Paper-17-Acta23.2010. 145KB Jun 04 2011 12:03:11 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 23/2010
pp. 159-178
SOME FAMILIES OF UNIVALENT FUNCTIONS ASSOCIATED
WITH SALAGEAN DERIVATIVE OPERATOR
S. B. Joshi and G. D. Shelake
Abstract. Making use of Salagean derivative operator, the authors have introλ (α, β, A, B) of normalized and univalent funcduced and studied new subclass Tn,k
tions in unit disk U = {z : |z| < 1}. Among other results we have established
λ (α, β, A, B). Finally, several applications involving
certain characterization of Tn,k
an integral operator and fractional calculus operators are also determined.
Keywords. Salagean operator, Hadmard product, integral operator, fractional
calculus operator.
2000 Mathematics Subject Classification: 30C45.
1. Introduction and Definitions
Let Ak denote the class of functions of the form
f (z) = z +
∞
X
aj z j
(k ∈ N := {1, 2, 3, . . . }),
(1.1)
j=k+1
which are analytic in open unit disk
U = {z : z ∈ C
and |z| < 1}.
Definition 1 [6]. We define the operator Dn : Ak → Ak , (n ∈ N0 := N ∪ {0})
by
D0 f (z) = f (z),
D1 f (z) = zf ′ (z),
Dn f (z) = D(Dn−1 f (z)).
159
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
The operator Dn is known as the Salagean derivative operator.
For the function f (z) given by (1.1), it follows form above definition that
n
D f (z) = z +
∞
X
j n aj z j
(n ∈ N0 )
(1.2)
j=k+1
with the help of the operator Dn we define, the subclass denoted by Aλn,k (α, β, A, B)
as follows.
Definition 2. We define the class Aλn,k (α, β, A, B) by
F
(z)
−
1
n,λ
λ
−1. Then the function F defined by
c+1
F (z) =
zc
Zz
tc−1 f (t)dt
0
169
(c > −1; f ∈ Ak )
(4.1)
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
λ (κ, β, A, B), where
belongs to the class Tn,k
κ=
(1 + βB){k + (c + 1)α} + β(B − A)(1 − α)
(1 + βB){k + c + 1} + β(B − A)(1 − α)
This result is sharp for the functions f defined by (2.8).
Proof. Form (4.1) we have,
∞
X
c+1
F (z) = z −
aj z j .
c+j
j=k+1
We need to find largest κ such that
{(j − 1)(1 + βB) + β(B − A)(1 − κ)}(c + 1)
(1 − κ)(c + j)
≤
(j − 1)(1 + βB) + β(B − A)(1 − α)
1−α
(j ≥ k + 1; k ∈ N)
or, equivalently,
κ≤
(1 + βB){(j − 1) + (c + 1)α} + β(B − A)(1 − α)
(1 + βB){c + j} + β(B − A)(1 − α)
(j ≥ k + 1; k ∈ N). (4.2)
The right hand side of (4.2) being an increasing function of j, we have
κ≤
(1 + βB){k + (c + 1)α} + β(B − A)(1 − α)
,
(1 + βB){k + c + 1} + β(B − A)(1 − α)
which completes the proof of Theorem 8.
Theorem 9. Let the function F given by
F (z) = z −
∞
X
dj z j
(dj ≥ 0; j = k + 1, k + 2, k + 3, . . . ;
k ∈ N)
j=k+1
λ (α, β, A, B), and let c be a real number such that c > −1. Then
be in the class Tn,k
the function f defined by
c+1
f (z) =
zc
Zz
tc−1 F (t)dt
0
170
(c > −1; F ∈ Ak )
(4.3)
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
is univalent in |z| < R, where
R = inf
j≥k+1
j n−1 (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}(c + 1)
β(B − A)(1 − α)(c + j)
1/(j−1)
(4.4)
The result (4.4) is sharp.
Proof. We find form (4.3) that,
∞
X
c+j
z 1−c (z c F (z))′
f (z) =
=z−
dj z j
c+1
c+1
j=k+1
In order to obtain desired result, it is sufficient to show that
|f ′ (z) − 1| < 1 whenever |z| < R,
where R is given by (4.4).
