Journal of Inequalities in Pure and
Applied Mathematics
SOME INEQUALITIES ASSOCIATED WITH A LINEAR OPERATOR
DEFINED FOR A CLASS OF ANALYTIC FUNCTIONS
volume 7, issue 1, article 6,
2006.
S. R. SWAMY
P. G. Department of Computer Science
R. V. College of Engineering
Bangalore 560 059, Karnataka, India.
Received 07 March, 2005;
accepted 25 July, 2005.
Communicated by: H. Silverman
EMail: mailtoswamy@rediffmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper, we give a sufficient condition on a linear operator Lp (a, c)g(z)
which can guarantee that for α a complex number with Re(α) > 0,
Lp (a, c)f(z)
Lp (a + 1, c)f(z)
Re (1 − α)
+α
> ρ, ρ < 1,
Lp (a, c)g(z)
Lp (a + 1, c)g(z)
in the unit disk E, implies
Lp (a, c)f(z)
0
Re
> ρ > ρ, z ∈ E.
Lp (a, c)g(z)
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
Some interesting applications of this result are also given.
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2000 Mathematics Subject Classification: 30C45.
Key words: Analytic functions, Differential subordination, Ruscheweyh derivatives,
Linear operator.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
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1.
Introduction
Let A(p, n) denote the class functions f normalized by
(1.1)
p
f (z) = z +
∞
X
ak z k
(p, n ∈ N = {1, 2, 3, ...}),
k=p+n
which are analytic in the open unit disk E = {z : z ∈ C, |z| < 1}.
In particular, we set A(p, 1) = Ap and A(1, 1) = A1 = A.
The Hadamard product (f ∗ g)(z) of two functions f (z) given by (1.1) and
g(z) given by
∞
X
g(z) = z p +
bk z k
(p, n ∈ N),
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
k=p+n
is defined, as usual, by
(f ∗ g)(z) = z p +
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∞
X
ak bk z k = (g ∗ f )(z).
k=p+n
The Ruscheweyh derivative of f (z) of order δ + p − 1 is defined by
(1.2) Dδ+p−1 f (z) =
zp
∗ f (z)
(1 − z)δ+p
(f ∈ A(p, n); δ ∈ R \ (−∞, −p])
Dδ+p−1 f (z) = z p +
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or, equivalently, by
(1.3)
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∞
X
k=p+n
δ+k−1
ak z k ,
k−p
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where f (z) ∈ A(p, n) and δ ∈ R \ (−∞, −p]. In particular, if δ = l ∈ N
we find from (1.2) or (1.3) that
Dl+p−1 f (z) =
S
{0},
dl+k−1 l−1
zp
z
f
(z)
.
(l + p − 1)! dz l+p−1
The author has proved the following result in [4].
Theorem A. Let α be a complex number satisfying Re(α) > 0 and ρ < 1. Let
δ > −p, f, g ∈ Ap and
Dδ+p−1 g(z)
Re α δ+p
> γ, 0 ≤ γ < Re(α), z ∈ E.
D g(z)
Then
Re
Dδ+p−1 f (z)
Dδ+p−1 g(z)
>
2ρ(δ + p) + γ
,
2(δ + p) + γ
z ∈ E,
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
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Contents
whenever
Dδ+p−1 f (z)
Dδ+p f (z)
Re (1 − α) δ+p−1
+ α δ+p
> ρ, z ∈ E.
D
g(z)
D g(z)
The Pochhammer symbol (λ)k or the shifted factorial is given by (λ)0 = 1
and (λ)k = λ(λ + 1)(λ + 2) · · · (λ + k − 1), k ∈ N. In terms of (λ)k , we now
define the function φp (a, c; z) by
p
φp (a, c; z) = z +
∞
X
(a)k
k=1
(c)k
z k+p ,
z ∈ E,
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where a ∈ R, c ∈ R \ z0− ; z0− = {0, −1, −2, . . . }.
Saitoh [3] introduced a linear operator LP (a, c), which is defined by
(1.4)
Lp (a, c)f (z) = φp (a, c, ; z) ∗ f (z),
z ∈ E,
or, equivalently by
(1.5)
p
Lp (a, c)f (z) = z +
∞
X
(a)k
k=1
(c)k
ak+p z k+p ,
z ∈ E,
where f (z) ∈ Ap , a ∈ R, c ∈ R \ z0− .
For f (z) ∈ A(p, n) and δ ∈ R \ (−∞, −p], we obtain
(1.6)
Lp (δ + p, 1)f (z) = Dδ+p−1 f (z),
which can easily be verified by comparing the definitions (1.3) and (1.5).
The main object of this paper is to present an extension of Theorem A to
hold true for a linear operator LP (a, c) associated with the class A(p, n).
The basic tool in proving our result is the following lemma.
