General Mathematics Vol. 14, No. 4 (2006), 51–70
On a Subclass of p-valent Functions whose
coefficients related to Beta Function 1
H. Ö. Güney
and
S. Sümer Eker
In memoriam of Associate Professor Ph. D. Luciana Lupaş
Abstract
In the present paper, by making use of the Beta function, we
introduce a subclass A∗s (p, A, B, α ) of functions with negative and
missing coefficients which are analytic and p-valent in the unit disc
U = { z : | z | < 1 }. We give basic properties for functions belonging to the class A∗s (p, A, B, α ) and obtain numerous sharp results
in terms of the Beta function including coefficient estimate, distortion theorems, closure theorems, integral operators and linear combinations of several functions belonging to A∗s (p, A, B, α ) . We also
obtain radii of close-to-convexity, starlikeness and convexity for functions belonging to A∗s (p, A, B, α ) . Furthermore, convolution properties of several functions belonging to the class A∗s (p, A, B, α ) are
studied here. Various distortion inequalities for fractional calculus of
functions in the A∗s (p, A, B, α ) are also given.
1
Received 1 October, 2006
Accepted for publication (in revised form) 12 December, 2006
51
52
H. Ö. Güney and S. Sümer Eker
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words and phrases: p-valent, coefficient, distortion, closure,
close-to-convex , starlike, convex, convolution, fractional calculus, Beta
function.
1
Introduction
Let Ap (p ≥ 2) denote the class of functions of the form
(1.1)
f (z) = z p +
∞
X
an z n
n=p+1
which are analytic and p-valent in the unit disc U = {z : |z| < 1}. A function f (z) belonging to the class Ap is said to be in the class A∗p (p, A, B, α)
if and only if
¾
α
f (p−1) (z)
>
Re
p!z
p
for −1 ≤ B < A ≤ 1, −1 ≤ B < 0, 0 ≤ α < p and all z ∈ U .
½
In the other words, f (z) ∈ A∗p (p, A, B, α) if and only if there exists a
function w(z) satisfying w(0) = 0 and |w(z)| < 1 for z ∈ U such that
¶
µ
f (p−1) (z)
α 1 + Aw(z) α
(1.2)
= 1−
+
p!z
p 1 + Bw(z) p
The condition (1.2) is equivalent to
¯
¯
(p−1)
¯
¯
p f p!z (z) − p
¯
¯
¯
¯ < 1,
(1.3)
¯
f (p−1) (z) ¯
¯ [pB + (A − B)(p − α)] − pB p!z ¯
z ∈ U.
Let As denote the subclass of Ap consisting of functions analytic and
p-valent which can be expressed in the form
(1.4)
f (z) = z p −
∞
X
n=k
ap+n z p+n
On a Subclass of p-valent Functions whose coefficients...
where ap+n > 0,
53
k ≥ 2.
Let us define
A∗s (p, A, B, α) = A∗p (p, A, B, α) ∩ As .
M.K. Aouf and H.E.Darwish [3], S.M.Sarangi and V.J.Patil [4] have
studied certain classes of p-valent functions with negative and missing coefficients. M.K. Aouf and H.E.Darwish [2], S.L.Shukla and Dastrath [5]have
studied certain classes of analytic functions with negative coefficients. Also
the class Ap is studied by M.Nunokawa [1]. In this paper, while we were obtaining coefficient estimates, distortion theorem, covering theorem, integral
operators, convolution properties and radii of close-to-convexity, starlikeness and convexity for functions belonging to A∗s (p, A, B, α) , we used the
beta function. Further it is shown that this class is closed under “arithmetic mean and “convex linear combinations. Also distortion theorems for
fractional calculus are shown.
2
Coefficient Estimates
Theorem 1 Let the function f (z) defined by (1.4). Then f (z) ∈ A∗s (p, A, B, α)
if and only if
(2.1)
(1 − B)
∞
X
n=k
1
ap+n ≤ (A − B)(p − α).
(n + 1)B(p, n + 1)
where B denotes the beta function. The result is sharp.