Now
′
|f (z) − 1| ≤
∞
X
j
c+j
c+1
j=k+1
c+j
c+1
dj |z|j−1 .
Thus we have |f ′ (z) − 1| < 1 if
∞
X
j
j=k+1
dj |z|j−1 < 1.
(4.5)
But, by Theorem 1, we know that
∞
X
j n (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}
dj ≤ 1.
β(B − A)(1 − α)
j=k+1
Hence (4.5) will be satisfied if
j(c + j) j−1 j n (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}
|z|
<
,
c+1
β(B − A)(1 − α)
That is, if
|z| <
j n−1 (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}(c + 1)
β(B − A)(1 − α)(c + j)
(j ≥ k + 1; k ∈ N).
1/(j−1)
(4.6)
171
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Therefore the function f given by (4.1) is univalent in |z| < R, where R is defined
by (4.4). The sharpness is follows if we take
f (z) = z −
β(B − A)(1 − α)(c + j)
zj
j n−1 (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}(c + 1)
(j ≥ k + 1; k ∈ N).
(4.7)
5. Applications of Fractional Calculus
In this section we prove distortion theorem for functions belonging to the class
which involve operators of fractional calculus defined as follows.
λ (α, β, A, B),
Tn,k
Definition 1 [5]. The fractional integral of order µ is defined, for a function f ,
by
1
Dz−µ f (z) =
Γ(µ)
Zz
f (ζ)
dζ
(z − ζ)1−µ
(µ > 0),
(5.1)
0
where f is analytic function in a simply connected region of the complex plane containing the origin, and the multiplicity of (z−ζ)1−µ is removed, by requiring log(z−ζ)
to be real when z − ζ > 0.
Definition 2 [5]. The fractional derivative of order µ is defined, for a function
f , by
Zz
d
f (ζ)
1
µ
dζ (0 ≤ µ < 1),
(5.2)
Dz f (z) =
Γ(1 − µ) dz
(z − ζ)µ
0
where f is constrained, and the multiplicity of (z − ζ)−µ is removed, as in Definition 1.
Definition 3 [5]. Under the hypothesis of Definition 2, the fractional derivative
of order n + µ is defined, for a function f , by
Dzn+µ f (z) =
dn
{Dµ f (z)}
dz n z
(0 ≤ µ < 1; n ∈ N0 ).
(5.3)
λ (α, β, A, B).
Theorem 10. Let the function f defined by (1.4) be in the class Tn,k
172
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Then
|Dz−µ (Di f (z))|
≥
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
r1+µ
1 −
rk
n−i
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n})
(5.4)
and
|Dz−µ (Di f (z))|
r1+µ
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
k
r
≤
1+
n−i
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n})
(5.5)
The results (5.4) and (5.5) are sharp.
Proof. We observe that
λ
f (z) ∈ Tn,k
(α, β, A, B)
and that
Di f (z) = z −
⇔
λ
Di f (z) ∈ Tn−i,k
(α, β, A, B)
∞
X
j i aj z j
(i ∈ N0 )
j=k+1
Then from Theorem 1, we have
(k + 1)n−i (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
∞
X
j i aj
j=k+1
≤
∞
X
j n (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}aj
j=k+1
≤ β(B − A)(1 − α),
173
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
so that
∞
X
j i aj ≤
j=k+1
(k +
1)n−i (1
β(B − A)(1 − α)
.
+ λk){(1 + βB)k + β(B − A)(1 − α)}
(5.6)
Consider the function G(z) defined by
G(z) = Γ(2 + µ)z −µ Dz−µ (Di f (z))
= z−
∞
X
Γ(j + 1)Γ(2 + µ) i j
j aj z
Γ(j + 1 + µ)
j=k+1
= z−
∞
X
Φ(j)j i aj z j ,
j=k+1
where
Φ(j) =
Γ(j + 1)Γ(2 + µ)
Γ(j + 1 + µ)
(j ≥ k + 1; k ∈ N; µ > 0).
