MATEMATIQKI VESNIK
originalni nauqni rad
research paper
62, 1 (2010), 23–35
March 2010
ON CERTAIN MULTIVALENT FUNCTIONS
WITH NEGATIVE COEFFICIENTS DEFINED
BY USING A DIFFERENTIAL OPERATOR
M. K. Aouf
Abstract. In this paper, we introduce the subclass Sj (n, p, q, α) of analytic and p-valent
functions with negative coefficients defined by new operator Dpn . In this paper we give some
properties of functions in the class Sj (n, p, q, α) and obtain numerous sharp results including
(for example) coefficient estimates, distortion theorem, radii of close-to-convexity, starlikeness
and convexity and modified Hadamard products of functions belonging to the class Sj (n, p, q, α).
Finally, several applications involving an integral operator and certain fractional calculus operators
are also considered.
1. Introduction
Let T (j, p) denote the class of functions of the form
f (z) = z p −
∞
P
ak z k
(ak ≥ 0; p, j ∈ N = {1, 2, . . . }),
(1.1)
k=j+p
which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. A function
f (z) ∈ T (j, p) is said to be p-valently starlike of order α if it satisfies the inequality
( 0
)
zf (z)
Re
> α (z ∈ U ; 0 ≤ α < p; p ∈ N ).
(1.2)
f (z)
We denote by Tj∗ (p, α) the class of all p-valently starlike functions of order α. Also
a function f (z) ∈ T (j, p) is said to be p-valently convex of order α if it satisfies the
inequality
(
)
00
zf (z)
> α (z ∈ U ; 0 ≤ α < p; p ∈ N ).
(1.3)
Re 1 + 0
f (z)
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Multivalent functions; differential operator; modified-Hadamard
product; fractional calculus.
23
24
M. K. Aouf
We denote by Cj (p, α) the class of all p-valently convex functions of order α. We
note that (see for example Duren [5] and Goodman [6])
0
f (z) ∈ Cj (p, α) ⇐⇒
zf (z)
∈ Tj∗ (p, α)
p
(0 ≤ α < p; p ∈ N ).
(1.4)
The classes Tj∗ (p, α) and Cj (p, α) were studied by Owa [12].
For each f (z) ∈ T (j, p), we have (see [3])
f (q) (z) =
∞
P
p!
k!
z p−q −
ak z k−q
(p − q)!
(k
−
q)!
k=j+p
(q ∈ N0 = N ∪ {0}; p > q). (1.5)
For a function f (z) in T (j, p), we define
Dp0 f (q) (z) = f (q) (z),
0
z
z
(f (q) (z)) =
f (1+q) (z)
(p − q)
(p − q)
µ
¶
∞
P
p!
k!
k−q
=
z p−q −
ak z k−q ,
(p − q)!
p−q
k=j+p (k − q)!
Dp1 f (q) (z) = Df (q) (z) =
Dp2 f (q) (z) = D(Dp1 f (q) (z))
=
∞
P
p!
k!
z p−q −
(p − q)!
(k
−
q)!
k=j+p
µ
k−q
p−q
(1.6)
¶2
ak z k−q ,
(1.7)
and
Dpn f (q) (z) = D(Dpn−1 f (q) (z)) (n ∈ N )
∞
P
p!
k!
=
z p−q −
(p − q)!
k=j+p (k − q)!
(p, j ∈ N ; q ∈ N0 ; p > q).
µ
k−q
p−q
¶n
ak z k−q
(1.8)
We note that, by taking q = 0 and p = 1, the differential operator D1n = Dn was
introduced by Salagean [13].
With the help of the differential operator Dpn , we say that a function f (z)
belonging to T (j, p) is in the class Sj (n, p, q, α) if and only if
(
)
0
z(Dpn f (q) (z))
Re
> α (p ∈ N ; q, n ∈ N0 )
(1.9)
Dpn f (q) (z)
for some α (0 ≤ α < p − q, p > q) and for all z ∈ U .