Proof : Assume that the inequality (2.1) holds true and let |z| = 1.
Then we obtain
¯
¯ ¯
¯ (p−1)
(p−1)
¯
¯ ¯
¯ f
(z)
f
(z)
¯
¯ − ¯pB + (A − B)(p − α) − pB
¯p
−
p
¯ ¯
¯
p!z
p!z ¯
54
H. Ö. Güney and S. Sümer Eker
¯
¯
∞
¯
¯ X
1
¯
¯
ap+n z n ¯ −
= ¯−
¯
¯
(n + 1)B(p, n + 1)
n=k
¯
¯
∞
¯
¯
X
1
¯
¯
ap+n z n ¯
− ¯(A − B)(p − α) + B
¯
¯
(n + 1)B(p, n + 1)
n=k
≤ (1 − B)
∞
X
n=k
1
ap+n − (A − B)(p − α) ;
(n + 1)B(p, n + 1)
B<0
≤0
by hypothesis. Hence, by the maximum modulus theorem, we have f (z) ∈
A∗s (p, A, B, α) . To prove the converse, assume that
¯
¯
(p−1)
¯
¯
p f p!z (z) − p
¯
¯
¯=
¯
(p−1)
¯
¯
¯ pB + (A − B)(p − α) − pB f p!z (z) ¯
¯
¯
P
1
n
¯
¯
− ∞
n=k (n+1)B(p,n+1) ap+n z
¯
¯
P∞
=¯
¯ < 1.
1
n
¯ (A − B)(p − α) + B n=k (n+1)B(p,n+1) ap+n z ¯
Since Re(z) ≤ |z| for all z, we have
(
P∞
1
(2.2)
Re
(A − B)(p
n
n=k (n+1)B(p,n+1) ap+n z
P
1
n
− α) + B ∞
n=k (n+1)B(p,n+1) ap+n z
Choose values of z on the real axis so that
f (p−1) (z)
p!z
)
< 1.
is real. Upon clearing
the denominator in (2.2) and letting z → 1− through real values, we obtain
(2.3)
(1 − B)
∞
X
n=k
1
ap+n ≤ (A − B)(p − α)
(n + 1)B(p, n + 1)
which obviously is required assertion (2.1).
Finally, sharpness follows if we take
(2.4)
f (z) = z p −
(A − B)(p − α)(n + 1)B(p, n + 1) p+n
z (n ≥ k, k ≥ 2).
(1 − B)
On a Subclass of p-valent Functions whose coefficients...
55
Corollary 1 Let the function f (z) defined by (1.4). If f (z) ∈ A∗s (p, A, B, α),
then
(2.5)
ap+n ≤
(A − B)(p − α)(n + 1)B(p, n + 1)
.
(1 − B)
The equality in (2.5) is attained for the function f (z) given by (2.4).
3
Distortion Properties
Theorem 2 If the function f (z) defined by (1.4) in the A∗s (p, A, B, α) then
for |z| = r < 1
rp −
(3.1)
(A − B)(p − α)(k + 1)B(p, k + 1) p+k
r
≤ |f (z)| ≤ rp +
(1 − B)
+
(A − B)(p − α)(k + 1)B(p, k + 1) p+k
r
(1 − B)
and
prp−1 −
(3.2)
(A − B)(p − α)k(k + 1)B(p, k) p+k−1
r
≤ |f ′ (z)| ≤ prp−1 +
(1 − B)
+
(A − B)(p − α)k(k + 1)B(p, k) p+k−1
r
(1 − B)
All the inequalities are sharp.
P
p+n
Proof : Let f (z) = z p − ∞
. From Theorem 1, we have
n=k ap+n z
∞
(1 − B)
X
1
ap+n ≤
(k + 1)B(p, k + 1) n=k
56
H. Ö. Güney and S. Sümer Eker
(3.3)
≤ (1 − B)
∞
X
1
ap+n ≤ (A − B)(p − α)
(n + 1)B(p, n + 1)
n=k
which immediately yields for n ≥ k
∞
X
(3.4)
ap+n ≤
n=k
(A − B)(p − α)(k + 1)B(p, k + 1)
(1 − B)
and
(3.5)
∞
X
(p + n)ap+n ≤
n=k
(A − B)(p − α)k(k + 1)B(p, k)
.
