General Mathematics Vol. 14, No. 2 (2006), 3–30
On New Classes of Sălăgean-type p-valent
Functions with Negative and Missing
Coefficients 1
H. Özlem Güney
Abstract
We define and investigate new classes of Sălăgean-type p-valent
functions with negative and missing coefficients. We obtain coefficient estimates, distortion bounds, integral operators of functions
belonging to these classes, extreme points, convex combinations and
radius of convexity for these classes of p-valent functions. Furthermore, we give modified Hadamard product of several functions and
some distortion theorems for fractional calculus of p-valent functions
with negative and missing coefficients belonging to these generalized
classes.
2000 Mathematics Subject Classification: 30C45
Key words and phrases: p-valent, coefficient, distortion, closure,
starlike, convex, extreme, fractional calculus, integral operator, Sălăgean
operator
1
Received April 4, 2006
Accepted for publication (in revised form) June 2 , 2006
3
4
1
H. Özlem Güney
Introduction
Let A be class of functions f (z)of the form
f (z) = z +
∞
X
ak z k
k=2
which are analytic in the unit disk U = {z : |z| < 1} .
For f (z) belong to A, Sălăgean [ 9 ] has introduced the following operator
called the Sălăgean operator:
D0 f (z) = f (z),
D1 f (z) = Df (z) = zf 0 (z),
Dn f (z) = D(Dn−1 f (z)), n ∈ N = {1, 2, · · ·} .
We note that
Dn f (z) = z +
∞
X
k n ak z k ; n ∈ N0 = {0} ∪ N.
k=2
Let Sp (p ≥ 1)denote the class of functions of the form
p
f (z) = z +
∞
X
ap+n z p+n
n=1
that are holomorphic and p-valent in the unit disc U.
Also, let Tp denote the subclass of Sp (p ≥ 1) consisting of functions that
can be expressed in the form,
p
f (z) = z −
∞
X
n=k
ap+n z p+n ; ap+n ≥ 0 , k ≥ 2.
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 5
We can write the following equalities for the functions f (z) belonging to
the class Tp :
D0 f (z) = f (z)
∞
X
z 0
(p + n)
1
p
D f (z) = Df (z) = f (z) = z −
ap+n z p+n ,
p
p
n=k
..
.
Dλ f (z) = D(Dλ−1 f (z)) = z p −
∞
X
(p + n)λ
n=k
pλ
ap+n z p+n
; λ ∈ N0 = {0}∪N.
Tp∗ (α, A, B, k, β, λ)
A function f (z) ∈ Tp is in
if and only if
¯
¯
λ
0
λ
¯
¯
z(D
f
(z))
−
pD
f
(z)
¯
¯
¯ [(A − B)(p − α) + pB]Dλ f (z) − Bz(Dλ f (z))0 ¯ < β,
for
λ ∈ N0 , 0 ≤ α < p, 0 < β ≤ 1, −1 ≤ B < A ≤ 1 , −1 ≤ B < 0, n ≥ k ≥ 2
and z ∈ U.
Further f said to belong to the class Cp (α, A, B, k, β, λ) if and only if
zf 0
p
∈ Tp∗ (α, A, B, k, β, λ).
We note that by specializing the parameters α, A, B, k, β and λ, we ob-
tain the following interesting subclasses including those that were studied
by various earlier authors.
(i) In [ 1 ], Ahuja and Jain defined T1∗ (α, −1, 1, q, β, 0), q ≥ 1, the subclass of starlike functions of order α and type-β and also defined
