A NEW SUBCLASS OF k-UNIFORMLY CONVEX
FUNCTIONS WITH NEGATIVE COEFFICIENTS
H. M. SRIVASTAVA
T. N. SHANMUGAM
Department of Mathematics and Statistics
University of Victoria
British Columbia 1V8W 3P4,
Canada
EMail: harimsri@math.uvic.ca
Department of Information Technology
Salalah College of Technology
PO Box 608, Salalah PC211,
Sultanate of Oman
EMail: drtns2001@yahoo.com
C. RAMACHANDRAN
S. SIVASUBRAMANIAN
Department of Mathematics
College of Engineering,
Anna University
Chennai 600025, Tamilnadu, India
EMail: crjsp2004@yahoo.com
Department of Mathematics
Easwari Engineering College,
Ramapuram, Chennai 600089,
Tamilnadu, India
EMail: sivasaisastha@rediffmail.com
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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Received:
31 May, 2007
Accepted:
15 June, 2007
Communicated by:
Th.M. Rassias
2000 AMS Sub. Class.:
Primary 30C45.
Key words:
Analytic functions; Univalent functions; Coefficient inequalities and coefficient
estimates; Starlike functions; Convex functions; Close-to-convex functions; kUniformly convex functions; k-Uniformly starlike functions; Uniformly starlike
functions; Hadamard product (or convolution); Extreme points; Radii of close-toconvexity, Starlikeness and convexity; Integral operators.
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Abstract:
Acknowledgements:
The main object of this paper is to introduce and investigate a subclass
U(λ, α, β, k) of normalized analytic functions in the open unit disk ∆,
which generalizes the familiar class of uniformly convex functions. The
various properties and characteristics for functions belonging to the class
U(λ, α, β, k) derived here include (for example) a characterization theorem, coefficient inequalities and coefficient estimates, a distortion theorem
and a covering theorem, extreme points, and the radii of close-to-convexity,
starlikeness and convexity. Relevant connections of the results, which are
presented in this paper, with various known results are also considered.
The present investigation was supported, in part, by the Natural Sciences
and Engineering Research Council of Canada under Grant OGP0007353.
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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Contents
1
Introduction and Motivation
4
2
A Characterization Theorem and Resulting Coefficient Estimates
6
3
Distortion and Covering Theorems for the Function Class U(λ, α, β, k)
11
4
Extreme Points of the Function Class U(λ, α, β, k)
16
5
Modified Hadamard Products (or Convolution)
20
6
Radii of Close-to-Convexity, Starlikeness and Convexity
23
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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7
Hadamard Products and Integral Operators
27
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1.
Introduction and Motivation
Let A denote the class of functions f normalized by
f (z) = z +
(1.1)
∞
X
an z n ,
n=2
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
which are analytic in the open unit disk
∆ = {z : z ∈ C
and |z| < 1}.
As usual, we denote by S the subclass of A consisting of functions which are univalent in ∆. Suppose also that, for 0 5 α < 1, S ∗ (α) and K(α) denote the classes of
functions in A which are, respectively, starlike of order α in ∆ and convex of order
α in ∆ (see, for example, [11]). Finally, let T denote the subclass of S consisting of
functions f given by
f (z) = z −
(1.2)
∞
X
an z n
(an = 0)
n=2
with negative coefficients. Silverman [9] introduced and investigated the following
subclasses of the function class T :
(1.3)
∗
∗
T (α) := S (α) ∩ T
and
C(α) := K(α) ∩ T
(0 5 α < 1).
Definition 1. A function f ∈ T is said to be in the class U(λ, α, β, k) if it satisfies
the following inequality:
0
0 zF (z)
zF (z)
(1.4)
<
> k
− 1 + β
F (z)
F (z)
(0 5 α 5 λ 5 1; 0 5 β < 1; k = 0),
vol. 8, iss. 2, art. 43, 2007
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where
(1.5)
F (z) := λαz 2 f 00 (z) + (λ − α)zf 0 (z) + (1 − λ + α)f (z).
