)
1 + Aw(z)
Dλ+1 f (z)
=
, z ∈ U,
Dλ f (z)
1 + Bw(z)
(6.1)
where q(z) is given by (1.2) and upon using the series:
Dλ f (z) = z +
∞
X
λ
(k + 1) ak+1 z k+1 , z ∈ U
k=1
and performing elementary calculations, we can write the series expansion
Dλ+1 f (z)
Dλ f (z)
µ−1
Dλ+1 f (z)
z
z
Dλ f (z)
q(z)
=
=
1 + (3 − µ) 2λ a2 z + (4 − µ) 3λ a3 − (µ − 1) 22λ−1 a22 z 2 + ... (6.2)
For the Schwarz function w(z), let φ ∈ P be defined by
φ(z) =
1 + w(z)
= 1 + c1 z + c2 z 2 + ....
1 − w(z)
(6.3)
Then
1 + Aw(z)
A−B
A−B
=1+
c1 z +
1 + Bw(z)
2
2
c2 −
B+1 2 2
c1 z + ...,
2
(6.4)
and from (6.2) and (6.4), we get
(3 − µ) 2λ a2
=
(4 − µ) 3λ a3 − (µ − 1) 22λ−1 a22
=
A−B
c1 ,
(6.5)
2 A−B
B+1 2
c2 −
c1 . (6.6)
2
2
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
363
In order to find in this section
sharp upper bound for |a2 | and for the Fekete
Szegö functional a3 − ρa22 (ρ ∈ C) , we use the following result from [12, p.
166] (see also [1, p. 41]).
Lemma 1. Let φ ∈ P be of the form φ(z) = 1 + c1 z + c2 z 2 + ..., then
c2 − c21 /2 ≤ 2 − |c1 |2 /2
and |ck | ≤ 2 for all k ∈ N.
Theorem 5. Let −1 ≤ B < A ≤ 1, µ ∈ (2, 3), and f ∈ A be of the form (1.1)
belong to the class T (λ, µ; A, B) , then
|a2 | ≤
A−B
,
(3 − µ) 2λ
and for all ρ ∈ C :
a3 − ρa22 ≤
)
( µ−1
(A − B) (4 − µ) 3λ
A−B
2λ−1
max 1, 2
−ρ
− B .
2
λ
λ
2λ
(4 − µ) 3
3
(3 − µ) 2
The result is sharp if B = −1 or if B = 0.
Proof. Let the function f (z) of the form (1.1) belong to the class T (λ, µ; A, B) ,
then using the Carathéodory condition: |c1 | ≤ 2 in (6.5), for the functions
φ ∈ P of the form (6.3), we get
|a2 | ≤
A−B
,
(3 − µ) 2λ
which by virtue of (6.5) and (6.6) gives
a3 − ρa22
2
(µ − 1) 22λ−1
(A − B)
−
ρ
c21 +
2
3λ
4 (3 − µ) 22λ
A−B
B+1 2
A−B
1 2
c
c
+
c
−
=
c
−
+
2
2
1
2 (4 − µ) 3λ
2
2 (4 − µ) 3λ
2 1
!
2
µ − 1 2λ−1
(A − B)
(A − B) B
+
2
−ρ
−
c21 .
2
3λ
4 (4 − µ) 3λ
(3 − µ) 22λ+2
=
By Lemma 1, it follows that
2
a3 − ρa22 ≤ F (|c1 |) = C + CD |c1 | ,
4
364
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
where
(A − B)
> 0, D = |E|−1, E =
C=
(4 − µ) 3λ
µ − 1 2λ−1
2
−ρ
3λ
(A − B) (4 − µ) 3λ
2
(3 − µ) 22λ
−B.
As |c1 | ≤ 2, we infer that
a3 − ρa22 ≤
F (0) = C,
|E| ≤ 1
.
F (2) = C |E| |E| ≥ 1
In the case when B = −1, the sharpness can be verified for the functions given
by
Dλ+1 f (z)
1 + Az 2
1 + Az
q(z) λ
=
or
,z ∈ U
D f (z)
1 − z2
1−z
and, in case when B = 0, the sharpness can be verified for functions given by
q(z)
Dλ+1 f (z)
= 1 + Az 2 (or 1 + Az) , z ∈ U,
Dλ f (z)
where q(z) is given by (1.2). This completes the proof of Theorem 5.
Remark 3. For λ = 0, µ = 2 + ν (0 < ν < 1) , Theorem 5 corresponds (for
A = 1 − 2α (0 ≤ α < 1) , B = −1) to Theorem 1, and (for A = k (0 < k ≤ 1) ,
B = 0) to Theorem 2 of Tuneski and Darus [22, pp. 64-65].
7
Integral Representations
Theorem 6. Let −1 ≤ B < A ≤ 1, 2 < µ ≤ 3 and f ∈ A be of the form
(1.1). If f ∈ T (λ, µ; A, B) , then for some Schwarz functions w1 (z) and w2 (z),
w1 (0) = 0 = w10 (0) − 1 (in case 2 < µ < 3):
z
λ
D f (z)
µ−2
= 1 − (µ − 2) (A − B) z
µ−2
Zz
tµ−1
w1 (t)
dt, z ∈ U,
(1 + Bw1 (t))
0
(7.1)
and w2 (0) = 0 = w20 (0) (in case µ = 3):
z
= 1 − 2λ a2 z − (A − B) z
Dλ f (z)
Zz
0
w2 (t)
dt, z ∈ U.
t2 (1 + Bw2 (t))
(7.2)
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
365
Proof. Let f ∈ A be of the form (1.1), then from (6.2), we have
q(z)
Dλ+1 f (z)
= 1 + (3 − µ) 2λ a2 z + (4 − µ) 3λ a3 − (µ − 1) 22λ−1 a22 z 2 + ...,
λ
D f (z)
where q(z) is given by (1.2). Hence, if f ∈ T (λ, µ; A, B) , the Schwarz function
w(z) in (6.1) is given by
w1 (z) : w1 (0) = 0 = w10 (0) − 1, if 2 < µ < 3,
w(z) =
w2 (z) : w2 (0) = 0 = w20 (0),
if
µ = 3.
It is easy to verify that
d
dz
1
(Dλ f (z))
µ−2
−
!
1
z µ−2
µ−2
= − µ−1
z
Dλ+1 f (z)
q(z) λ
−1 ,
D f (z)
where q(z) is given by (1.2) and therefore by (6.1), we get
!
1
d
1
(µ − 2) (A − B) w(z)
− µ−2 = − µ−1
, z ∈ U.
µ−2
dz (Dλ f (z))
z
z
(1 + Bw(z))
(7.3)
From (1.5), we also have
1
µ−2
(Dλ f (z))
−
1
z µ−2
=
1
z µ−2
[q(z) − 1] =
∞
X
bk z k−µ+2 ,
k=1
which yields that
1
µ−2
(Dλ f (z))
−
1
z µ−2
!
=
b1 ,
µ=3
.
0, 2 < µ < 3
z=0
By using (1.5) and (6.2), and equating the coefficient of z on both the sides
of (2.3), we find that b1 = − (µ − 2) 2λ a2 . Now, upon integrating (7.3), we
obtain the desired representations given by (7.1) and (7.2).
Remark 4. For µ = 3 and λ = 0, the above representation (7.2) corresponds
to the representation due to Shanmugam and Gangadharan in [17, Theorem
2.1, pp. 2-3] and corresponds to Theorem 1 of Obradovic et al. [10] if A = 1
and B = 0.
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
366
Corollary 3. Let −1 ≤ B ≤ 0 < A ≤ 1 and f ∈ A be of the form (1.1). If
f ∈ T (λ, µ; A, B) , then for µ ∈ (2, 3) :
µ−2
z
(7.4)
−
1
Dλ f (z)
(
(µ−2)A
3−µ |z| , B = 0, z ∈ U,
≤
|z|
(µ − 2) (A − B) 3−µ 2 F1 (1, 3 − µ, 4 − µ; |B| |z|) , −1 ≤ B < 0, z ∈ U,
and for µ = 3 :