C1 (α, −1, 1, q, β, 0) , q ≥ 1, the class of convex functions of order α and
type-β.
(ii) The subclass T1∗ (α, A, B, q, β, 0), q ≥ 1 has been studied by Aouf
[6].
6
H. Özlem Güney
(iii) The subclasses T1∗ (α, −1, 1, 1, β, 0) and C1 (α, −1, 1, 1, β, 0), espe-
cially the class T1∗ (α, (2α − 1)β, β, 1, 1, 0), have been studied by Gupta and
Jain [2].
(iv)
The classes T ∗ (α)
=
T1∗ (α, −1, 1, 1, 1, 0) and C(α)
=
= C1 (α, −1, 1, 1, 1, 0) which are subclasses starlike of order α and convex
of of order α, respectively, have been studied by Silverman [3]. Evidently
T ∗ (0) = T1∗ (0, −1, 1, 1, 1, 0).
(v) The subclass Tp∗ (0, A, B, 1, 1, 0) = Tp∗ (A, B) was defined by Goel
and Sohi [7].
(vi) The subclass Tp∗ (0, A, B, k, 1, 0) = Pk (p, A, B, 0) has been investigated by Sarangi and Patel [5].
(vii) The subclass Tp∗ (α, A, B, k, 1, 0) = Pk (p, A, B, α) was defined by
Aouf and Darwish [4].
Finally, we generalized the results of Aouf and Darwish [4] and investigated the class Cp (α, A, B, k, β, λ) which is generalization of the results of
Ahuja and Jain [1] ,Gupta and Jain [2] and Silverman [3]. Furthermore, we
give modified Hadamard product of several functions and some distortion
theorems for fractional calculus of analytic functions with negative and missing coefficients belonging to a certain generalized classes Tp∗ (α, A, B, k, β, λ)
and Cp (α, A, B, k, β, λ).
2
Coefficients Estimates
Theorem 2.1. A function
p
f (z) = z −
∞
X
n=k
ap+n z p+n ; ap+n ≥ 0,
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 7
belongs to Tp∗ (α, A, B, k, β, λ) if and only if
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ ap+n ≤ (A − B)pλ β(p − α)
(1)
n=k
The result is sharp.
Proof. Assume that the inequality (1) holds true and let |z| = 1. Then
we obtain
¯
¯
¯
¯
¯z(Dλ f (z))0 − pDλ f (z)¯−β ¯[(A − B)(p − α) + pB]Dλ f (z) − Bz(Dλ f (z))0 ¯ =
¯
¯
∞
¯ X
¯
¯
λ+1
λ
p+n ¯
[(p
+
n)
−
p(p
+
n)
]a
z
−
¯
¯−
p+n
¯
¯
n=k
¯
∞
¯
X
¯
λ p
−β ¯(A − B)(p − α)p z −
[(A − B)(p − α)(p + n)λ +
¯
n=k
¯
+pB(p + n)λ − B(p + n)λ+1 ]ap+n z p+n ¯ ≤
∞
X
[(p + n)λ+1 − p(p + n)λ + (A − B)β(p − α)(p + n)λ − Bβn(p + n)λ ]ap+n −
≤
n=k
−(A − B)β(p − α)pλ =
=
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ ap+n − (A − B)pλ β(p − α) ≤ 0
n=k
by hypothesis. Hence, by the maximum modulus theorem, we have
f ∈ Tp∗ (α, A, B, k, β, λ).
Conversely, assume that
¯
¯
z(Dλ f (z))0 − pDλ f (z)
¯
¯ [(A − B)(p − α) + pB]Dλ f (z) − Bz(Dλ f (z))0
¯
¯
¯
¯
8
H. Özlem Güney
¯
∞
P
λ+1
λ
¯
¯
−
[ (p+n)
− (p+n)
]ap+n z p+n
pλ
pλ−1
¯
n=k
= ¯¯
∞
λ
¯ (A − B)(p − α)z p − P [(A − B)(p − α) − Bn] (p+n)
ap+n z p+n
¯
pλ
n=k
Since |Re(z)| ≤| z | for all z, we have