The above-defined function class U(λ, α, β, k) is of special interest and it contains many well-known as well as new classes of analytic univalent functions. In
particular, U(λ, α, β, 0) is the class of functions with negative coefficients, which
was introduced and studied recently by Kamali and Kadıoğlu [3], and U(λ, 0, β, 0)
is the function class which was introduced and studied by Srivastava et al. [12]
(see also Aqlan et al. [1]). We note that the class of k-uniformly convex functions
was introduced and studied recently by Kanas and Wiśniowska [4]. Subsequently,
Kanas and Wiśniowska [5] introduced and studied the class of k-uniformly starlike
functions. The various properties of the above two function classes were extensively
investigated by Kanas and Srivastava [6]. Furthermore, we have [cf. Equation (1.3)]
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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(1.6)
U(0, 0, β, 0) ≡ T ∗ (α)
and
U(1, 0, β, 0) ≡ C(α).
We remark here that the classes of k-uniformly starlike functions and k-uniformly
convex functions are an extension of the relatively more familiar classes of uniformly
starlike functions and uniformly convex functions investigated earlier by (for example) Goodman [2], Rønning [8], and Ma and Minda [7] (see also the more recent
contributions on these function classes by Srivastava and Mishra [10]).
In our present investigation of the function class U(λ, α, β, k), we obtain a characterization theorem, coefficient inequalities and coefficient estimates, a distortion
theorem and a covering theorem, extreme points, and the radii of close-to-convexity,
starlikeness and convexity for functions belonging to the class U(λ, α, β, k).
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2.
A Characterization Theorem and Resulting Coefficient
Estimates
We employ the technique adopted by Aqlan et al. [1] to find the coefficient estimates for the function class U(λ, α, β, k). Our main characterization theorem for
this function class is stated as Theorem 1 below.
Theorem 1. A function f ∈ T given by (1.2) is in the class U(λ, α, β, k) if and only
if
(2.1)
∞
X
{n(k + 1) − (k + β)} {(n − 1)(nλα + λ − α) + 1} an 5 1 − β
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
n=2
(0 5 α 5 λ 5 1; 0 5 β < 1; k = 0).
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The result is sharp for the function f (z) given by
1−β
z n (n = 2).
(2.2) f (z) = z −
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
Proof. By Definition 1, f ∈ U(λ, α, β, k) if and only if the condition (1.4) is satisfied. Since it is easily verified that
<(ω) > k|ω − 1| + β ⇐⇒ < ω(1 + keiθ ) − keiθ > β
(−π 5 θ < π; 0 5 β < 1; k = 0),
the inequality (1.4) may be rewritten as follows:
0
zF (z)
iθ
iθ
(2.3)
<
(1 + ke ) − ke
>β
F (z)
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or, equivalently,
<
(2.4)
zF 0 (z)(1 + keiθ ) − F (z)keiθ
F (z)
> β.
Now, by setting
G(z) = zF 0 (z)(1 + keiθ ) − F (z)keiθ ,
(2.5)
the inequality (2.4) becomes equivalent to
|G(z) + (1 − β)F (z)| > |G(z) − (1 + β)F (z)|
(0 5 β < 1),
where F (z) and G(z) are defined by (1.5) and (2.5), respectively. We thus observe
that
|G(z) + (1 − β)F (z)|
∞
X
n
= |(2 − β)z| − (n − β + 1){(n − 1)(nλα + λ − α) + 1}an z n=2
∞
iθ X
n
− ke
(n − 1){(n − 1)(nλα + λ − α) + 1}an z n=2
= (2 − β)|z| −
∞
X
(n − β + 1){(n − 1)(nλα + λ − α) + 1}an |z|n
n=2
−k
∞
X
n=2
= (2 − β)|z| −
n=2
vol. 8, iss. 2, art. 43, 2007
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(n − 1){(n − 1)(nλα + λ − α) + 1}an |z|n
∞
X
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
{(n(k + 1) − (k + β) + 1}{(n − 1)(nλα + λ − α) + 1}an |z|n .
Similarly, we obtain
|G(z) − (1 + β)F (z)|
∞
X
5 β|z| +
{(n(k + 1) − (k + β) − 1}{(n − 1)(nλα + λ − α) + 1}an |z|n .
n=2
Therefore, we have
|G(z) + (1 − β)F (z)| − |G(z) − (1 + β)F (z)|
∞
X
= 2(1 − β)|z| − 2
{(n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}an |z|n
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
n=2
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= 0,
which implies the inequality (2.1) asserted by Theorem 1.
