COEFFICIENT INEQUALITIES FOR CERTAIN
CLASSES OF RUSCHEWEYH TYPE ANALYTIC
FUNCTIONS
Coefficient Inequalities
S. Latha
vol. 9, iss. 2, art. 52, 2008
S. LATHA
Department of Mathematics and Computer Science
Maharaja’s College
University of Mysore
Mysore - 570 005, INDIA.
EMail: drlatha@gmail.com
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Received:
18 January, 2007
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Accepted:
5 May, 2008
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Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
30C45.
Key words:
Convolution, Ruscheweyh derivative, Uniformly starlike and Uniformly convex.
Abstract:
A class of univalent functions which provides an interesting transition from starlike functions to convex functions is defined by making use of the Ruscheweyh
derivative. Some coefficient inequalities for functions in these classes are discussed which generalize the coefficient inequalities considered by Owa, Polatoğlu and Yavuz.
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Contents
1
Introduction
3
2
Main Results
5
Coefficient Inequalities
S. Latha
vol. 9, iss. 2, art. 52, 2008
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1.
Introduction
Let N denote the class of functions of the form
∞
X
(1.1)
f (z) = z +
an z n
n=2
which are analytic in the open unit disc U = {z ∈ C : |z| < 1}.
We designate V(β, b, δ) as the subclass of N consisting of functions f obeying
the condition
2 2 Dδ+1 f (z)
(1.2)
< 1− +
>β
b b Dδ f (z)
vol. 9, iss. 2, art. 52, 2008
where, b 6= 0, δ > −1, 0 ≤ β < 1 and Dδ f is the Rushceweyh derivative of f
[5] given by,
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(1.3)
∞
X
z
δ
D f (z) =
∗ f (z) = z +
an Bn (δ)z n ,
(1 − z)1+δ
n=2
where ∗ stands for the convolution or Hadamard product of two power series and
(δ + 1)(δ + 2) · · · (δ + n − 1)
.
Bn (δ) =
(n − 1)!
This class is obtained by putting k = 2 and λ = 0 in the class Vkλ (β, b, δ) introduced by Latha and Nanjunda Rao [2]. The class Vkλ (β, b, δ) is of special interest
for it contains many well known as well as new classes of analytic univalent functions studied in literature. It provides a transition from starlike functions to convex
functions. More specifically, V20 (β, 2, 0) is the family of starlike functions of order β and V20 (β, 1, 1) is the class of convex functions of order β. Shams, Kulkarni
Coefficient Inequalities
S. Latha
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and Jahangiri [6] introduced the subclass SD(α, β) of N consisting of functions f
satisfying
0
0 zf (z)
zf (z)
(1.4)
<
> α
− 1 + β
f (z)
f (z)
for some α ≥ 0, 0 ≤ β < 1 and z ∈ U.
The class KD(α, β), another subclass of N , is defined as the set of all functions
f obeying
00
zf (z)
zf 00 (z)
> α 0
− 1 + β
(1.5)
< 1+ 0
f (z)
f (z)
Coefficient Inequalities
S. Latha
vol. 9, iss. 2, art. 52, 2008
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for some α ≥ 0, 0 ≤ β < 1 and z ∈ U.
We introduce the class VD(α, β, b, δ) as the subclass of N consisting of functions f which satisfy
2 Dδ+1 f (z)
2 2 Dδ+1 f (z)
+β
>
α
−
1
< 1− +
b Dδ f (z)
b b Dδ f (z)
where, b 6= 0, α ≥ 0, and 0 ≤ β < 1.
For the parametric values b = 2, δ = 0 and b = δ = 1 we obtain the classes
SD(α, β) and KD(α, β) respectively.
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2.
Main Results
We prove some coefficient inequalities for functions in the class VD(α, β, b, δ).
Theorem 2.1. If f (z) ∈ VD(α, β, b, δ) with 0 ≤ α ≤ β, or, α >
, b, δ .
V β−α
1−α
Proof. Since <{ω} ≤ |ω| for any complex number ω, f (z)
plies that
2 Dδ+1 f (z)
2 2 Dδ+1 f (z)
(2.1)
< 1− +
>
α
b Dδ f (z) −
b b Dδ f (z)
1+β
,
2
then f (z) ∈
∈ VD(α, β, b, δ) imCoefficient Inequalities
S. Latha
2 + β.
b
Equivalently,
(2.2)
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2 2 Dδ+1 f (z)
β−α
< 1− +
>
,
δ
b b D f (z)
1−α
If 0 ≤ α ≤ β, we have, 0 ≤
α−β
≤ 0.
α−1
β−α
1−α
< 1, and if α >
1+β
,
2
then we have −1 <
Corollary 2.3. The parametric values b = δ = 1, yield the Corollary 2.2 in [3]
stated as:
β−α
If f (z) ∈ KD(α, β) with 0 ≤ α ≤ β, or, α > 1+β
,
then
f
(z)
∈
K
.
2
1−α
Theorem 2.4. If f (z) ∈ VD(α, β, b, δ) then,
|a2 | ≤
b(1 − β)
|1 − α|
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(z ∈ U).
Corollary 2.2. For the parametric values b = 2 and δ = 0, we get Theorem 2.1 in
[3] which reads as:
∗ β−α
If f (z) ∈ SD(α, β) with 0 ≤ α ≤ β, or, α > 1+β
,
then
f
(z)
∈
S
.
2
1−α
(2.3)
vol. 9, iss. 2, art. 52, 2008
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and
(2.4)
n−2 b(1 − β)(δ + 1) Y
b(δ + 1)(1 − β)
|an | ≤
1+
,
(n − 1)|1 − α|Bn (δ) j=1
j|1 − α|
Proof. We note that for f (z) ∈ VD(α, β, b, δ),
2 2 Dδ+1 f (z)
β−α
< 1− +
>
,
δ
b b D f (z)
1−α
(n ≥ 3).