∞
P
λ
λ+1


− (p+n)
]ap+n z p+n
[ (p+n)

pλ
pλ−1
Re
n=k
¯
¯
¯
¯
¯ < β.
¯
¯
¯




< β.
∞
P
λ

p+n 


[(A − B)(p − α) − Bn] (p+n)
a
z
 (A − B)(p − α)z p −

p+n
pλ
n=k
(2)
Choose values of z on the real axis and letting z → 1− through real
values, we obtain
∞
X
[(p + n)λ+1 − p(p + n)λ ]ap+n ≤ (A − B)β(p − α)pλ −
n=k
∞
X
[(A − B)β(p − α) − Bβn](p + n)λ ap+n
−
n=k
which obviously is required assertion (1).
Finally, the function
(3)
p
f (z) = z −
∞
X
n=k
(A − B)β(p − α)pλ
z p+n .
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
is an extremal function.
Corollary 2.1. If f ∈ Tp∗ (α, A, B, k, β, λ), then
ap+n ≤
(A − B)β(p − α)pλ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
with equality only for functions of the form
f (z) = z p −
(A − B)β(p − α)pλ
z p+n .
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 9
Theorem 2.2. A function
p
f (z) = z −
∞
X
an+p z p+n
n=k
belongs to Cp (α, A, B, k, β, λ) if and only if
∞
X
(4)
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ+1 ap+n ≤
n=k
≤ (A − B)pλ+1 β(p − α).
The result is sharp.
Proof.
f ∈ Cp (α, A, B, k, β, λ) is equivalent
Since
∈ Tp∗ (α, A, B, k, β, λ).
¶
∞ µ
X
p+n
zf 0 (z)
p
=z −
an+p z p+n ,
p
p
n=k
we may replace ap+n by
3
zf 0
p
p+n
ap+n
p
in Theorem 2.1.
Distortion Properties
Theorem 3.1. If f ∈ Tp∗ (α, A, B, k, β, λ), then for | z |= r < 1
(5)
rp −
(A − B)β(p − α)pλ
rp+k ≤ |f (z)| ≤
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
≤ rp +
(A − B)β(p − α)pλ
rp+k
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
and
(6) prp−1 −
(A − B)β(p − α)pλ
rp+k−1 ≤ |f 0 (z)| ≤
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ−1
≤ prp−1 +
(A − B)β(p − α)pλ
rp+k−1 .
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ−1
10
H. Özlem Güney
All the inequalities are sharp.
∞
P
Proof. Let f (z) = z p −
ap+n z p+n , ap+n ≥ 0.
n=k
From Theorem 2.1, we have
λ
[(1 − Bβ)k + (A − B)β(p − α)](p + k)
∞
X
ap+n ≤
n=k
≤
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ ap+n ≤ (A − B)β(p − α)pλ
n=k
which implies that
∞
X
(7)
ap+n ≤
n=k
(A − B)β(p − α)pλ
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
and
∞
X
(8)
(p + n)ap+n ≤
n=k
(A − B)β(p − α)pλ
.
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ−1
Consequently, for | z |= r < 1, we obtain
p
| f (z) |≤| z | +
∞
X
| ap+n || z |
p+n
p
≤r +r
n=k
≤ rp +
p+k
∞
X
ap+n ≤
n=k
(A − B)β(p − α)pλ
rp+k
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
and
p
| f (z) |≥| z | −
∞
X
| ap+n || z |p+n ≥
n=k
p
≥r −r
p+1
∞
X
ap+n ≥
n=k
≥ rp −
(A − B)β(p − α)pλ
rp+k
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 11
which prove that the assertion (5) of Theorem 3.1.
Furthermore, for | z |= r < 1 and (8) , we have
0
| f (z) |≤ p | z |
p−1
+
∞
X
(p + n) | ap+n || z |p+n−1 ≤
n=k
≤ pr
p−1
+r
p+k−1
∞
X
(p + n)ap+n ≤
n=k
≤ prp−1 +
(A − B)β(p − α)pλ
rp+k−1
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ−1
and
0
| f (z) |≥ p | z |
p−1
∞
X
−
(p + n) | ap+n || z |p+n−1 ≥
n=k
≥ pr
p−1
∞
X
(p + n)ap+n ≥
−r
p
n=k
≥ prp−1 −
(A − B)β(p − α)pλ
rp+k−1
λ−1
[(1 − Bβ)k + (A − B)β(p − α)](p + k)
which prove that the assertion (6) of Theorem 3.1.
The bounds in (5) and (6) are attained for the function f given by
(9)
f (z) = z p −
(A − B)β(p − α)pλ
z p+k .
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
Letting r → 1− in the left hand side of (5), we have the following:
Corollary 3.1.
Let f ∈ Tp∗ (α, A, B, k, β, λ). Then the unit disk U is
mapped by f onto a domain that contains the disk
(p + k)λ (1 − Bβ)k + (A − B)β(p − α)[(p + k)λ − pλ ]
.
|w| <
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
The result is sharp with the extremal function f being given by (6).
Theorem 3.2. Let f ∈ Cp (α, A, B, k, β, λ), then | z |= r < 1