Since Φ(j) is a decreasing function of j, we get
0 < Φ(j) ≤ Φ(k + 1) =
Γ(k + 2)Γ(2 + µ)
Γ(k + 2 + µ)
(j ≥ k + 1; k ∈ N; µ > 0).
(5.7)
Thus by using (5.6) and (5.7), we see that
|G(z)| ≥ r − Φ(k + 1)rk+1
∞
X
j i aj
j=k+1
≥ r−
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n})
and
|G(z)| ≤ r + Φ(k + 1)rk+1
∞
X
j i aj
j=k+1
≤ r+
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n}),
174
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
which proves the inequalities (5.4) and (5.5) of Theorem 10.
The inequalities (5.4) and (5.5) are attained for the function f (z) given by
Di f (z) = z −
β(B − A)(1 − α)
z k+1
(k + 1)n−i (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
(k ∈ N).
(5.8)
This completes the proof of Theorem 10.
λ (α, β, A, B).
Corollary 2. Let the function f defined by (1.4) be in the class Tn,k
Then
|Dz−µ f (z)|
(|z| = r < 1; µ > 0)
(5.9)
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
r1+µ
k
r
1−
≥
n
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
and
|Dz−µ f (z)|
(|z| = r < 1; µ > 0)
(5.10)
r1+µ
β(B
−
A)(1
−
α)Γ(2
+
k)Γ(2
+
µ)
k
1 +
≤
r
n
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
The estimates in (5.9) and (5.10) are sharp for the function f given by (5.8) with
i = 0.
λ (α, β, A, B).
Theorem 11. Let the function f defined by (1.4) be in the class Tn,k
175
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Then
|Dzµ (Di f (z))|
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
r1−µ
k
≥
r
1
−
Γ(2 − µ)
(k + 1)n−i (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1})
(5.11)
and
|Dzµ (Di f (z))|
β(B
−
A)(1
−
α)Γ(2
+
k)Γ(2
−
µ)
r1−µ
k
1 +
r
≤
n−i
Γ(2 − µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1}).
(5.12)
The results (5.11) and (5.12) are sharp.
Proof. Consider the function H(z) defined by
H(z) = Γ(2 − µ)z µ Dzµ (Di f (z))
= z−
∞
X
Γ(j + 1)Γ(2 − µ) i j
j aj z
Γ(j + 1 − µ)
j=k+1
= z−
∞
X
Ψ(j)j i aj z j ,
j=k+1
where
Ψ(j) =
Γ(j + 1)Γ(2 − µ)
Γ(j + 1 − µ)
(j ≥ k + 1; k ∈ N; 0 ≤ µ < 1).
Since Ψ(j) is a decreasing function of j, we get
0 < Ψ(j) ≤ Ψ(k + 1) =
Γ(k + 2)Γ(2 − µ)
Γ(k + 2 − µ)
176
(j ≥ k + 1; k ∈ N; 0 ≤ µ < 1).
(5.13)
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Thus by using (5.6) and (5.13), we see that
|H(z)| ≥ r − Ψ(k + 1)r
k+1
∞
X
j i aj
j=k+1
≥ r−
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1})
and
|H(z)| ≤ r + Ψ(k + 1)rk+1
∞
X
j i aj
j=k+1
≤ r+
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1}),
The inequalities (5.11) and (5.11) are attained for the function f (z) given by (5.8).
This completes the proof of Theorem 11.
λ (α, β, A, B).
Corollary 3. Let the function f defined by (1.4) be in the class Tn,k
Then
|Dzµ f (z)|
r1−µ
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
k
≥
r
1−
n
Γ(2 − µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1)
(5.14)
177
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
and
|Dzµ f (z)|
β(B
−
A)(1
−
α)Γ(2
+
k)Γ(2
−
µ)
r1−µ
k
1 +
≤
r
n
Γ(2 − µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1)
(5.15)
The estimates in (5.14) and (5.15) are sharp for the function f (z) given by (5.8)
with i = 0.
References
[1] O. Altintas, On subclass of certain starlike functions with negative coefficients,
Math. Japon., Vol. 36, (1991), 489–495.
[2] M. K. Aouf, On fractional derivative and fractional integrals of certain subclasses of starlike and convex functions, Math. Japon., Vol. 35, (1990), 831–837.