Lemma 1.1 (cf. Miller and Mocanu [2, p. 35, Theorem 2.3 i(i)]). Let Ω be
a set in the complex plane C. Suppose that the function Ψ : C 2 × E −→ C
satisfies the condition Ψ(ix2 , y1 ; z) ∈
/ Ω for all z ∈ E and for all real x2 and y1
such that
1
(1.7)
y1 ≤ − n(1 + x22 ).
2
n
If p(z) = 1 + cn z + · · · is analytic in E and for z ∈ E, Ψ(p(z), zp0 (z); z) ⊂ Ω,
then Re(p(z)) > 0 in E.
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
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2.
Main Results
Theorem 2.1. Let α be a complex number satisfying Re(α) > 0 and ρ < 1. Let
a > 0, f, g ∈ A(p, n) and
Lp (a, c)g(z)
(2.1)
Re α
> γ,
0 ≤ γ < Re(α), z ∈ E.
Lp (a + 1, c)g(z)
Then
Re
Lp (a, c)f (z)
Lp (a, c)g(z)
>
2aρ + nγ
,
2a + nγ
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
z ∈ E,
whenever
(2.2)
Lp (a + 1, c)f (z)
Lp (a, c)f (z)
Re (1 − α)
+α
> ρ,
Lp (a, c)g(z)
Lp (a + 1, c)g(z)
S. R. Swamy
z ∈ E.
Title Page
Proof. Let τ = (2aρ + nγ)/(2a + nγ) and define the function p(z) by
Lp (a, c)f (z)
−1
(2.3)
p(z) = (1 − τ )
−τ .
Lp (a, c)g(z)
Then, clearly, p(z) = 1 + cn z n + cn+1 z n+1 + · · · and is analytic in E. We set
u(z) = αLp (a, c)g(z)/Lp (a + 1, c)g(z) and observe from (2.1) that Re(u(z)) >
γ, z ∈ E. Making use of the familiar identity
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z(Lp (a, c)f (z))0 = aLp (a + 1, c)f (z) − (a − p)Lp (a, c)f (z),
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we find from (2.3) that
(2.4) (1 − α)
Lp (a, c)f (z)
Lp (a + 1, c)f (z)
+α
Lp (a, c)g(z)
Lp (a + 1, c)g(z)
u(z) 0
= τ + (1 − τ ) p(z) +
zp (z) .
a
If we define Ψ(x, y; z) by
(2.5)
u(z)
Ψ(x, y; z) = τ + (1 − τ ) x +
y ,
a
then, we obtain from (2.2) and (2.4) that
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
0
{Ψ(p(z), zp (z); z) : |z| < 1} ⊂ Ω = {w ∈ C : Re(w) > ρ}.
Now for all z ∈ E and for all real x2 and y1 constrained by the inequality (1.7),
we find from (2.5) that
(1 − τ )
y1 Re(u(z))
a
(1 − τ )nγ
≤τ−
≡ ρ.
2a
Re{Ψ(ix2 , y1 ; z)} = τ +
Hence
y1 ; z) ∈
/ Ω. Thus by Lemma 1.1, Re(p(z)) > 0 and hence
n Ψ(ix2 , o
Lp (a,c)f (z)
Re Lp (a,c)g(z) > τ in E. This proves our theorem.
Remark 1. Theorem A is a special case of Theorem 2.1 obtained by taking
a = δ + p and c = n = 1, which reduces to Theorem 2.1 of [1], when p = 1.
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Corollary 2.2. Let α be a real number with α ≥ 1 and ρ < 1. Let a > 0, f,
g ∈ A(p, n) and
Lp (a, c)g(z)
Re
> γ, 0 ≤ γ < 1, z ∈ E.
Lp (a + 1, c)g(z)
Then
α(2aρ + nγ) − (1 − ρ)nγ
,
α(2a + nγ)
z ∈ E,
Lp (a, c)f (z)
Lp (a + 1, c)f (z)
Re (1 − α)
+α
> ρ,
Lp (a, c)g(z)
Lp (a + 1, c)g(z)
z ∈ E.
Re
Lp (a + 1, c)f (z)
Lp (a + 1, c)g(z)
>
whenever
Proof. Proof follows from Theorem 2.1 (Since α ≥ 1).
In its special case when α = 1, Theorem 2.1 yields:
n
o
p (a,c)g(z)
Corollary 2.3. Let a > 0, f, g ∈ A(p, n) and Re LLp (a+1,c)g(z)
> γ, 0 ≤ γ <
1, then for ρ < 1,
Lp (a + 1, c)f (z)
> ρ, z ∈ E,
Re
Lp (a + 1, c)g(z)
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
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implies
Re
Lp (a, c)f (z)
Lp (a, c)g(z)
>
2aρ + nγ
,
2a + nγ
z ∈ E.