(1 − B)
Consequently, for |z| = r < 1, we obtain
|f (z)| ≤ |z|p +
∞
X
|ap+n ||z|p+n ≤ rp + rp+k
n=k
+
∞
X
ap+n ≤ rp +
n=k
(A − B)(p − α)(k + 1)B(p, k + 1) p+k
r
(1 − B)
and
p
|f (z)| ≥ |z| −
∞
X
p+n
|ap+n ||z|
p
≥r −r
p+k
n=k
∞
X
ap+n ≥ rp −
n=k
(A − B)(p − α)(k + 1)B(p, k + 1) p+k
r
(1 − B)
which prove that the assertion (3.1) of Theorem 2.
−
Furthermore, for |z| = r < 1 and (3.5), we have
′
p−1
|f (z)| ≤ p|z|
∞
∞
X
X
p+n−1
p−1
p+k−1
(p + n)ap+n
(p + n)|ap+n ||z|
≤ pr
+r
+
n=k
n=k
≤ prp−1 +
(A − B)(p − α)k(k + 1)B(p, k) p+k−1
r
(1 − B)
and
|f ′ (z)| ≥ p|z|p−1 −
∞
X
n=k
(p + n)|ap+n ||z|p+n−1 ≥ prp−1 − rp+k−1
∞
X
n=k
(p + n)ap+n
57
On a Subclass of p-valent Functions whose coefficients...
(A − B)(p − α)k(k + 1)B(p, k) p+k−1
r
(1 − B)
which prove that the assertion (3.2) of Theorem 2.
≥ prp−1 −
The bounds in (3.1) and (3.2) are attained for the function f (z) given
by
(3.6)
f (z) = z p −
(A − B)(p − α)(k + 1)B(p, k + 1) p+k
z
(1 − B)
; z = ∓r.
Letting r → 1− in the left hand side of (3.1), we have the following:
Corollary 2 If f (z) ∈ A∗s (p, A, B, α), then the disc |z| < 1 is mapped
by f (z) onto a domain that contains the disc
|w| <
(1 − B) − (k + 1)B(p, k + 1)(A − B)(p − α)
.
(1 − B)
The result is sharp with the extremal function f (z) being given by (3.6).
Putting α = 0 in Theorem 2 and Corollary 2, we get
Corollary 3 If the function f (z) defined by (1.4) in the A∗s (p, A, B, 0)
then for |z| = r
rp −
(A − B)p(k + 1)B(p, k + 1) p+k
r
≤ |f (z)| ≤ rp +
(1 − B)
+
(A − B)p(k + 1)B(p, k + 1) p+k
r
(1 − B)
and
prp−1 −
(A − B)pk(k + 1)B(p, k) p+k−1
r
≤ |f ′ (z)| ≤ prp−1 +
(1 − B)
(A − B)pk(k + 1)B(p, k) p+k−1
r
.
(1 − B)
The result is sharp with the extremal function
+
(3.7)
f (z) = z p −
(A − B)p(k + 1)B(p, k + 1) p+k
z
(1 − B)
; z = ∓r.
58
H. Ö. Güney and S. Sümer Eker
Corollary 4 If f (z) ∈ A∗s (p, A, B, α), then the disc |z| < 1 is mapped
by f (z) onto a domain that contains the disc
|w| <
(1 − B) − p(k + 1)B(p, k + 1)(A − B)
.
(1 − B)
The result is sharp with the extremal function f (z) being given by (3.7).
4
Radii Of Close-To-Convexity, Starlikeness
And Convexity
Theorem 3 : Let the function f (z) defined by (1.4) in the class A∗s (p, A, B, α).
Then f (z) is p-valent close-to-convex of order δ
(0 ≤ δ < p) in |z| < R1 ,
where
(4.1)
R1 = inf
n≥2
(·
(1 − B)
(A − B)(p − α)(n + 1)B(p, n + 1)
µ
p−δ
p+n
¶¸ n1 )
Theorem 4 : Let the function f (z) defined by (1.4) in the class A∗s (p, A, B, α).