[3] M. K. Aouf and H. M. Srivastava, Some families of starlike functions with
negative coefficients, J. Math. Anal. Appl., Vol. 203, (1996), 762–690.
[4] S. K. Chatterjea, On starlike functions, J. Pure Math., Vol. 1, (1981), 23–26.
[5] B. Ross, A brief history and exposition of the fundamental theory of fractional
calculus, Fractional Calculus and its Applications (Ed. B. Ross), Springer-Verlag,
Berlin, Heidelberg and New York (1975).
[6] Gr. St. Salagean, Subclasses of univalent functions, Complex Analysis, Fifth
Romanian-Finnish Sem., Lect. Notes in Math., Vol. 1013, Springer-Verlag, (1983),
362–372.
[7] A. Schild and H. Silverman, Convolution of univalent functions with negative
coefficients, Ann. Univ. Mariae Curie-Sklodowska Sect., Vol. A 29, (1975), 99–106.
[8] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., Vol. 51, (1975), 109–116.
[9] H. M. Srivastava, S. Owa and S. K. Chatterjea, A note on certain classes of
starlike functions, Rend, Sem. Mat. Univ. Padova, Vol. 77, (1987), 115–124.
[10] H. M. Srivastava, J. Patel and P. Sahoo, Some families of analytic functions
with negative coefficients, Math. Slovaca, Vol. 51, No. 4, (2001), 421–439.
S. B. Joshi and G. D. Shelake
Department of Mathematics
Walchand College of Engineering
Sangli (M.S), India 416 415
e-mails: joshisb@hotmail.com, shelakegd@rediffmail.com
178
ISSN: 1582-5329
No. 23/2010
pp. 159-178
SOME FAMILIES OF UNIVALENT FUNCTIONS ASSOCIATED
WITH SALAGEAN DERIVATIVE OPERATOR
S. B. Joshi and G. D. Shelake
Abstract. Making use of Salagean derivative operator, the authors have introλ (α, β, A, B) of normalized and univalent funcduced and studied new subclass Tn,k
tions in unit disk U = {z : |z| < 1}. Among other results we have established
λ (α, β, A, B). Finally, several applications involving
certain characterization of Tn,k
an integral operator and fractional calculus operators are also determined.
Keywords. Salagean operator, Hadmard product, integral operator, fractional
calculus operator.
2000 Mathematics Subject Classification: 30C45.
1. Introduction and Definitions
Let Ak denote the class of functions of the form
f (z) = z +
∞
X
aj z j
(k ∈ N := {1, 2, 3, . . . }),
(1.1)
j=k+1
which are analytic in open unit disk
U = {z : z ∈ C
and |z| < 1}.
Definition 1 [6]. We define the operator Dn : Ak → Ak , (n ∈ N0 := N ∪ {0})
by
D0 f (z) = f (z),
D1 f (z) = zf ′ (z),
Dn f (z) = D(Dn−1 f (z)).
159
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
The operator Dn is known as the Salagean derivative operator.
For the function f (z) given by (1.1), it follows form above definition that
n
D f (z) = z +
∞
X
j n aj z j
(n ∈ N0 )
(1.2)
j=k+1
with the help of the operator Dn we define, the subclass denoted by Aλn,k (α, β, A, B)
as follows.
Definition 2. We define the class Aλn,k (α, β, A, B) by
F
(z)
−
1
n,λ
λ
−1. Then the function F defined by
c+1
F (z) =
zc
Zz
tc−1 f (t)dt
0
169
(c > −1; f ∈ Ak )
(4.1)
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
λ (κ, β, A, B), where
belongs to the class Tn,k
κ=
(1 + βB){k + (c + 1)α} + β(B − A)(1 − α)
(1 + βB){k + c + 1} + β(B − A)(1 − α)
This result is sharp for the functions f defined by (2.8).