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If we set
Lp (a + 1, c)f (z)
v(z) =
−
Lp (a + 1, c)g(z)
1
Lp (a, c)f (z)
−1
,
α
Lp (a, c)g(z)
then for a > 0, α > 0 and ρ = 0, Theorem 2.1 reduces to
z∈E
Re(v(z)) > 0,
implies
(2.6)
Re
Lp (a, c)f (z)
Lp (a, c)g(z)
>
nαγ
,
2a + nαγ
z ∈ E,
whenever Re(Lp (a, c)g(z)/Lp (a + 1, c)g(z)) > γ, 0 ≤ γ < 1. Let α → ∞.
Then (2.6) is equivalent to
Lp (a + 1, c)f (z) Lp (a, c)f (z)
Re
−
> 0 in E
Lp (a + 1, c)g(z)
Lp (a, c)g(z)
implies
Re
Lp (a, c)f (z)
Lp (a, c)g(z)
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
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> 1 in E,
whenever Re(Lp (a, c)g(z)/Lp (a + 1, c)g(z)) > γ, 0 ≤ γ < 1.
In the following theorem we shall extend the above result, the proof of which
is similar to that of Theorem 2.1.
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n
o
p (a,c)g(z)
Theorem 2.4. Let a > 0, ρ < 1, f, g ∈ A(p, n) and Re Lpl(a+1,c)g(z))
> γ,
0 ≤ γ < 1.
If
Lp (a + 1, c)f (z) Lp (a, c)f (z)
nγ(1 − ρ)
Re
−
>−
,
z ∈ E,
Lp (a + 1, c)g(z)
Lp (a, c)g(z)
2a
then
Re
and
Re
Lp (a, c)f (z)
Lp (a, c)g(z)
Lp (a + 1, c)f (z)
Lp (a + 1, c)g(z)
>
> ρ,
z ∈ E,
ρ(2a + nγ) − nγ
,
2a
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
z ∈ E.
S. R. Swamy
Using Theorem 2.1 and Theorem 2.4, we can generalize and improve several
other interesting results available in the literature by taking g(z) = z p . We
illustrate a few in the following theorem.
Theorem 2.5. Let a > 0, ρ < 1 and f (z) ∈ A(p, n). Then
(a) for α a complex number satisfying Re(α) > 0, we have
Lp (a, c)f (z)
Lp (a + 1, c)f (z)
+α
> ρ,
Re (1 − α)
zp
zp
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z ∈ E,
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implies
Re
Lp (a, c)f (z)
zp
>
2aρ + n Re(α)
,
2a + n Re(α)
z ∈ E.
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(b) for α real and α ≥ 1, we have
Lp (a, c)f (z)
Lp (a + 1, c)f (z)
+α
> ρ,
Re (1 − α)
zp
zp
in E
implies
Re
Lp (a + 1, c)f (z)
zp
>
(2a + n)ρ + n(α − 1)
2a + nα
in E
(c) for z ∈ E,
Re
Lp (a + 1, c)f (z) Lp (a, c)f (z)
−
zp
zp
>−
n(1 − ρ)
2a
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
implies
Re
Lp (a + 1, c)f (z)
zp
>
(2a + n)ρ − n
.
2a
Remark 2. By taking a = δ + p, c = n = 1 in Theorem 2.5 we obtain Theorem
1.6 of the author [4], which when p = 1 reduces to Theorem 2.3 of Bhoosnurmath and Swamy [1].
In a manner similar to Theorem 2.1, we can easily prove the following, which
when r = 1 reduces to part (a) of Theorem 2.5.
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Theorem 2.6. Let a > 0, r > 0, ρ < 1 and f (z) ∈ A(p, n).Then for α a
complex number with Re(α) > 0, we have
r Lp (a, c)f (z)
2aρr + n Re(α)
Re
>
,
z ∈ E,
p
z
2ar + n Re(α)
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whenever
r
Lp (a, c)f (z)
Re (1 − α)
zp
r−1 )
Lp (a + 1, c)f (z)
Lp (a, c)f (z)
+α
> ρ,
p
z
zp
z ∈ E.
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
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References
[1] S.S. BHOOSNARMATH AND S.R. SWAMY, Differential subordination
and conformal mappings, J. Math. Res. Expo., 14(4) (1994), 493–498.
[2] S.S. MILLER AND P.T. MOCANU, Differential Subordinations: Theory
and Applications, Series on Monographs and Text Books in Pure and Applied Mathematics (No. 225), Marcel Dekker, New York and Besel, 2000.
[3] H. SAITOH, A linear operator and its applications of first order differential
subordinations, Math. Japon., 44 (1996), 31–38.
[4] S.R. SWAMY, Some studies in univalent functions, Ph.D thesis, Karnatak
University, Dharwad, India, 1992, unpublished.
Some Inequalities Associated
with a Linear Operator Defined
for a Class of Analytic
Functions
S. R. Swamy
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