Conversely, by setting
0 5 |z| = r < 1,
and choosing the values of z on the positive real axis, the inequality (2.3) reduces to
the following form:


∞
P
iθ
n−1
(1−β)−n=2{(n−β)−ke (n−1)}{(n−1)(nλα+λ−α)+1}an r 
=0,
(2.6) <
∞


P
n−1
1− (n−1){(n−1)(nλα+λ−α)+1}an r
n=2
which, in view of the elementary identity:
<(−eiθ ) = −|eiθ | = −1,
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becomes
(2.7)


∞
P
n−1

(1−β)−n=2{(n−β)−k(n−1)}{(n−1)(nλα+λ−α)+1}an r
 = 0.
<
∞


P
n−1
1− (n−1){(n−1)(nλα+λ−α)+1}an r
n=2
Finally, upon letting r → 1− in (2.7), we get the desired result.
By taking α = 0 and k = 0 in Theorem 1, we can deduce the following corollary.
Corollary 1. Let f ∈ T be given by (1.2). Then f ∈ U(λ, 0, β, 0) if and only if
∞
X
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
(n − β){(n − 1)λ + 1}an 5 1 − β.
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n=2
By setting α = 0, λ = 1 and k = 0 in Theorem 1, we get the following corollary.
Corollary 2 (Silverman [9]). Let f ∈ T be given by (1.2). Then f ∈ C(β) if and
only if
∞
X
n(n − β)an 5 1 − β.
n=2
The following coefficient estimates for f ∈ U(λ, α, β, k) is an immediate consequence of Theorem 1.
Theorem 2. If f ∈ U(λ, α, β, k) is given by (1.2), then
(2.8)
an 5
1−β
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
(0 5 α 5 λ 5 1; 0 5 β < 1; k = 0).
Equality in (2.8) holds true for the function f (z) given by (2.2).
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(n = 2)
By taking α = k = 0 in Theorem 2, we obtain the following corollary.
Corollary 3. Let f ∈ T be given by (1.2). Then f ∈ U(λ, 0, β, 0) if and only if
(2.9)
an 5
1−β
(n − β){(n − 1)λ + 1}
(n = 2).
Equality in (2.9) holds true for the function f (z) given by
(2.10)
f (z) = z −
1−β
zn
(n − β){(n − 1)λ + 1}
(n = 2).
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
Lastly, if we set α = 0, λ = 1 and k = 0 in Theorem 1, we get the following
familiar result.
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Corollary 4 (Silverman [9]). Let f ∈ T be given by (1.2). Then f ∈ C(β) if and
only if
(2.11)
1−β
an 5
n(n − β)
(n = 2).
Equality in (2.11) holds true for the function f (z) given by
(2.12)
1−β
f (z) = z −
zn
n(n − β)
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(n = 2).
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3.
Distortion and Covering Theorems for the Function Class
U(λ, α, β, k)
Theorem 3. If f ∈ U(λ, α, β, k), then
(3.1)
r−
1−β
r2
(2 + k − β)(2λα + λ − α)
5 |f (z)|
1−β
5r+
r2
(2 + k − β)(2λα + λ − α)
(|z| = r < 1).
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
Equality in (3) holds true for the function f (z) given by
1−β
f (z) = z −
z2.
(2 + k − β)(2λα + λ − α)
(3.2)
Proof. We only prove the second part of the inequality in (3), since the first part can
be derived by using similar arguments. Since f ∈ U(λ, α, β, k), by using Theorem
1, we find that
(2 + k − β)(2λα + λ − α + 1)
∞
X
=
an
5
(2 + k − β)(2λα + λ − α + 1)an
n=2
∞
X
{n(k + 1) − (k + β)} {(n − 1)(nλα + λ − α) + 1} an
n=2
5 1 − β,
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n=2
∞
X
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which readily yields the following inequality:
∞
X
(3.3)
an 5
n=2
1−β
.