Proof. Form (4.1) we have,
∞
X
c+1
F (z) = z −
aj z j .
c+j
j=k+1
We need to find largest κ such that
{(j − 1)(1 + βB) + β(B − A)(1 − κ)}(c + 1)
(1 − κ)(c + j)
≤
(j − 1)(1 + βB) + β(B − A)(1 − α)
1−α
(j ≥ k + 1; k ∈ N)
or, equivalently,
κ≤
(1 + βB){(j − 1) + (c + 1)α} + β(B − A)(1 − α)
(1 + βB){c + j} + β(B − A)(1 − α)
(j ≥ k + 1; k ∈ N). (4.2)
The right hand side of (4.2) being an increasing function of j, we have
κ≤
(1 + βB){k + (c + 1)α} + β(B − A)(1 − α)
,
(1 + βB){k + c + 1} + β(B − A)(1 − α)
which completes the proof of Theorem 8.
Theorem 9. Let the function F given by
F (z) = z −
∞
X
dj z j
(dj ≥ 0; j = k + 1, k + 2, k + 3, . . . ;
k ∈ N)
j=k+1
λ (α, β, A, B), and let c be a real number such that c > −1. Then
be in the class Tn,k
the function f defined by
c+1
f (z) =
zc
Zz
tc−1 F (t)dt
0
170
(c > −1; F ∈ Ak )
(4.3)
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
is univalent in |z| < R, where
R = inf
j≥k+1
j n−1 (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}(c + 1)
β(B − A)(1 − α)(c + j)
1/(j−1)
(4.4)
The result (4.4) is sharp.
Proof. We find form (4.3) that,
∞
X
c+j
z 1−c (z c F (z))′
f (z) =
=z−
dj z j
c+1
c+1
j=k+1
In order to obtain desired result, it is sufficient to show that
|f ′ (z) − 1| < 1 whenever |z| < R,
where R is given by (4.4).
Now
′
|f (z) − 1| ≤
∞
X
j
c+j
c+1
j=k+1
c+j
c+1
dj |z|j−1 .
Thus we have |f ′ (z) − 1| < 1 if
∞
X
j
j=k+1
dj |z|j−1 < 1.
(4.5)
But, by Theorem 1, we know that
∞
X
j n (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}
dj ≤ 1.
β(B − A)(1 − α)
j=k+1
Hence (4.5) will be satisfied if
j(c + j) j−1 j n (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}
|z|
<
,
c+1
β(B − A)(1 − α)
That is, if
|z| <
j n−1 (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}(c + 1)
β(B − A)(1 − α)(c + j)
(j ≥ k + 1; k ∈ N).
1/(j−1)
(4.6)
171
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Therefore the function f given by (4.1) is univalent in |z| < R, where R is defined
by (4.4). The sharpness is follows if we take
f (z) = z −
β(B − A)(1 − α)(c + j)
zj
j n−1 (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}(c + 1)
(j ≥ k + 1; k ∈ N).
(4.7)
5. Applications of Fractional Calculus
In this section we prove distortion theorem for functions belonging to the class
which involve operators of fractional calculus defined as follows.
λ (α, β, A, B),
Tn,k
Definition 1 [5]. The fractional integral of order µ is defined, for a function f ,
by
1
Dz−µ f (z) =
Γ(µ)
Zz
f (ζ)
dζ
(z − ζ)1−µ
(µ > 0),
(5.1)
0
where f is analytic function in a simply connected region of the complex plane containing the origin, and the multiplicity of (z−ζ)1−µ is removed, by requiring log(z−ζ)
to be real when z − ζ > 0.
Definition 2 [5]. The fractional derivative of order µ is defined, for a function
f , by
Zz
d
f (ζ)
1
µ
dζ (0 ≤ µ < 1),
(5.2)
Dz f (z) =
Γ(1 − µ) dz
(z − ζ)µ
0
where f is constrained, and the multiplicity of (z − ζ)−µ is removed, as in Definition 1.
Definition 3 [5]. Under the hypothesis of Definition 2, the fractional derivative
of order n + µ is defined, for a function f , by
Dzn+µ f (z) =
dn
{Dµ f (z)}
dz n z
(0 ≤ µ < 1; n ∈ N0 ).
(5.3)
λ (α, β, A, B).