We note that, by specializing the parameters j, p, n, q and α, we obtain the
following subclasses studied by various authors:
(i) Sj (0, p, q, α) = Sj (p, q, α) and Sj (1, p, q, α) = Cj (p, q, α) (Chen et al. [3]);
(ii) Sj (n, 1, 0, α) = P (j, α, n) (j ∈ N ; n ∈ N0 ; 0 ≤ α < 1) (Aouf and Srivastava [1]);
25
On certain multivalent functions defined by a differential operator
(iii) S1 (n, 1, 0, α) = T (n, α) (n ∈ N0 ; 0 ≤ α < 1) (Hur and Oh [7]);
½ ∗
Tj (p, α) (Owa [12]),
(iv) Sj (0, p, 0, α) =
(p, j ∈ N ; 0 ≤ α < p).
Tα (p, j) (Yamakawa [20])
½
Cj (p, α)
(Owa [12]),
(v) Sj (1, p, 0, α) =
(p, j ∈ N ; 0 ≤ α < p).
CTα (p, j) (Yamakawa [20])
(vi) S1 (0, p, 0, α) = T ∗ (p, α) and S1 (1, p, 0, α) = C(p, α) (p ∈ N ; 0 ≤ α < p)
(Owa [11] and Salagean et al. [14]);
(vii) Sj (0, 1, 0, α) = Tα (j) and Sj (1, 1, 0, α) = Cα (j) (j ∈ N ; 0 ≤ α < 1)
(Srivastava et al. [19]);
(viii) Sj (n, p, 0, α) = Sj (n, p, α) (p, j ∈ N ; n ∈ N0 ; 0 ≤ α < p), where
Sj (n, p, α) represents the class of functions f (z) ∈ T (j, p) satisfying the inequality
(
Re
z(Dpn f (z))
Dpn f (z)
0
)
>α
(z ∈ U ).
(1.10)
In our present paper, we shall make use of the familiar integral operator Jc,p
defined by (cf. [2] , [8] and [9]; see also [18])
Z
c + p z c−1
(Jc,p f )(z) =
t f (t) dt
(1.11)
zc
0
(f ∈ T (j, p); c > −p; p ∈ N ) as well as the fractional calculus operator Dzµ for
which it is well known that (see for details [10] and [16]; see also Section 5 below)
Dzµ {z ρ } =
Γ(ρ + 1)
z ρ−µ
Γ(ρ + 1 − µ)
(ρ > −1; µ ∈ R)
(1.12)
in terms of Gamma functions.
2. Coefficient estimates
Theorem 1. Let the function f (z) be defined by (1.1).
Sj (n, p, q, α) if and only if
∞
P
k=j+p
µ
k−q
p−q
Then f (z) ∈
¶n
(k − q − α)δ(k, q)ak ≤ (p − q − α)δ(p, q)
(2.1)
(0 ≤ α < p − q; p, j ∈ N ; q, n ∈ N0 ; p > q) where
p!
=
δ(p, q) =
(p − q)!
½
p(p − 1) · . . . · (p − q + 1),
1,
q=
6 0,
q = 0.
(2.2)
26
M. K. Aouf
Proof. Assume that inequality (2.1) holds true. Then we find that
∞
P
¯
¯
¯ z(Dn f (q) (z))0
¯
¯
¯
p
−
(p
−
q)
¯
¯≤
¯ Dpn f (q) (z)
¯
k=j+p
k−p
n
(k − p)( k−q
p−q ) δ(k, q)ak |z|
∞
P
δ(p, q) −
k=j+p
∞
P
k=j+p
≤
n
( k−q
p−q ) δ(k, q)ak |z|
k−p
n
(k − p)( k−q
p−q ) δ(k, q)ak
≤ p − q − α.
∞
P
δ(p, q) −
k=j+p
n
( k−q
p−q ) δ(k, q)ak
This shows that the values of the function
0
z(Dpn f (q) (z))
Φ(z) =
Dpn f (q) (z)
(2.3)
lie in a circle which is centered at w = (p − q) and whose radius is (p − q − α).
Hence f (z) satisfies the condition (1.9).