Then f (z) is p-valent starlike of order δ (0 ≤ δ < p) in |z| < R2 , where
(·
µ
¶¸ n1 )
p−δ
(1 − B)
.
(4.2) R2 = inf
n≥2
(A − B)(p − α)(n + 1)B(p, n + 1) p + n − δ
Theorem 5 : Let the function f (z) defined by (1.4) in the class A∗s (p, A, B, α).
Then f (z) is p-valent convex function of order δ
(0 ≤ δ < p) in |z| < R3 ,
where
(4.3)
R3 = inf
n≥2
(·
(1 − B)
(A − B)(p − α)(n + 1)B(p, n + 1)
µ
p(p − δ)
(p + n)(p + n − δ)
¶¸ n1 )
.
59
On a Subclass of p-valent Functions whose coefficients...
The results in Theorem 3,4,5 are sharp with the extremal function f (z)
given by (2.4). Furthermore, taking δ = 0 in Theorem 3,4,5, we obtain
radius of close-to-convexity , starlikeness and convexity, respectively.
5
Integral Operators
Theorem 6
Let c be a real number such that c > −p.
If f (z) ∈
A∗s (p, A, B, α), then the function F (z) defined by
Z
c + p z c−1
t f (t)dt
(5.1)
F (z) =
zc 0
also belongs to A∗s (p, A, B, α).
P
p+n
Proof : Let f (z) = z p − ∞
. Then from representation of
n=k ap+n z
³
´
P
c+p
p+n
b
z
where
b
=
F (z), it follows that F (z) = z p − ∞
ap+n .
p+n
n=k p+n
c+p+n
Therefore using Theorem 1 for the coefficients of F (z) we have
(1 − B)
∞
X
n=k
= (1 − B)
∞
X
n=k
since
c+p
c+p+n
1
bp+n =
(n + 1)B(p, n + 1)
1
(n + 1)B(p, n + 1)
µ
c+p
c+p+n
¶
ap+n ≤ (A − B)(p − α)
< 1 and f (z) ∈ A∗s (p, A, B, α) . Hence F (z) ∈ A∗s (p, A, B, α).
Theorem 7 Let c be a real number such that c > −p . If F (z) ∈
A∗s (p, A, B, α), then the function f (z) defined by (5.1) is p-valent in |z| < R∗ ,
where
(5.2)
R∗ = inf
n≥2
(µ
c+p
c+p+n
The result is sharp.
¶·
(1 − B)
(A − B)(p − α)(n + 1)B(p, n + 1)
µ
p
p+n
¶¸ n1 )
.
60
H. Ö. Güney and S. Sümer Eker
Proof : Let F (z) = z p −
P∞
n=k
ap+n z p+n . It follows from (5.1)
¶
∞ µ
X
c+p+n
z 1−c d c
p
[z F (z)] = z −
ap+n z p+n .
f (z) =
c + p dz
c
+
p
n=k
¯
¯ ′
¯
¯ (z)
− p¯ < p
In order to obtain the required result it sufficient to show that ¯ fzp−1
for |z| < R∗ where R∗ is defined by (5.2). Now
¯ ′
¯ X
µ
¶
∞
¯ f (z)
¯
c+p+n
¯
¯
(p + n)
ap+n |z|n .
¯ z p−1 − p¯ ≤
c
+
p
n=k
¯
¯ ′
¯
¯ f (z)
Thus ¯ zp−1 − p¯ < p if
∞
X
(5.3)
(p + n)
n=k
µ
c+p+n
c+p
¶
ap+n |z|n < p.