Theorem 10. Let the function f defined by (1.4) be in the class Tn,k
172
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Then
|Dz−µ (Di f (z))|
≥
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
r1+µ
1 −
rk
n−i
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n})
(5.4)
and
|Dz−µ (Di f (z))|
r1+µ
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
k
r
≤
1+
n−i
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n})
(5.5)
The results (5.4) and (5.5) are sharp.
Proof. We observe that
λ
f (z) ∈ Tn,k
(α, β, A, B)
and that
Di f (z) = z −
⇔
λ
Di f (z) ∈ Tn−i,k
(α, β, A, B)
∞
X
j i aj z j
(i ∈ N0 )
j=k+1
Then from Theorem 1, we have
(k + 1)n−i (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
∞
X
j i aj
j=k+1
≤
∞
X
j n (1 − λ + λj){(j − 1)(1 + βB) + β(B − A)(1 − α)}aj
j=k+1
≤ β(B − A)(1 − α),
173
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
so that
∞
X
j i aj ≤
j=k+1
(k +
1)n−i (1
β(B − A)(1 − α)
.
+ λk){(1 + βB)k + β(B − A)(1 − α)}
(5.6)
Consider the function G(z) defined by
G(z) = Γ(2 + µ)z −µ Dz−µ (Di f (z))
= z−
∞
X
Γ(j + 1)Γ(2 + µ) i j
j aj z
Γ(j + 1 + µ)
j=k+1
= z−
∞
X
Φ(j)j i aj z j ,
j=k+1
where
Φ(j) =
Γ(j + 1)Γ(2 + µ)
Γ(j + 1 + µ)
(j ≥ k + 1; k ∈ N; µ > 0).
Since Φ(j) is a decreasing function of j, we get
0 < Φ(j) ≤ Φ(k + 1) =
Γ(k + 2)Γ(2 + µ)
Γ(k + 2 + µ)
(j ≥ k + 1; k ∈ N; µ > 0).
(5.7)
Thus by using (5.6) and (5.7), we see that
|G(z)| ≥ r − Φ(k + 1)rk+1
∞
X
j i aj
j=k+1
≥ r−
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n})
and
|G(z)| ≤ r + Φ(k + 1)rk+1
∞
X
j i aj
j=k+1
≤ r+
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 + µ)
(|z| = r < 1; µ > 0; i ∈ {0, 1, . . . , n}),
174
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
which proves the inequalities (5.4) and (5.5) of Theorem 10.
The inequalities (5.4) and (5.5) are attained for the function f (z) given by
Di f (z) = z −
β(B − A)(1 − α)
z k+1
(k + 1)n−i (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
(k ∈ N).
(5.8)
This completes the proof of Theorem 10.
λ (α, β, A, B).
Corollary 2. Let the function f defined by (1.4) be in the class Tn,k
Then
|Dz−µ f (z)|
(|z| = r < 1; µ > 0)
(5.9)
β(B − A)(1 − α)Γ(2 + k)Γ(2 + µ)
r1+µ
k
r
1−
≥
n
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
and
|Dz−µ f (z)|
(|z| = r < 1; µ > 0)
(5.10)
r1+µ
β(B
−
A)(1
−
α)Γ(2
+
k)Γ(2
+
µ)
k
1 +
≤
r
n
Γ(2 + µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 + µ)
The estimates in (5.9) and (5.10) are sharp for the function f given by (5.8) with
i = 0.
λ (α, β, A, B).
Theorem 11. Let the function f defined by (1.4) be in the class Tn,k
175
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Then
|Dzµ (Di f (z))|
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
r1−µ
k
≥
r
1
−
Γ(2 − µ)
(k + 1)n−i (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1})
(5.11)
and
|Dzµ (Di f (z))|
β(B
−
A)(1
−
α)Γ(2
+
k)Γ(2
−
µ)
r1−µ
k
1 +
r
≤
n−i
Γ(2 − µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1}).
(5.12)
The results (5.11) and (5.12) are sharp.