Conversely, assume that the function f (z) is in the class Sj (n, p, q, α). Then
we have
(
)
0
z(Dpn f (q) (z))
=
Re
Dpn f (q) (z)


´n
³
∞
P
k−q
k−p 

δ(k,
q)a
z
(p
−
q)δ(p,
q)
−
(k
−
q)


k
p−q


Re



δ(p, q) −
k=j+p
∞ ³
P
k=j+p
k−q
p−q
´n
δ(k, q)ak z
k−p



> α,
(2.4)
for some α (0 ≤ α < p − q), p, j ∈ N, q, n ∈ N0 , p > q and z ∈ U . Choose
values of z on the real axis so that Φ(z) given by (2.3) is real. Upon clearing the
denominator in (2.4) and letting z → 1− through real values, we can see that
¶n
µ
∞
P
k−q
δ(k, q)ak ≥
(p − q)δ(p, q) −
(k − q)
p−q
k=j+p
(
)
µ
¶n
∞
P
k−q
≥ α δ(p, q) −
δ(k, q)ak . (2.5)
k=j+p p − q
Thus we have the inequality (2.1).
Corollary 1. Let the function f (z) defined by (1.1) be in the class Sj (n, p, q, α).
Then
ak ≤ ³
(p − q − α)δ(p, q)
´n
(k − q − α)δ(k, q)
k−q
p−q
(k ≥ j + p; p, j ∈ N ; q, n ∈ N0 ; p > q). (2.6)
On certain multivalent functions defined by a differential operator
27
The result is sharp for the function f (z) given by
f (z) = z p − ³
(p − q − α)δ(p, q)
´n
zk
(k − q − α)δ(k, q)
k−q
p−q
(2.7)
(k ≥ j + p; p, j ∈ N ; q, n ∈ N0 ; p > q).
Remark 1. (i) Putting n = 0 in Theorem 1, we obtain the result obtained by
Chen et al. [3, Theorem 1].
(ii) Putting n = 1 in Theorem 1, we obtain the result obtained by Chen et al.
[3, Theorem 2].
3. Distortion theorem
Theorem 3. If the function f (z) defined by (1.1) is in the class Sj (n, p, q, α),
then
(
)
(p − q − α)δ(p, q)(j + p − q)!
p!
j
p−m
−
|z| |z|
n
(p − m)! ( j+p−q
p−q ) (j + p − q − α)(j + p − m)!
¯
¯
¯
¯
≤ ¯f (m) (z)¯ ≤
(
)
p!
(p − q − α)δ(p, q)(j + p − q)!
j
p−m
+
|z| |z|
n (j + p − q − α)(j + p − m)!
(p − m)! ( j+p−q
)
p−q
(3.1)
(z ∈ U ; 0 ≤ α < p − q; p, j ∈ N ; q, n, m ∈ N0 ; p > max{q, m}). The result is
sharp for the function f (z) given by
f (z) = z p −
(p − q − α)δ(p, q)
z j+p
+ p − q − α)δ(j + p, q)
n
( j+p−q
p−q ) (j
(3.2)
(p, j ∈ N ; q, n ∈ N0 ; p > q).
Proof. In view of Theorem 1, we have
n
( j+p−q
p−q ) (j
∞
+ p − q − α)δ(j + p, q) P
k!ak
(p − q − α)δ(p, q)(j + p)!
k=j+p
≤
n
∞
( k−q
P
p−q ) (k − q − α)δ(k, q)
ak ≤ 1
(p − q − α)δ(p, q)
k=j+p
which readily yields
∞
P
k=j+p
k!ak ≤
(p − q − α)δ(p, q)(j + p − q)!
.
n
( j+p−q
p−q ) (j + p − q − α)
Now, by differentiating both sides of (1.1) m times, we have
∞
P
k!
p!
z p−m −
ak z k−m
f (m) (z) =
(p − m)!
(k
−
m)!
k=j+p
(3.3)
(3.4)
(k ≥ j + p; p, j ∈ N ; q, m ∈ N0 ; p > max{q, m}) and Theorem 2 follows from (3.3)
and (3.4).
28
M. K. Aouf
Remark 2. (i) Putting n = 0 in Theorem 2, we obtain the result obtained by
Chen et al. [3, Theorem 7].
(ii) Putting n = 1 in Theorem 2, we obtain the result obtained by Chen et al.
[3, Theorem 8].
4. Radii of close-to-convexity, starlikeness and convexity
Theorem 3. Let the function f (z) defined by (1.1) be in the class Sj (n, p, q, α).
Then
(i) f (z) is p-valently close-to-convex of order ϕ (0 ≤ ϕ < p) in |z| < r1 , where
(
r1 = inf
n
( k−q
p−q ) (k − q − α)δ(k, q)
µ
(p − q − α)δ(p, q)
k
p−ϕ
k
1
¶) k−p
(4.1)
(k ≥ j + p; p, j ∈ N ; q, n ∈ N0 ; p > q).