But Theorem 1 confirms that
¸
·
∞
X
(1 − B)
ap+n ≤ p.
p
(A
−
B)(p
−
α)(n
+
1)
B(p,
n
+
1)
n=k
Hence (5.3) will be satisfied if
¶
·
¸
µ
(1 − B)
c+p+n
n
ap+n |z| ≤ p
ap+n
(p + n)
c+p
(A − B)(p − α)(n + 1)B(p, n + 1)
or if
|z| ≤
½µ
c+p
c+p+n
¶
(1 − B)
(A − B)(p − α)(n + 1)B(p, n + 1)
µ
p
p+n
¶¾ n1
Therefore f (z) is p-valent in |z| < R∗ .
Sharpness follows if we take
¶
µ
c + p + n (A − B)(p − α)(n + 1)B(p, n + 1) p+n
p
z .
f (z) = z −
c+p
(1 − B)
.
61
On a Subclass of p-valent Functions whose coefficients...
6
Closure Properties
In this section we show that the class A∗s (p, A, B, α) is closed under “arithmetic mean and “convex linear combinations.
P
p+n
Theorem 8 Let fj (z) = z p − ∞
j = 1, 2, ... and h(z) =
n=k ap+n,j z
P
P
P
∞
∞
z p − n=k bp+n z p+n , where bp+n = j=1 λj ap+n,j , λj > 0 and ∞
j=1 λj = 1.
If fj (z) ∈ A∗s (p, A, B, α) for each j = 1, 2, ..., then h(z) ∈ A∗s (p, A, B, α) .
Proof : If fj (z) ∈ A∗s (p, A, B, α), then we have from Theorem 1 that
(1 − B)
∞
X
n=k
Therefore
1
ap+n,j ≤ (A − B)(p − α) j = 1, 2, ....
(n + 1)B(p, n + 1)
∞
X
1
bp+n =
(n
+
1)B(p,
n
+
1)
n=k
"
!#
̰
∞
X
X
1
= (1 − B)
≤ (A − B)(p − α).
λj ap+n,j
(n + 1)B(p, n + 1) j=1
n=k
(1 − B)
Hence, by Theorem 1, h(z) ∈ A∗s (p, A, B, α).
Theorem 9 The class A∗s (p, A, B, α) is closed under convex linear combinations.
Proof : Let f (z) = z p −
P∞
n=k
ap+n z p+n and g(z) = z p −
P∞
n=k bp+n z
p+n
(k ≥ p, ap+n > 0, bp+n > 0), be any two functions of the class A∗s (p, A, B, α).
For λ (0 ≤ λ ≤ 1), it is sufficient to show that h(z) = (1 − λ) f (z)+λg(z), z ∈
U is also a function of A∗s (p, A, B, α). Now,
p
h(z) = z −
∞
X
[(1 − λ) ap+n + λbp+n ] z p+n .
n=k
Applying Theorem 1 to f, g ∈ A∗s (p, A, B, α), we have
(1 − B)
∞
X
n=k
1
[(1 − λ) ap+n + λbp+n ] =
(n + 1)B(p, n + 1)
62
H. Ö. Güney and S. Sümer Eker
= (1 − λ)(1 − B)
∞
X
n=k
+λ(1 − B)
∞
X
n=k
1
ap+n +
(n + 1)B(p, n + 1)
1
bp+n ≤
(n + 1)B(p, n + 1)
≤ (1 − λ)(A − B)(p − α) + λ(A − B)(p − α) = (A − B)(p − α).
Then h(z) ∈ A∗s (p, A, B, α).
Theorem 10 Let fp (z) = z p and fp+n (z) = z p − (A−B)(p−α)(n+1)B(p,n+1)
z p+n
(1−B)
(n ≥ k, k ≥ 2). Then f (z) ∈ A∗s (p, A, B, α) if and only if it can be expressed
in the form
f (z) = λp fp (z) +
∞
X
λn fp+n (z),
z∈U
n=k
where λn ≥ 0 and λp = 1 −
P∞
n=k
λn .
Proof : Let us assume that
f (z) = λp fp (z) +
∞
X
λn fp+n (z)
n=k
∞
X
∞
X
¾
½
(A − B)(p − α)(n + 1)B(p, n + 1) p+n
p
z
λn z −
λn ]z +
= [1 −
(1 − B)
n=k
n=k
p
p
=z −
∞
X
(A − B)(p − α)(n + 1)B(p, n + 1)
n=k
(1 − B)
λn z p+n .
