Volume 9 (2008), Issue 2, Article 52, 5 pp.
COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF RUSCHEWEYH
TYPE ANALYTIC FUNCTIONS
S. LATHA
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE
M AHARAJA’ S C OLLEGE
U NIVERSITY OF M YSORE
M YSORE - 570 005
INDIA.
drlatha@gmail.com
Received 18 January, 2007; accepted 5 May, 2008
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Convolution, Ruscheweyh derivative, Uniformly starlike and Uniformly convex.
2000 Mathematics Subject Classification. 30C45.
1. I NTRODUCTION
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− +
b b Dδ f (z)
(1.2)
>β
where, b 6= 0, δ > −1, 0 ≤ β < 1 and Dδ f is the Rushceweyh derivative of f [5] given by,
(1.3)
031-07
∞
X
z
D f (z) =
∗ f (z) = z +
an Bn (δ)z n ,
1+δ
(1 − z)
n=2
δ
2
S. L ATHA
where ∗ stands for the convolution or Hadamard product of two power series and
Bn (δ) =
(δ + 1)(δ + 2) · · · (δ + n − 1)
.
(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 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)
(1.5)
< 1+ 0
> α 0
− 1 + β
f (z)
f (z)
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
< 1− +
>
α
−
1
b b Dδ f (z)
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.
2. M AIN R ESULTS
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 >
V β−α
,
b,
δ
.
1−α
1+β
,
2
then f (z) ∈
Proof. Since <{ω} ≤ |ω| for any complex number ω, f (z) ∈ VD(α, β, b, δ) implies that
2 Dδ+1 f (z) 2 2 2 Dδ+1 f (z)
(2.1)
< 1− +
> α − + β.
b b Dδ f (z)
b Dδ f (z)
b
Equivalently,
(2.2)
2 2 Dδ+1 f (z)
β−α
< 1− +
>
,
δ
b b D f (z)
1−α
If 0 ≤ α ≤ β, we have, 0 ≤
β−α
1−α
< 1, and if α >
1+β
,
2
(z ∈ U).
then we have −1 <
α−β
α−1
≤ 0.
Corollary 2.2. For the parametric values b = 2 and δ = 0, we get Theorem 2.1 in [3] which
reads as:
, then f (z) ∈ S ∗ β−α
.
If f (z) ∈ SD(α, β) with 0 ≤ α ≤ β, or, α > 1+β
2
1−α
Corollary 2.3. The parametric values b = δ = 1, yield the Corollary 2.2 in [3] stated as:
If f (z) ∈ KD(α, β) with 0 ≤ α ≤ β, or, α > 1+β
, then f (z) ∈ K β−α
.
2
1−α
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 52, 5 pp.
http://jipam.vu.edu.au/
C OEFFICIENT I NEQUALITIES
3
Theorem 2.4. If f (z) ∈ VD(α, β, b, δ) then,
|a2 | ≤
(2.3)
b(1 − β)
|1 − α|
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
p(z) =
(2.5)
h
(1 − α) 1 − 2b +
2 Dδ+1 f (z)
b Dδ f (z)
i
− (β − α)
(1 − β)
,
(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
=1+
pn z n .
(2.6)
1− +
b b Dδ f (z)
1 − α n=1
That is,
2(Dδ+1 f (z) − Dδ f (z)) = bDδ f (z)
!
∞
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)
Applying the coefficient estimates |pn | ≤ 2 for Carathéodory functions [1], we obtain,
(2.7)
|an | ≤
b(1 − β)(δ + 1)
[1 + B2 (δ)|a2 | + B3 (δ)|a3 | + · · · + Bn−1 (δ)|an−1 |] .
|1 − α|(n − 1)Bn (δ)
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.
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 52, 5 pp.
http://jipam.vu.edu.au/
4
S. L ATHA
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)
b(1 − β)(δ + 1) Y
+··· +
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.
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)
n−2 2(1 − β) Y
2(1 − β)
|an | ≤
1+
,
(n − 1)|1 − α| j=1
j|1 − α|
(n ≥ 3).
Corollary 2.6. Putting α = 0 in Corollary 2.5, we get
Qn
j=1 (j − 2β)
(2.10)
|an | ≤
, (n ≥ 2),
(n − 1)!
a result by Robertson [4].
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,
(2.11)
|a2 | ≤
(1 − β)
|1 − α|
and
(2.12)
n−2
Y
2(1 − β)
2(1 − β)
|an | ≤
1+
,
n(n − 1)|1 − α| j=1
j|1 − α|
(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!
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 52, 5 pp.
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C OEFFICIENT I NEQUALITIES
5
R EFERENCES
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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.
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 52, 5 pp.
http://jipam.vu.edu.au/