(ii) f (z) is p-valently starlike of order ϕ (0 ≤ ϕ < p) in |z| < r2 , where
(
r2 = inf
n
( k−q
p−q ) (k − q − α)δ(k, q)
(p − q − α)δ(p, q)
k
µ
p−ϕ
k−ϕ
1
¶) k−p
(4.2)
(k ≥ j + p; p, j ∈ N ; q, n ∈ N0 ; p > q).
(iii) f (z) is p-valently convex of order ϕ (0 ≤ ϕ < p) in |z| < r3 , where
(
r3 = inf
k
n
( k−q
p−q ) (k − q − α)δ(k, q) p(p − ϕ)
(p − q − α)δ(p, q)
1
) k−p
k(k − ϕ)
(4.3)
(k ≥ j + p; p, j ∈ N ; q, n ∈ N0 ; p > q). Each of these results is sharp for the
function f (z) given by (2.7).
Proof. It is sufficient to show that
¯ 0
¯
¯ f (z)
¯
¯
¯
¯ p−1 − p¯ ≤ p − ϕ (|z| < r1 ; 0 ≤ ϕ < p; p ∈ N ),
¯z
¯
¯
¯ 0
¯
¯ zf (z)
¯
¯
− p¯ ≤ p − ϕ (|z| < r2 ; 0 ≤ ϕ < p; p ∈ N ),
¯
¯
¯ f (z)
and that
¯
¯
00
¯
¯
zf (z)
¯
¯
− p¯ ≤ p − ϕ
¯1 + 0
¯
¯
f (z)
(|z| < r3 ; 0 ≤ ϕ < p; p ∈ N )
(4.4)
(4.5)
(4.6)
for a function f (z) ∈ Sj (n, p, q, α), where r1 , r2 and r3 are defined by (4.1), (4.2)
and (4.3), respectively. The details involved are fairly straightforward and may be
omitted.
On certain multivalent functions defined by a differential operator
29
5. Modified Hadamard products
For the functions
∞
P
fν (z) = z p −
ak,ν z k
(ak,ν ≥ 0; ν = 1, 2)
(5.1)
k=j+p
we denote by (f1 ~ f2 )(z) the modified Hadamard product (or convolution) of the
functions f1 (z) and f2 (z), where
∞
P
(f1 ~ f2 )(z) = z p −
ak,1 · ak,2 z k .
(5.2)
k=j+p
Theorem 4. Let the functions fν (z) (ν = 1, 2) defined by (5.1) be in the class
Sj (n, p, q, α). Then (f1 ~ f2 )(z) ∈ Sj (n, p, q, γ), where
γ = (p − q) −
j(p − q − α)2 δ(p, q)
.
n
2
2
( j+p−q
p−q ) (j + p − q − α) δ(j + p, q) − (p − q − α) δ(p, q)
(5.3)
The result is sharp for the functions fν (z) (ν = 1, 2) given by
fν (z) = z p −
(p − q − α)δ(p, q)
z j+p
+ p − q − α)δ(j + p, q)
n
( j+p−q
p−q ) (j
(5.4)
(p, j ∈ N ; q, n ∈ N0 ; p > q; ν = 1, 2).
Proof. Employing the technique used earlier by Schild and Silverman [15], we
need to find the largest γ such that
n
∞
( k−q
P
p−q ) (p − q − γ)δ(k, q)
ak,1 · ak,2 ≤ 1
(p − q − γ)δ(p, q)
k=j+p
(5.5)
(fν (z) ∈ Sj (n, p, q, α), ν = 1, 2). Since fν (z) ∈ Sj (n, p, q, α) (ν = 1, 2), we readily
see that
n
∞
( k−q
P
p−q ) (p − q − α)δ(k, q)
ak,ν ≤ 1 (ν = 1, 2).
(5.6)
(p − q − α)δ(p, q)
k=j+p
Therefore, by the Cauchy-Schwarz inequality, we obtain
n
∞
( k−q
P
p−q ) (k − q − α)δ(k, q) √
ak,1 · ak,2 ≤ 1.