Then from Theorem 1 we have
(1 − B)
∞
X
n=k
1
(A − B)(p − α)(n + 1)B(p, n + 1)
λn
(n + 1)B(p, n + 1)
(1 − B)
= (A − B)(p − α)
∞
X
n=k
Hence f (z) ∈ A∗s (p, A, B, α).
λn ≤ (A − B)(p − α).
63
On a Subclass of p-valent Functions whose coefficients...
Conversely, let f (z) ∈ A∗s (p, A, B, α). It follows from Corollary 1 that
ap+n ≤
(A − B)(p − α)(n + 1)B(p, n + 1)
.
(1 − B)
Setting
(1 − B)
ap+n ,
(A − B)(p − α)(n + 1)B(p, n + 1)
P
and λp = 1 − ∞
n=k λn , we have
λn =
p
f (z) = z −
∞
X
n = k, k + 1, . . . , k ≥ 2
ap+n z p+n
n=k
∞
X
∞
X
∞
X
(A − B)(p − α)(n + 1)B(p, n + 1) p+n
z
(1 − B)
n=k
n=k
n=k
½
¾
∞
∞
X
X
(A − B)(p − α)(n + 1)B(p, n + 1) p+n
p
p
= [1 −
λn ]z +
λn z −
z
(1
−
B)
n=k
n=k
p
=z −
p
λn z +
p
λn z −
λn
= λp fp (z) +
∞
X
λn fp+n (z).
n=k
This completes the proof of Theorem 10.
7
Convolution Properties
P
P
p+n
p+n
Theorem 11 If f1 (z) = z p − ∞
and f2 (z) = z p − ∞
n=k ap+n z
n=k bp+n z
P
p+n
are in the class A∗s (p, A, B, α) then (f1 ∗ f2 )(z) = z p − ∞
n=k ap+n bp+n z
is in the class A∗s (p, A, B, ψ), where
ψ =p−
3(A − B)(p − α)2 B(p, 3)
.
(1 − B)
The result is best possible for f1 (z) and f2 (z) given by
3(A − B)(p − α)2 B(p, 3) p+2
fj (z) = z −
z
(1 − B)
p
j = 1, 2.
64
H. Ö. Güney and S. Sümer Eker
Proof : In order to prove our theorem, we have to find the largest ψ =
ψ(p, A, B, α) such that
∞
X
n=k
(1 − B)
ap+n bp+n ≤ 1
(A − B)(p − ψ)(n + 1)B(p, n + 1)
for f1 (z) and f2 (z) in the class A∗s (p, A, B, α). Since f1 (z) and f2 (z) are in
the class A∗s (p, A, B, α), in view of Theorem 1,
∞
X
(1 − B)
ap+n ≤ 1
(A − B)(p − α)(n + 1)B(p, n + 1)
∞
X
(1 − B)
bp+n ≤ 1.
(A − B)(p − α)(n + 1)B(p, n + 1)
n=k
and
n=k
Therefore, by the Cauchy-Schwarz inequality, we obtain
(7.1)
∞
X
n=k
p
(1 − B)
ap+n bp+n ≤ 1
(A − B)(p − α)(n + 1)B(p, n + 1)
Thus it is sufficient to show that
(1 − B)
ap+n bp+n ≤
(A − B)(p − ψ)(n + 1)B(p, n + 1)
≤
or
Note that
p
(1 − B)
ap+n bp+n
(A − B)(p − α)(n + 1)B(p, n + 1)
p
ap+n bp+n ≤
p−ψ
.
p−α
p
(A − B)(p − α)(n + 1)B(p, n + 1)
.
ap+n bp+n ≤
(1 − B)
Hence, we need only to prove that
(7.2)
(A − B)(p − α)(n + 1)B(p, n + 1)
p−ψ
≤
(1 − B)
p−α
On a Subclass of p-valent Functions whose coefficients...
65
or, equivalently, that
ψ ≤p−
(A − B)(p − α)2 (n + 1)B(p, n + 1)
.