(2 + k − β)(2λα + λ − α + 1)
Moreover, it follows from (1.2) and (3.3) that
∞
X
n
|f (z)| = z −
an z n=2
5 |z| + |z|2
∞
X
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
an
n=2
5 r + r2
∞
X
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an
Contents
n=2
5r+
1−β
r2 ,
(2 + k − β)(2λα + λ − α + 1)
which proves the second part of the inequality in (3).
1−
2(1 − β)
r
(2 + k − β)(2λα + λ − α)
5 |f 0 (z)|
2(1 − β)
51+
r
(2 + k − β)(2λα + λ − α)
Equality in (3.4) holds true for the function f (z) given by (3.2).
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Theorem 4. If f ∈ U(λ, α, β, k), then
(3.4)
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(|z| = r < 1).
Proof. Our proof of Theorem 4 is much akin to that of Theorem 3. Indeed, since
f ∈ U(λ, α, β, k), it is easily verified from (1.2) that
(3.5)
0
|f (z)| 5 1 +
∞
X
n−1
nan |z|
51+r
n=2
∞
X
nan
n=2
and
(3.6)
0
|f (z)| = 1 −
∞
X
n−1
nan |z|
51+r
n=2
∞
X
nan .
n=2
The assertion (3.4) of Theorem 4 would now follow from (3.5) and (3.6) by means
of a rather simple consequence of (3.3) given by
(3.7)
∞
X
nan 5
2(1 − β)
.
(2 + k − β)(2λα + λ − α + 1)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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This completes the proof of Theorem 4.
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Theorem 5. If f ∈ U(λ, α, β, k), then f ∈ T ∗ (δ), where
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n=2
δ := 1 −
1−β
.
(2 + k − β)(2λα + λ − α) − (1 − β)
The result is sharp with the extremal function f (z) given by (3.2).
Proof. It is sufficient to show that (2.1) implies that
(3.8)
∞
X
n=2
(n − δ)an 5 1 − δ,
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that is, that
(3.9)
n−δ
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
5
1−δ
1−β
(n = 2),
Since (3.9) is equivalent to the following inequality:
(n − 1)(1 − β)
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} − (1 − β)
=: Ψ(n),
δ 51−
(n = 2)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
and since
Ψ(n) 5 Ψ(2)
(n = 2),
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(3.9) holds true for
n = 2, 0 5 λ 5 1, 0 5 α 5 1, 0 5 β < 1 and k = 0.
This completes the proof of Theorem 5.
By setting α = k = 0 in Theorem 5, we can deduce the following result.
Corollary 5. If f ∈ U(λ, α, β, k), then
λ(2 − β) + β
∗
f ∈T
.
λ(2 − β) + 1
This result is sharp for the extremal function f (z) given by
f (z) = z −
1−β
z2.
(λ + 1)(2 − β)
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For the choices α = 0, λ = 1 and k = 0 in Theorem 5, we obtain the following
result of Silverman [9].
Corollary 6. If f ∈ C(β), then
f ∈T
∗
2
3−β
.
This result is sharp for the extremal function f (z) given by
f (z) = z −
1−β
z2.
2(2 − β)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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4.
Extreme Points of the Function Class U(λ, α, β, k)
Theorem 6. Let
f1 (z) = z
and
fn (z) = z −
1−β
zn
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
(n = 2).
Then f ∈ U(λ, α, β, k) if and only if it can be represented in the form:
!
∞
∞
X
X
(4.1)
f (z) =
µn fn (z)
µn = 0;
µn = 1 .
n=1
vol. 8, iss. 2, art. 43, 2007
n=1
Proof. Suppose that the function f (z) can be written as in (4.1). Then
∞
X
1−β
n
f (z) =
z
µn z −
{n(k
+
1)
−
(k
+
β)}{(n
−
1)(nλα
+
λ
−
α)
+
1}
n=1
∞
X
1−β
=z−
µn
zn.
{n(k
+
1)
−
(k
+
β)}{(n
−
1)(nλα
+
λ
−
α)
+
1}
n=2
Now
∞
X
n=2
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
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µn
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}(1 − β)
(1 − β){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
=
∞
X
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µn
n=2
= 1 − µ1
5 1,
which implies that f ∈ U(λ, α, β, k).
Conversely, we suppose that f ∈ U(λ, α, β, k). Then, by Theorem 2, we have
an 5
1−β
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
(n = 2).