(p − q − α)δ(p, q)
k=j+p
(5.7)
Thus we only need to show that
(k − q − γ)
(k − q − α) √
ak,1 · ak,2 ≤
ak,1 · ak,2
(p − q − γ)
(p − q − α)
(5.8)
(k ≥ j + p; p, j ∈ N ), or, equivalently, that
√
ak,1 .ak,2 ≤
(p − q − γ)(k − q − α)
(p − q − α)(k − q − γ)
(5.9)
30
M. K. Aouf
(k ≥ j + p; p, j ∈ N ). Hence, in the light of inequality (5.7), it is sufficient to prove
that
(p − q − γ)(k − q − α)
(p − q − α)δ(p, q)
≤
(5.10)
k−q n
(p
− q − α)(k − q − γ)
( p−q ) (k − q − α)δ(k, q)
(k ≥ j + p; p, j ∈ N ). It follows from (5.10) that
γ ≤ (p − q) −
(k − p)(p − q − α)2 δ(p, q)
n
2
2
( k−q
p−q ) (k − q − α) δ(k, q) − (p − q − α) δ(p, q)
(5.11)
(k ≥ j + p; p, j ∈ N ).
Now, defining the function G(k) by
G(k) = (p − q) −
n
( k−q
p−q ) (k
(k − p)(p − q − α)2 δ(p, q)
− q − α)2 δ(k, q) − (p − q − α)2 δ(p, q)
(5.12)
(k ≥ j + p; p, j ∈ N ), we see that G(k) is an increasing function of k. Therefore,
we conclude that
j(p − q − α)2 δ(p, q)
,
n
2
2
( j+p−q
p−q ) (j + p − q − α) δ(j + p, q) − (p − q − α) δ(p, q)
(5.13)
which evidently completes the proof of Theorem 4.
Putting n = 0 and n = 1 in Theorem 4, we obtain
γ ≤ G(j + p) = (p − q) −
Corollary 2. Let the functions fν (z) (ν = 1, 2) defined by (5.1) be in the
class Sj (p, q, α). Then (f1 ~ f2 )(z) ∈ Sj (p, q, γ), where
γ = (p − q) −
j(p − q − α)2 δ(p, q)
.
(j + p − q − α)2 δ(j + p, q) − (p − q − α)2 δ(p, q)
(5.14)
The result is sharp.
Remark 3. We note that the result obtained by Chen et al. [3, Theorem 5] is
not correct. The correct result is given by (5.14).
Corollary 3. Let the functions fν (z)(ν = 1, 2)) defined by (5.1) be in the
class Cj (p, q, α). Then (f1 ~ f2 )(z) ∈ Cj (p, q, γ), where
γ = (p − q) −
(j + p − q −
j(p − q − α)2 δ(p, q + 1)
.
+ p, q + 1) − (p − q − α)2 δ(p, q + 1)
α)2 δ(j
(5.15)
The result is sharp.
Remark 4. We note that the result obtained by Chen et al. [3, Theorem 6] is
not correct. The correct result is given by (5.15).
Using arguments similar to those in the proof of Theorem 4, we obtain the
following results.
On certain multivalent functions defined by a differential operator
31
Theorem 5. Let the functions f1 (z), resp. f2 (z) defined by (5.1) be in the
class Sj (n, p, q, α), resp. Sj (n, p, q, τ ). Then (f1 ~ f2 )(z) ∈ Sj (n, p, q, ζ), where
j(p − q − α)(p − q − τ )δ(p, q)
ζ = (p − q) − j+p−q
, (5.16)
n
( p−q ) (j + p − q − α)(j + p − q − τ )δ(j + p, q) − Ωδ(p, q)
where
Ω = (p − q − α)(p − q − τ ).
(5.17)
The result is the best possible for the functions
(p − q − α)δ(p, q)
z j+p (p, j ∈ N ; q, n ∈ N0 ; p > q)
f1 (z) = z p − j+p−q
n
( p−q ) (j + p − q − α)δ(j + p, q)
(5.18)
f2 (z) = z p −
(p − q − τ )δ(p, q)
z j+p
+ p − q − τ )δ(j + p, q)
n
( j+p−q
p−q ) (j
(p, j ∈ N ; q, n ∈ N0 ; p > q).