z
Dλ f (z) − 1 ≤  2λ a2 |z| +
2
2λ a2 |z| +A |z| , B =0, z ∈ U,
√
1+|z| |B|
(A−B)|z|
√
√
log
, −1 ≤ B < 0, z ∈ U.
2
|B|
1−|z|
|B|
(7.5)
Proof. From (7.1) when µ ∈ (2, 3) , and on substituting t = zu and noting
that
|w1 (zu)| ≤ |z| u, we get
µ−2
Z1
z
|z|
du, z ∈ U.
−
1
≤
(µ
−
2)
(A
−
B)
Dλ f (z)
uµ−2 (1 − |B| |z| u)
0
Now if B = 0, the above integral gives simply
Z1
uµ−2
|z|
|z|
du =
, z ∈ U,
(1 − |B| |z| u)
3−µ
0
and if −1 ≤ B < 0, making use of the known integral representation of the
Gaussian hypergeometric function mentioned, for instance, see [8, p. 7], we
get
Z1
uµ−2
|z|
|z|
du =
2 F1 (1, 3 − µ, 4 − µ; |B| |z|)
(1 − |B| |z| u)
3−µ
0
and hence, we have the inequality (7.4). Also, from (7.2), by substituting
2
t = zu and noting that |w2 (zu)| ≤ |z| u2 , we get
Z1
2
z
|z|
λ
−
1
≤
2
a
|z|
+
(A
−
B)
du, z ∈ U.
2
2
Dλ f (z)
1 − |B| |z| u2
0
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
367
Using now the following integral (for −1 ≤ B ≤ 0):

2
Z1
2
|z|
B = 0, z ∈ U,

|z|
√ du
=
|B|
1+|z|
|z|
2
√
 √ log
, −1 ≤ B < 0, z ∈ U,
1 − |B| |z| u2
0
2
|B|
1−|z|
|B|
we are lead to the second inequality (7.5) of Corollary 3.
Acknowledgment. The authors thank the reviewer for his/her valuable suggestions.
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R. K. RAINA,
M. P. University of Agriculture and Technology,
Udaipur 313001, Rajasthan, India,
Current address: 10/11 Ganpati Vihar, Opposite Sector 5, Udaipur 313002,
Rajasthan, India.
Email: rkraina 7@hotmail.com
Poonam SHARMA,
Department of Mathematics &, Astronomy
University of Lucknow,
Lucknow 226007, India,.
Email: sharma poonam@lkouniv.ac.in
Grigore Stefan. SĂLĂGEAN,
Department of Mathematics,
Faculty of Mathematics and Computer Science,
Babes-Bolyai University,
Str. M. Kogalniceanu Nr. 1, 400084 Cluj-Napoca, Romania.
Email: salagean@math.ubbcluj.ro
SOME CHARACTERISTIC PROPERTIES OF ANALYTIC FUNCTIONS
370