Proof. Consider the function H(z) defined by
H(z) = Γ(2 − µ)z µ Dzµ (Di f (z))
= z−
∞
X
Γ(j + 1)Γ(2 − µ) i j
j aj z
Γ(j + 1 − µ)
j=k+1
= z−
∞
X
Ψ(j)j i aj z j ,
j=k+1
where
Ψ(j) =
Γ(j + 1)Γ(2 − µ)
Γ(j + 1 − µ)
(j ≥ k + 1; k ∈ N; 0 ≤ µ < 1).
Since Ψ(j) is a decreasing function of j, we get
0 < Ψ(j) ≤ Ψ(k + 1) =
Γ(k + 2)Γ(2 − µ)
Γ(k + 2 − µ)
176
(j ≥ k + 1; k ∈ N; 0 ≤ µ < 1).
(5.13)
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
Thus by using (5.6) and (5.13), we see that
|H(z)| ≥ r − Ψ(k + 1)r
k+1
∞
X
j i aj
j=k+1
≥ r−
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1})
and
|H(z)| ≤ r + Ψ(k + 1)rk+1
∞
X
j i aj
j=k+1
≤ r+
(k +
1)n−i (1
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
rk+1
+ λk){(1 + βB)k + β(B − A)(1 − α)}Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1; i ∈ {0, 1, . . . , n − 1}),
The inequalities (5.11) and (5.11) are attained for the function f (z) given by (5.8).
This completes the proof of Theorem 11.
λ (α, β, A, B).
Corollary 3. Let the function f defined by (1.4) be in the class Tn,k
Then
|Dzµ f (z)|
r1−µ
β(B − A)(1 − α)Γ(2 + k)Γ(2 − µ)
k
≥
r
1−
n
Γ(2 − µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1)
(5.14)
177
S. B. Joshi, G. D. Shelake - Some families of univalent functions associated with...
and
|Dzµ f (z)|
β(B
−
A)(1
−
α)Γ(2
+
k)Γ(2
−
µ)
r1−µ
k
1 +
≤
r
n
Γ(2 − µ)
(k + 1) (1 + λk){(1 + βB)k + β(B − A)(1 − α)}
Γ(k + 2 − µ)
(|z| = r < 1; 0 ≤ µ < 1)
(5.15)
The estimates in (5.14) and (5.15) are sharp for the function f (z) given by (5.8)
with i = 0.
References
[1] O. Altintas, On subclass of certain starlike functions with negative coefficients,
Math. Japon., Vol. 36, (1991), 489–495.
[2] M. K. Aouf, On fractional derivative and fractional integrals of certain subclasses of starlike and convex functions, Math. Japon., Vol. 35, (1990), 831–837.
[3] M. K. Aouf and H. M. Srivastava, Some families of starlike functions with
negative coefficients, J. Math. Anal. Appl., Vol. 203, (1996), 762–690.
[4] S. K. Chatterjea, On starlike functions, J. Pure Math., Vol. 1, (1981), 23–26.
[5] B. Ross, A brief history and exposition of the fundamental theory of fractional
calculus, Fractional Calculus and its Applications (Ed. B. Ross), Springer-Verlag,
Berlin, Heidelberg and New York (1975).
[6] Gr. St. Salagean, Subclasses of univalent functions, Complex Analysis, Fifth
Romanian-Finnish Sem., Lect. Notes in Math., Vol. 1013, Springer-Verlag, (1983),
362–372.
[7] A. Schild and H. Silverman, Convolution of univalent functions with negative
coefficients, Ann. Univ. Mariae Curie-Sklodowska Sect., Vol. A 29, (1975), 99–106.
[8] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., Vol. 51, (1975), 109–116.
[9] H. M. Srivastava, S. Owa and S. K. Chatterjea, A note on certain classes of
starlike functions, Rend, Sem. Mat. Univ. Padova, Vol. 77, (1987), 115–124.
[10] H. M. Srivastava, J. Patel and P. Sahoo, Some families of analytic functions
with negative coefficients, Math. Slovaca, Vol. 51, No. 4, (2001), 421–439.
S. B. Joshi and G. D. Shelake
Department of Mathematics
Walchand College of Engineering
Sangli (M.S), India 416 415
e-mails: joshisb@hotmail.com, shelakegd@rediffmail.com
178