(5.19)
Theorem 6. Let the functions fν (z) (ν = 1, 2) defined by (5.1) be in the class
Sj (n, p, q, α). Then the function
∞
P
h(z) = z p −
(a2k,1 + a2k,2 )z k
(5.20)
k=j+p
belongs to the class Sj (n, p, q, ξ), where
ξ = (p − q) −
2j(p − q − α)2 δ(p, q)
. (5.21)
n
2
2
( j+p−q
p−q ) (j + p − q − α) δ(j + p, q) − 2(p − q − α) δ(p, q)
The result is the sharp for the functions fν (z) (ν = 1, 2) defined by (5.4).
6. Applications of fractional calculus
Various operators of fractional calculus (that is, fractional integral and fractional derivatives) have been studied in the literature rather extensively (cf., e.g.,
[4], [10], [17] and [18]; see also the various references cited therein). For our present
investigation, we recall the following definitions.
Definition 1. The fractional integral of order µ is defined, for a function
f (z), by
Z z
1
f (ζ)
Dz−µ f (z) =
dζ (µ > 0),
(6.1)
Γ(µ) 0 (z − ζ)1−µ
where the function f (z) is analytic in a simply-connected domain of the complex zplane containing the origin and the multiplicity of (z−ζ)µ−1 is removed by requiring
log(z − ζ) to be real when z − ζ > 0.
Definition 2. The fractional derivative of order µ is defined, for a function
f (z), by
Z z
f (ζ)
1
dζ (0 ≤ µ < 1),
(6.2)
Dzµ f (z) =
Γ(1 − µ) 0 (z − ζ)µ
where the function f (z) is constrained, and the multiplicity of (z −ζ)−µ is removed,
as in Definition 1.
32
M. K. Aouf
Definition 3. Under the hypotheses of Definition 2, the fractional derivative
of order n + µ is defined, for a function f (z), by
Dzn+µ f (z) =
dn
{Dzµ f (z)}
dz n
(0 ≤ µ < 1; n ∈ N0 ).
(6.3)
In this section, we shall investigate the growth and distortion properties of
functions in the class Sj (n, p, q, α), involving the operators Jc,p and Dzµ . In order
to derive our results, we need the following lemma given by Chen et al. [4].
Lemma 1. [4] Let the function f (z) be defined by (1.1). Then
Dzµ {(Jc,p f )(z)} =
∞
P
Γ(p + 1)
(c + p)Γ(p + 1)
z p−µ −
ak z k−µ
Γ(p + 1 − µ)
(c
+ k)Γ(p + 1 − µ)
k=j+p
(6.4)
(µ ∈ R; c > −p; p, j ∈ N ) and
Jc,p (Dzµ {f (z)}) =
∞
P
(c + p)Γ(p + 1)
(c + p)Γ(k + 1)
z p−µ −
ak z k−µ
(c + p − µ)Γ(p + 1 − µ)
(c
+
k
−
µ)Γ(k
+
1
−
µ)
k=j+p
(6.5)
(µ ∈ R; c > −p; p, j ∈ N ), provided that no zeros appear in the denominators in
(6.4) and (6.5).
Theorem 7. Let the function f (z) defined by (1.1) be in the class Sj (n, p, q, α).
Then
½
¯ −µ
¯
Γ(p + 1)
¯Dz {(Jc,p f )(z)}¯ ≥
−
Γ(p + 1 + µ)
)
(c + p)Γ(j + p + 1)(p − q − α)δ(p, q)
j
p+µ
|z| |z|
(6.6)
n (j + p − q − α)δ(j + p, q)
)
(c + j + p)Γ(j + p + 1 + µ)( j+p−q
p−q
(z ∈ U ; 0 ≤ α < p − q; µ > 0; c > −p; p, j ∈ N ; q, n ∈ N0 ; p > q) and
½
¯ −µ
¯
Γ(p + 1)
¯Dz {(Jc,p f )(z)}¯ ≤
+
Γ(p + 1 + µ)
)
(c + p)Γ(j + p + 1)(p − q − α)δ(p, q)
j
p+µ
|z| |z|
j+p−q n
(c + j + p)Γ(j + p + 1 + µ)( p−q ) (j + p − q − α)δ(j + p, q)
(6.7)
(z ∈ U ; 0 ≤ α < p − q; µ > 0; c > −p; p, j ∈ N ; q, n ∈ N0 ; p > q). Each of the
assertions (6.6) and (6.7) is sharp.