Therefore, we may write
µn =
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
an
1−β
(n = 2)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
and
µ1 = 1 −
∞
X
µn .
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n=2
Then
f (z) =
∞
X
Contents
µn fn (z),
n=1
with fn (z) given as in (4.1). This completes the proof of Theorem 6.
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Corollary 7. The extreme points of the function class f ∈ U(λ, α, β, k) are the
functions
f1 (z) = z
and
fn (z) = z −
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1−β
zn
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
For α = k = 0 in Corollary 7, we have the following result.
(n = 2).
Corollary 8. The extreme points of f ∈ U(λ, 0, β, 0) are the functions
f1 (z) = z
and
fn (z) = z −
1−β
zn
{n − β}{(n − 1)λ + 1}
(n = 2).
By setting α = 0, λ = 1 and k = 0 in Corollary 7, we obtain the following result
obtained by Silverman [9].
Corollary 9. The extreme points of the class C(β) are the functions
f1 (z) = z
fn (z) = z −
and
1−β
zn
n(n − β)
(n = 2).
Theorem 7. The class U(λ, α, β, k) is a convex set.
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
Title Page
Proof. Suppose that each of the functions fj (z) (j = 1, 2) given by
(4.2)
fj (z) = z −
∞
X
an,j z n
(an,j = 0; j = 1, 2)
n=2
is in the class U(λ, α, β, k). It is sufficient to show that the function g(z) defined by
g(z) = µf1 (z) + (1 − µ)f2 (z)
(0 5 µ 5 1)
is also in the class U(λ, α, β, k). Since
g(z) = z −
∞
X
n=2
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[µan,1 + (1 − µ)an,2 ]z n ,
with the aid of Theorem 1, we have
∞
X
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
n=2
× [µan,1 + (1 − µ)an,2 ]
∞
X
5µ
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}an,1
n=2
+ (1 − µ)
∞
X
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}an,2
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
n=2
5 µ(1 − β) + (1 − µ)(1 − β)
(4.3) 5 1 − β,
which implies that g ∈ U(λ, α, β, k). Hence U(λ, α, β, k) is indeed a convex set.
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5.
Modified Hadamard Products (or Convolution)
For functions
f (z) =
∞
X
an z
n
and
g(z) =
n=0
∞
X
bn z n ,
n=0
the Hadamard product (or convolution) (f ∗g)(z) is defined, as usual, by
∞
X
(f ∗g)(z) :=
(5.1)
an bn z n =: (g∗f )(z).
n=0
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
On the other hand, for functions
fj (z) = z −
∞
X
an,j z n
(j = 1, 2)
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n=2
in the class T , we define the modified Hadamard product (or convolution) as follows:
(f1 •f2 )(z) := z −
(5.2)
∞
X
n
an,1 an,2 z =: (f2 •f1 )(z).
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n=2
Then we have the following result.
Theorem 8. If fj (z) ∈ U(λ, α, β, k) (j = 1, 2), then
(f1 •f2 )(z) ∈ U(λ, α, β, k, ξ),
where
(2 − β){2 + k − β}{2λα + λ − α + 1} − 2(1 − β)2
.
(2 − β){2 + k − β}{2λα + λ − α + 1} − (1 − β)2
The result is sharp for the functions fj (z) (j = 1, 2) given as in (3.2).
ξ :=
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Proof. Since fj (z) ∈ U(λ, α, β, k) (j = 1, 2), we have
(5.3)
∞
X
{n(k+1)−(k+β)}{(n−1)(nλα+λ−α)+1}an,j 5 1−β
(j = 1, 2),
n=2
which, in view of the Cauchy-Schwarz inequality, yields
(5.4)
∞
X
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} √
an,1 an,2 5 1.
1−β
n=2
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
We need to find the largest ξ such that
(5.5)
∞
X
{n(k + 1) − (k + ξ)}{(n − 1)(nλα + λ − α) + 1}
n=2
1−ξ
vol. 8, iss. 2, art. 43, 2007
an,1 an,2 5 1.
Title Page
Thus, in light of (5.4) and (5.5), whenever the following inequality:
n − ξ√
n−β
an,1 an,2 5
1−ξ
1−β
Contents
(n = 2)
holds true, the inequality (5.5) is satisfied. We find from (5.4) that
(5.6)
√
an,1 an,2
1−β
5
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
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(n = 2).