(1 − B)
Defining the function Ξ(n) by
(7.3)
Ξ(n) = p −
(A − B)(p − α)2 (n + 1)B(p, n + 1)
,
(1 − B)
we see that Ξ(n) is an increasing function of n. Therefore, letting n = 2 in
(7.3), we obtain
ψ ≤ Ξ(2) = p −
3(A − B)(p − α)2 B(p, 3)
(1 − B)
which completes the assertion of theorem.
8
Definitions And Applications Of The Fractional Calculus
In this section, we shall prove several distortion theorems in terms of the
beta function for functions to general class A∗s (p, A, B, α). Each of these
theorems would involve certain operators of fractional calculus we find it to
be convenient to recall here the following definition which were used recently
by Owa [6] (and more recently, by Owa & Srivastava [7], and Srivastava &
Owa [8], ; see also Srivastava et all. [9] )
Definition 1 The fractional integral of order λ is defined, for a function
f (z), by
(8.1)
Dz−λ f (z)
1
=
Γ(λ)
Z
0
z
f (ζ)
dζ
(z − ζ)1−λ
(λ > 0)
66
H. Ö. Güney and S. Sümer Eker
where f (z) is an analytic function in a simply connected region of the z
-plane containing the origin, and the multiplicity of (z − ζ)λ−1 is removed
by requiring log(z − ζ) to be real when z − ζ > 0.
Definition 2
The fractional derivative of order λ is defined, for a
function f (z), by
(8.2)
Dzλ f (z)
d
1
=
Γ(1 − λ) dz
Z
z
0
f (ζ)
dζ
(z − ζ)λ
(0 ≤ λ < 1)
where f (z) is constrained, and the multiplicity of (z − ζ)−λ is removed, as
in Definition 1.
Definition 3
Under the hypotheses of Definition 2, the fractional
derivative of order (n + λ) is defined by
(8.3)
Dzn+λ f (z) =
dn λ
D f (z)
dz n z
where 0 ≤ λ < 1 and n ∈ N0 = N ∪ {0}.
From Definition 2, we have
Dz0 f (z) = f (z)
(8.4)
which, in view of Definition 3 yields,
(8.5)
Dzn+0 f (z) =
dn 0
Dz f (z) = f n (z).
n
dz
Thus, it follows from (8.4) and (8.5) that
lim Dz−λ f (z) = f (z)
λ→0
and
lim Dz1−λ f (z) = f ′ (z).
λ→0
67
On a Subclass of p-valent Functions whose coefficients...
Theorem 12 Let the function f (z) defined by (1.4) be in the class
A∗s (p, A, B, α). Then for z ∈ U and λ > 0,
½
¯ −λ
¯
¯Dz f (z)¯ ≥
Γ(p + 1)
|z|p+λ ·
Γ(λ + p + 1)
(A − B)(p − α)(k + 1)B(p + λ + 1, k)B(p, k + 1) k
|z|
· 1−
B(p + 1, k)(1 − B)
and
¯
¯ −λ
¯Dz f (z)¯ ≤
¾
Γ(p + 1)
|z|p+λ ·
Γ(λ + p + 1)
¾
½
(A − B)(p − α)(k + 1)B(p + λ + 1, k)B(p, k + 1) k
|z| .
· 1+
B(p + 1, k)(1 − B)
The result is sharp.
Proof : Let
Γ(p + 1 + λ) −λ −λ
z Dz f (z) =
Γ(p + 1)
F (z) =
p
=z −
∞
X
B(p + λ + 1, n)
n=k
p
=z −
B(p + 1, n)
∞
X
ap+n z p+n
ψ(n)ap+n z p+n
n=k
where
ψ(n) =
B(p + λ + 1, n)
B(p + 1, n)
(n ≥ k).