Proof. In view of Theorem 1, we have
n
( j+p−q
p−q ) (j
∞
+ p − q − α)δ(j + p, q) P
ak
(p − q − α)δ(p, q)
k=j+p
n
∞
( k−q
P
p−q ) (k − q − α)δ(k, q)
≤
ak ≤ 1,
(p − q − α)δ(p, q)
k=j+p
(6.8)
33
On certain multivalent functions defined by a differential operator
which readily yields
∞
P
ak ≤
k=j+p
(p − q − α)δ(p, q)
.
+ p − q − α)δ(j + p, q)
n
( j+p−q
p−q ) (j
(6.9)
Consider the function F (z) defined in U by
Γ(p + 1 + µ) −µ −µ
z Dz {(Jc,p f )(z)}
Γ(p + 1)
∞
P
(c + p)Γ(k + 1)Γ(p + 1 + µ)
= zp −
ak z k
k=j+p (c + k)Γ(k + 1 + µ)Γ(p + 1)
∞
P
= zp −
Φ(k)ak z k (z ∈ U )
F (z) =
k=j+p
where
Φ(k) =
(c + p)Γ(k + 1)Γ(p + 1 + µ)
(c + k)Γ(k + 1 + µ)Γ(p + 1)
(k ≥ j + p; p, j ∈ N ; µ > 0).
(6.10)
Since Φ(k) is a decreasing function of k when µ > 0, we get
0 < Φ(k) ≤ Φ(j + p) =
(c + p)Γ(j + p + 1)Γ(p + 1 + µ)
(c + j + p)Γ(j + p + 1 + µ)Γ(p + 1)
(6.11)
(c > −p; p, j ∈ N ; µ > 0). Thus, by using (6.9) and (6.11), we deduce that
∞
p
j+p P
p
|F (z)| ≥ |z| − Φ(j + p) |z|
ak ≥ |z| −
k=j+p
(c + p)Γ(j + p + 1)Γ(p + 1 + µ)(p − q − α)δ(p, q)
j+p
|z|
j+p−q n
(c + j + p)Γ(j + p + 1 + µ)Γ(p + 1)( p−q ) (j + p − q − α)δ(j + p, q)
(z ∈ U ) and
p
|F (z)| ≤ |z| + Φ(j + p) |z|
j+p
∞
P
p
ak ≤ |z| +
k=j+p
(c + p)Γ(j + p + 1)Γ(p + 1 + µ)(p − q − α)δ(p, q)
j+p
|z|
n (j + p − q − α)δ(j + p, q)
(c + j + p)Γ(j + p + 1 + µ)Γ(p + 1)( j+p−q
)
p−q
(z ∈ U ), which yield the inequalities (6.6) and (6.7) of Theorem 7. The equalities
in (6.6) and (6.7) are attained for the function f (z) given by
½
Γ(p + 1)
−µ
Dz {(Jc,p f )(z)} =
−
Γ(p + 1 + µ)
)
(c + p)Γ(j + p + 1)(p − q − α)δ(p, q)
z j z p+µ (6.12)
n
(c + j + p)Γ(j + p + 1 + µ)( j+p−q
p−q ) (j + p − q − α)δ(j + p, q)
or, equivalently, by
(Jc,p f )(z) = z p −
(c + p)(p − q − α)δ(p, q)
z j+p .
n (j + p − q − α)δ(j + p, q)
(c + j + p)( j+p−q
)
p−q
Thus we complete the proof of Theorem 7.
(6.13)
34
M. K. Aouf
Using arguments similar to those in the proof of Theorem 7, we obtain the
following result.
Theorem 8. Let the function f (z) defined by (1.1) be in the class Sj (n, p, q, α).