Thus, if
1−β
n−β
n−ξ
5
1−ξ
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
1−β
(n − β){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} − n(1 − β)2
(n − β){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} − (1 − β)2
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(n = 2),
or, if
ξ5
JJ
(n = 2),
Close
then (5.4) is satisfied. Setting
Φ(n) :=
(n − β){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} − n(1 − β)2
(n − β){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} − (1 − β)2
(n = 2),
we see that Φ(n) is an increasing function for n = 2. This implies that
(2 − β){2 + k − β}{2λα + λ − α + 1} − 2(1 − β)2
ξ 5 Φ(2) =
.
(2 − β){2 + k − β}{2λα + λ − α + 1} − (1 − β)2
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
Finally, by taking each of the functions fj (z) (j = 1, 2) given as in (3.2), we see
that the assertion of Theorem 8 is sharp.
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6.
Radii of Close-to-Convexity, Starlikeness and Convexity
Theorem 9. Let the function f (z) defined by (1.2) be in the class U(λ, α, β, k). Then
f (z) is close-to-convex of order ρ (0 5 ρ < 1) in |z| < r1 (λ, α, β, ρ, k), where
r1 (λ, α, β, ρ, k)
1
(1 − ρ){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} n−1
:= inf
n
n(1 − β)
(n = 2).
The result is sharp for the function f (z) given by (2.2).
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
Proof. It is sufficient to show that
|f 0 (z) − 1| 5 1 − ρ
0 5 ρ < 1; |z| < r1 (λ, α, β, ρ, k) .
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Since
(6.1)
Contents
∞
∞
X
X
0
n−1 |f (z) − 1| = −
nan z 5
nan |z|n−1 ,
n=2
n=2
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we have
0
|f (z) − 1| 5 1 − ρ
(0 5 ρ < 1),
if
(6.2)
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∞ X
n
an |z|n−1 5 1.
1−ρ
n=2
Hence, by Theorem 1, (6.2) will hold true if
n
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
|z|n−1 5
,
1−ρ
1−β
Close
that is, if
(6.3) |z| 5
(1−ρ){n(k+1)−(k+β)}{(n−1)(nλα+λ−α)+1}
n(1−β)
1
n−1
(n = 2).
The assertion of Theorem 9 would now follow easily from (6.3).
Theorem 10. Let the function f (z) defined by (1.2) be in the class U(λ, α, β, k).
Then f (z) is starlike of order ρ (0 5 ρ < 1) in |z| < r2 (λ, α, β, ρ, k), where
r2 (λ, α, β, ρ, k)
1
(1 − ρ){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} n−1
:= inf
n
(n − ρ)(1 − β)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
(n = 2).
Title Page
The result is sharp for the function f (z) given by (2.2).
Proof. It is sufficient to show that
0
zf (z)
51−ρ
−
1
f (z)
Contents
0 5 ρ < 1; |z| < r2 (λ, α, β, ρ, k) .
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I
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0
zf (z)
5
−
1
f (z)
∞
P
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(n − 1)an |z|n−1
n=2
1−
∞
P
,
an
z n−1
n=2
we have
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Page 24 of 29
Since
(6.4)
JJ
0
zf (z)
51−ρ
−
1
f (z)
(0 5 ρ < 1),
Close
if
∞ X
n−ρ
(6.5)
n=2
1−ρ
an |z|n−1 5 1.
Hence, by Theorem 1, (6.5) will hold true if
n−ρ
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
|z|n−1 5
,
1−ρ
1−β
that is, if
vol. 8, iss. 2, art. 43, 2007
(6.6) |z| 5
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
(1−ρ){n(k+1)−(k+β)}{(n−1)(nλα+λ−α)+1}
(n−ρ)(1−β)
1
n−1
(n = 2).
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The assertion of Theorem 10 would now follow easily from (6.6).
Theorem 11. Let the function f (z) defined by (1.2) be in the class U(λ, α, β, k).
Then f (z) is convex of order ρ (0 5 ρ < 1) in |z| < r3 (λ, α, β, ρ, k), where
r3 (λ, α, β, ρ, k)
1
(1 − ρ){n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1} n−1
:= inf
n
n(n − ρ)(1 − β)
The result is sharp for the function f (z) given by (2.2).