Since
0 < ψ(n) ≤ ψ(k) =
B(p + λ + 1, k)
,
B(p + 1, k)
we have, with the help of (3.4).
|F (z)| ≥ |z|p − ψ(k) |z|p+k
∞
X
n=k
ap+n ≥
68
H. Ö. Güney and S. Sümer Eker
≥ |z|p −
B(p + λ + 1, k)(A − B)(p − α)(k + 1)B(p, k + 1) p+k
|z|
B(p + 1, k)(1 − B)
and
p
|F (z)| ≤ |z| + ψ(k) |z|
p+k
∞
X
ap+n ≤
n=k
B(p + λ + 1, k)(A − B)(p − α)(k + 1)B(p, k + 1) p+k
|z|
B(p + 1, k)(1 − B)
which prove the inequalities of Theorem 12. Further equalities are attained
≤ |z|p +
for the function
f (z) = z p −
(8.6)
(A − B)(p − α)(k + 1)B(p, k + 1) p+k
z
(1 − B)
Theorem 13 Let the function f (z) defined by (1.4) be in the class
A∗s (p, A, B, α). Then for 0 ≤ λ < 1,
½
¯ λ
¯
¯Dz f (z)¯ ≥
Γ(p + 1)
|z|p−λ ·
Γ(p − λ + 1)
(A − B)(p − α)k(k + 1)B(p − λ + 1, k)B(p, 1) k
· 1−
|z|
(1 − B)
and
¯
¯ λ
¯Dz f (z)¯ ≤
¾
Γ(p + 1)
|z|p−λ ·
Γ(p − λ + 1)
¾
½
(A − B)(p − α)k(k + 1)B(p − λ + 1, k)B(p, 1) k
|z| .
· 1+
(1 − B)
The result is sharp for the function f (z) given by (8.6).
The proof of Theorem 13 is obtained by using the same technique as
in the proof of Theorem 12. Setting λ = 0 in Theorem 13, we obtain the
following Corollary:
Corollary 8 If f (z) ∈ A∗s (p, A, B, α), then
½
¾
(A − B)(p − α)k(k + 1)B(p + 1, k) k
p
|f (z)| ≥ |z| 1 −
|z|
(1 − B)p
69
On a Subclass of p-valent Functions whose coefficients...
and
|f (z)| ≤ |z|
p
½
(A − B)(p − α)k(k + 1)B(p + 1, k) k
|z|
1+
(1 − B)p
¾
for k ≥ 2, p ∈ N and for all z ∈ U.
References
[1] M. NUNOKAWA ; A note on multivalent Functions, Tsukuba J.
Math.Vol.13 No.2 (1989),453-455.
[2] M.K. AOUF & H.E. DARWISH ; Basic Properties and characterizations of a certain class of Analytic func. with negative coefficients II,
Utilitas Mathematica 46 (1994),167-177.
[3] M.K. AOUF & H.E. DARWISH ; On p-valent func. with negative and
missing coefficients II, Commun. Fac. Sci. Univ. Ank. Series A, V. 44
(1995) pp. 1-14.
[4] S.M.SARANGI & V.J.PATIL ; On multivalent functions with negative
and missing coefficients, J.Math.Res.Exposition 10 (1990), no.3, 341348.
[5] S.L.SHUKLA & DASTRATH ; On certain classes of multivalent functions with negative coefficients, Pure and Appl. Math.Sci.20 (1984),
1-2, 63-72.
[6] S.OWA, On the distortion theorems, I.Kyunpook Math.J. 18 (1978),
53-59.
70
H. Ö. Güney and S. Sümer Eker
[7] S.OWA & H.M.SRIVASTAVA, Univalent and Starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987), 1057-1077.
[8] H.M.SRIVASTAVA & S.OWA , Some Characterization and distortion
theorems involving fractional calculus, linear operators and certain subclasses of analytic functions, Nagoya Math. J.106 (1987), 1-28.
[9] H.M.SRIVASTAVA & M.SAIGO & S.OWA, A class of distortion theorems involving certain operators of fractional calculus,
J.Math.Anal.Appl.131 (1988), 412-420.
Dicle University
Faculty of Science and Letters
Department of Mathematics
21280 - Diyarbakir, Turkey
E-mail address: ozlemg@dicle.edu.tr
E-mail address: sevtaps@dicle.edu.tr