Then
½
Γ(p + 1)
|Dzµ {(Jc,p f )(z)}| ≥
−
Γ(p + 1 − µ)
)
(c + p)Γ(j + p + 1)(p − q − α)δ(p, q)
j
p−µ
|z| |z|
j+p−q n
(c + j + p)Γ(j + p + 1 − µ)( p−q ) (j + p − q − α)δ(j + p, q)
(6.14)
(z ∈ U ; 0 ≤ α < p − q; 0 ≤ µ < 1; c > −p; p, j ∈ N ; q, n ∈ N0 ; p > q) and
½
Γ(p + 1)
+
|Dzµ {(Jc,p f )(z)}| ≤
Γ(p + 1 − µ)
)
(c + p)Γ(j + p + 1)(p − q − α)δ(p, q)
j
p−µ
|z| |z|
n (j + p − q − α)δ(j + p, q)
(c + j + p)Γ(j + p + 1 − µ)( j+p−q
)
p−q
(6.15)
(z ∈ U ; 0 ≤ α < p − q; 0 ≤ µ < 1; c > −p; p, j ∈ N ; q, n ∈ N0 ; p > q). Each of
the assertions (6.14), and (6.15) is sharp.
Remark 5. Putting n = 0 and n = 1 in Theorem 7 and Theorem 8, we obtain
the corresponding results for the classes Sj (p, q, α) and Cj (p, q, α), respectively.
REFERENCES
[1] M.K. Aouf, H.M. Srivastava, Some families of starlike functions with negative coefficients,
J. Math. Anal. Appl. 203 (1996), 762–790.
[2] S.D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969),
429–446.
[3] M.-P. Chen, H. Irmak, H.M. Srivastava, Some multivalent functions with negative coefficients, defined by using differential operator, PanAmer. Math. J. 6 (1996), 55–64.
[4] M.-P. Chen, H. Irmak, H.M. Srivastava, Some families of multivalently analytic functions
with negative coefficients, J. Math. Anal. Appl. 214 (1997), 674–690.
[5] P.L. Duren, Univalent Functions, Grundlehen der Mathematischen Wissenschaften 259,
Springer-Verlag, New York, Berlin, Heidelberg, Tokoyo, 1983.
[6] A.W. Goodman, Univalent Functions, Vols. I and II, Polygonal Publishing House, Washington, New Jersey, 1983.
[7] M.D. Hur, G.H. Oh, On certain class of analytic functions with negative coefficients, Pusan
Kyongnam Math. J. 5 (1989), 69–80.
[8] R.J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1969),
755–758.
[9] A.E. Livingston, On the radius of univalence of cerain analytic functions, Proc. Amer. Math.
Soc. 17 (1966), 352–357.
[10] S. Owa, On the distortion theorems. I, Kyungpook Math. J. 18 (1978), 55–59.
[11] S. Owa, On certain classes of p-valent functions with negative coefficients, Simon Stevin 59
(1985), 385–402.
[12] S. Owa, The quasi-Hadamard products of certain analytic functions, in: H. M. Srivastava,
S. Owa (Eds.) Current Topics in Analytic Function Theory, World Scientific Publishing
Company, Singapore, New Jersey, Lnodon, and Hong Kong, 1992, 234–251.
On certain multivalent functions defined by a differential operator
35
[13] G.S. Salagean, Subclasses of univalent functions, Lecture Notes in Math. (Springer-Verlag)
1013 (1983), 362–372.
[14] G.S. Salagean, H.M. Hossen, M.K. Aouf, On certain classes of p-valent functions with negative coefficients. II, Studia Univ. Babes-Bolyai 69 (2004), 77–85.
[15] A. Schild, H. Silverman, Convolultions of univalent functions with negative coefficients, Ann.
Univ. Mariae-Curie Sklodowska Sect. A 29 (1975), 99–107.
[16] H.M. Srivastava, M.K.Aouf, A certain fractional derivative operator and its applications to a
new class of analytic and multivalent functions with negative coefficients. I and II, J. Math.
Anal. Appl. 171 (1992), 1–13; ibid. 192 (1995), 673–688.
[17] H.M. Srivastava, S. Owa (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New
York, Chichester, Brisbane, Toronto, 1989.
[18] H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, Hong Kong, 1992.
[19] H.M. Srivastava, S. Owa, S.K. Chatterjea, A note on certain classes of starlike functions,
Rend. Sem. Mat. Univ. Padova 77 (1987), 115–124.
[20] R. Yamakawa, Certain subclasses of p-valently starlike functions with negative coefficients,
in: H.M. Srivastava, S. Owa (Eds.) Current Topics in Analytic Function Theory, World
Scientific Publishing Company, Singapore, New Jersey, London, Hong Kong, 1992, 393–402.
(received 11.10.2008)
Faculty of Science, Mansoura University, Mansoura 35516, Egypt
E-mail: mkaouf127@yahoo.com