Proof. It is sufficient to show that
00 zf (z) f 0 (z) 5 1 − ρ
0 5 ρ < 1; |z| < r3 (λ, α, β, ρ, k) .
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(n = 2).
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Since
(6.7)
00 zf (z) f 0 (z) 5
∞
P
n(n − 1)an |z|n−1
n=2
∞
P
1−
,
nan
|z|n−1
n=2
we have
00 zf (z) f 0 (z) 5 1 − ρ
(0 5 ρ < 1),
if
(6.8)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
∞ X
n(n − ρ)
n=2
1−ρ
an |z|n−1 5 1.
Title Page
Contents
Hence, by Theorem 1, (6.8) will hold true if
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}
n(n − ρ)
|z|n−1 5
,
1−ρ
1−β
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that is, if
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1
(1−ρ){n(k+1)−(k+β)}{(n−1)(nλα+λ−α)+1} n−1
(6.9) |z| 5
(n = 2).
n(n−ρ)(1−β)
Theorem 11 now follows easily from (6.9).
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7.
Hadamard Products and Integral Operators
Theorem 12. Let f ∈ U(λ, α, β, k). Suppose also that
∞
X
(7.1)
g(z) = z +
gn z n
(0 5 gn 5 1).
n=2
Then
f ∗g ∈ U(λ, α, β, k).
Proof. Since 0 5 gn 5 1 (n = 2),
∞
X
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}an gn
n=2
5
∞
X
vol. 8, iss. 2, art. 43, 2007
Title Page
{n(k + 1) − (k + β)}{(n − 1)(nλα + λ − α) + 1}an
n=2
(7.2)
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
5 1 − β,
which completes the proof of Theorem 12.
Corollary 10. If f ∈ U(λ, α, β, k), then the function F(z) defined by
Z
1 + c z c−1
(7.3)
F(z) :=
t f (t)dt
(c > −1)
zc 0
is also in the class U(λ, α, β, k).
Proof. Since
F(z) = z +
∞ X
c+1
n=2
c+n
z
n
c+1
<1 ,
0<
c+n
the result asserted by Corollary 10 follows from Theorem 12.
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References
[1] E. AQLAN, J.M. JAHANGIRI AND S.R. KULKARNI, Classes of k-uniformly
convex and starlike functions, Tamkang J. Math., 35 (2004), 1–7.
[2] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56
(1991), 87–92.
[3] M. KAMALI AND E. KADIOĞLU, On a new subclass of certain starlike functions with negative coefficients, Atti Sem. Mat. Fis. Univ. Modena, 48 (2000),
31–44.
[4] S. KANAS AND A. WIŚNIOWSKA, Conic regions and k-uniform convexity,
J. Comput. Appl. Math., 105 (1999), 327–336.
[5] S. KANAS AND A. WIŚNIOWSKA, Conic domains and starlike functions,
Rev. Roumaine Math. Pures Appl., 45 (2000), 647–657.
[6] S. KANAS AND H.M. SRIVASTAVA, Linear operators associated with kuniformly convex functions, Integral Transform. Spec. Funct., 9 (2000), 121–
132.
[7] W. MA AND D. MINDA, Uniformly convex functions. II, Ann. Polon. Math.,
58 (1993), 275–285.
[8] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.
[9] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51 (1975), 109–116.
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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Contents
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[10] H.M. SRIVASTAVA AND A.K. MISHRA, Applications of fractional calculus
to parabolic starlike and uniformly convex functions, Comput. Math. Appl., 39
(3–4) (2000), 57–69.
[11] H.M. SRIVASTAVA AND S. OWA (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey,
London and Hong Kong, 1992.
[12] H.M. SRIVASTAVA, S. OWA AND S.K. CHATTERJEA, A note on certain
classes of starlike functions, Rend. Sem. Mat. Univ Padova, 77 (1987), 115–
124.
k-Uniformly Convex
Functions
H. M. Srivastava, T. N. Shanmugam,
C. Ramachandran and
S. Sivasubramanian
vol. 8, iss. 2, art. 43, 2007
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