(z ∈ U).
We define the function p(z) by
i
h
2
2 Dδ+1 f (z)
(1 − α) 1 − b + b Dδ f (z) − (β − α)
(2.5)
p(z) =
,
(1 − β)
S. Latha
vol. 9, iss. 2, art. 52, 2008
(z ∈ U).
Then, p(z) is analytic in U with p(0) = 1 and <{p(z)} > 0 and z ∈ U.
Let p(z) = 1 + p1 z + p1 z 2 + · · · . We have
∞
2 2 Dδ+1 f (z)
1−β X
(2.6)
1− +
=1+
pn z n .
δ
b b D f (z)
1 − α n=1
That is,
2(D
δ+1
δ
δ
f (z) − D f (z)) = bD f (z)
Coefficient Inequalities
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∞
1−β X
pn z n .
1 − α n=1
which implies that
2Bn (δ)(n − 1)an
(δ + 1)
b(1 − β)
=
[pn−1 + B2 (δ) + a2 pn−2 + B3 (δ)a3 pn−3 + · · · + Bn−1 (δ)an−1 p1 ] .
(1 − α)
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Applying the coefficient estimates |pn | ≤ 2 for Carathéodory functions [1], we
obtain,
(2.7) |an | ≤
b(1 − β)(δ + 1)
|1 − α|(n − 1)Bn (δ)
× [1 + B2 (δ)|a2 | + B3 (δ)|a3 | + · · · + Bn−1 (δ)|an−1 |] .
For n = 2, |a2 | ≤
For n = 3,
b(1−β)
,
|1−α|
which proves (2.3).
b(1 − β)(δ + 1)
b(1 − β)(δ + 1)
|a3 | ≤
1+
.
2|1 − α|B3 (δ)
|1 − α|
Therefore (2.4) holds for n = 3.
Assume that (2.4) is true for n = k.
Consider,
b(1 − β)(δ + 1)
b(1 − β)(δ + 1)
|ak+1 | ≤
1+
kBk+1 (δ)
|1 − α|
b(1 − β)(δ + 1)
b(1 − β)(δ + 1)
+
1+
|1 − α|B2 (δ)
|1 − α|
)
k−2 b(1 − β)(δ + 1) Y
b(1 − β)(δ + 1)
+··· +
1+
(k − 1)!|1 − α|Bk (δ) j=1
j(|1 − α|)
k−1 b(1 − β)(δ + 1) Y
b(1 − β)(δ + 1)
=
1+
.
kBk+1 (δ)
j(|1 − α|)
j=1
Therefore, the result is true for n = k + 1. Using mathematical induction, (2.4)
holds true for all n ≥ 3.
Coefficient Inequalities
S. Latha
vol. 9, iss. 2, art. 52, 2008
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Corollary 2.5. The parametric values b = 2 and δ = 0 yield Theorem 2.3 in [3]
which states that:
If f (z) ∈ SD(α, β), then
|a2 | ≤
(2.8)
2(1 − β)
|1 − α|
and
(2.9)
Coefficient Inequalities
n−2 2(1 − β)
2(1 − β) Y
1+
,
|an | ≤
(n − 1)|1 − α| j=1
j|1 − α|
S. Latha
(n ≥ 3).
vol. 9, iss. 2, art. 52, 2008
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Corollary 2.6. Putting α = 0 in Corollary 2.5, we get
Qn
j=1 (j − 2β)
(2.10)
|an | ≤
, (n ≥ 2),
(n − 1)!
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a result by Robertson [4].
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Corollary 2.7. For the parametric values b = δ = 1 we obtain Corollary 2.5 in [3]
given by:
If f (z) ∈ KD(α, β) then,
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(2.11)
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(1 − β)
|a2 | ≤
|1 − α|
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and
(2.12)
n−2
Y
2(1 − β)
2(1 − β)
|an | ≤
1+
,
n(n − 1)|1 − α| j=1
j|1 − α|
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(n ≥ 3).
Corollary 2.8. Letting α = 0 in Corollary 2.7, we get the inequality by Robertson
[4] given by:
Qn
j=1 (j − 2β)
(2.13)
|an | ≤
, (n ≥ 2).
n!
Coefficient Inequalities
S. Latha
vol. 9, iss. 2, art. 52, 2008
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References
[1] C. CARATHÉODORY, Über den variabilitätsbereich der Fourier’schen konstanten von possitiven harmonischen funktionen, Rend. Circ. Palermo.,32 (1911),
193–217.
[2] S. LATHA AND S. NANJUNDA RAO, Convex combinations of n analytic functions in generalized Ruscheweyh class, Int. J. Math, Educ. Sci. Technology.,
25(6) (1994), 791–795.
[3] S. OWA, Y. POLATOǦLU AND E. YAVUZ, Cofficient inequalities fo classes
of uniformly starlike and convex functions, J. Ineq. in Pure and Appl. Math.,
7(5) (2006), Art. 160. [ONLINE: http://jipam.vu.edu.au/article.
php?sid=778].
[4] M.S. ROBERTSON, On the theory of univalent functions, Ann. Math., 37
(1936), 374–408.
[5] S. RUSCHEWEYH, A new criteria for univalent function, Proc. Amer. Math.
Soc., 49(1) (1975), 109–115.
[6] S. SHAMS, S.R. KULKARNI AND J. M. JAHANGIRI, Classes of uniformly
starlike convx functions, Internat. J. Math. and Math. Sci., 55 (2004), 2959–
2961.
Coefficient Inequalities
S. Latha
vol. 9, iss. 2, art. 52, 2008
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