12
H. Özlem Güney
(10)
rp −
(A − B)β(p − α)pλ+1
rp+k ≤ |f (z)| ≤
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1
≤ rp +
(A − B)β(p − α)pλ+1
rp+k
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1
and
(11)
prp−1 −
(A − B)β(p − α)pλ+1
rp+k−1 ≤ |f 0 (z)| ≤
λ
[(1 − Bβ)k + (A − B)β(p − α)](p + k)
(A − B)β(p − α)pλ+1
rp+k−1
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
≤ prp−1 +
All the inequalities are sharp with the extremal function
(12)
f (z) = z p −
(A − B)β(p − α)pλ+1
z p+k .
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1
Proof. Using the arguments as in the Theorem 3.1, the required results
for Cp (α, A, B, k, β, λ) is established.
Letting r → 1− in the left hand side of (10), we have:
Corollary 3.2.
Let f ∈ Cp (α, A, B, k, β, λ). Then the unit disk U is
mapped by f onto a domain that contains the disk
|w| <
(p + k)λ+1 (1 − Bβ)k + (A − B)β(p − α)[(p + k)λ+1 − pλ+1 ]
.
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1
The result is sharp with the extremal function f being given by (12).
4
Integral Operators
Theorem 4.1. Let c be a real number such that c > −p.
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 13
If, f ∈ Tp∗ (α, A, B, k, β, λ), then the function F defined by
c+p
F (z) =
zc
(13)
Zz
tc−1 f (t)dt
0
also belongs to Tp∗ (α, A, B, k, β, λ).
∞
P
an+p z p+n ; ap+n ≥ 0. Then from representation
Proof. Let f (z) = z p −
of F , it follows that
n=k
F (z) = z p −
∞
X
bp+n z p+n
; bp+n ≥ 0
n=k
where
µ
bp+n =
c+p
c+p+n
¶
ap+n .
Therefore using Theorem 2.1 for the coefficients of F , we have
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ bp+n =
n=k
=
∞
X
µ
λ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
n=k
≤
c+p
c+p+n
¶
ap+n ≤
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ ap+n ≤ (A − B)pλ β(p − α)
n=k
since
c+p
c+p+n
< 1 and f ∈ Tp∗ (α, A, B, k, β, λ).
Hence F ∈ Tp∗ (α, A, B, k, β, λ).
Theorem 4.2. Let c be a real number such that c > −p.
If F ∈ Tp∗ (α, A, B, k, β, λ), then the function f (z) = z p −
ap+n ≥ 0 is p-valent in | z |< R∗ , where
½·µ
¶
c+p
∗
R = inf
(14)
ap+n ≤
n≥k≥2
c+p+n
∞
P
n=k
ap+n z p+n ,
14
H. Özlem Güney
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ−1
≤
(A − B)pλ−1 β(p − α)
The result is sharp.
Proof. Let F (z) = z p −
∞
P
¸ n1 )
.
ap+n z p+n ; ap+n ≥ 0. It follows then from (13)
n=k
that
¶
∞ µ
X
z 1−c d c
c+p+n
p
f (z) =
[z F (z)] = z −
ap+n z p+n .
c + p dz
c+p
n=k
In order to obtain the required result it sufficient to show that
¯ 0
¯
¯ f (z)
¯
¯
¯ <p
−
p
¯ z p−1
¯
for | z |< R∗ where R∗ is defined by (14).
Now
¯
¯ 0
¯ ¯¯ ∞
¶
µ
¯
¯ f (z)
¯ ¯ X
c+p+n
n¯
¯
¯
ap+n z ¯ ≤
(p + n)
¯ z p−1 − p¯ = ¯¯−
¯
c+p
n=k
≤
∞
X
n=k
µ
(p + n)
c+p+n
c+p
¶
|ap+n | |z|n .
Thus
¯ 0
¯
µ
¶
∞
X
¯ f (z)
¯
c
+
p
+
n
¯
¯
(15)
(p + n)
ap+n |z|n < p.
¯ z p−1 − p¯ < p if
c
+
p
n=k
But Theorem 2.1 confirms that
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
n=k
(A − B)pλ−1 β(p − α)
Hence (15) will be satisfied if
µ
¶
c+p+n
(p + n)
ap+n |z|n ≤
c+p
ap+n ≤ p.
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 15
≤
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
ap+n ;
(A − B)pλ−1 β(p − α)
n ≥ k ≥ 2 or if
½µ
¶
¾1
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ−1 n
c+p
;
| z |≤
c+p+n
(A − B)pλ−1 β(p − α)
n ≥ k ≥ 2.
Therefore f is p-valent in | z |< R∗ .
Sharpness follows if we take
F (z) = z p −
(A − B)β(p − α)pλ
z p+n ;
λ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
n ≥ k ≥ 2.
5
Closure Properties
In this section we show that the classes Tp∗ (α, A, B, k, β, λ) and Cp (α, A, B, k, β, λ)
are closed under “arithmetic mean” and “convex linear combinations”.
Theorem 5.1. The class Tp∗ (α, A, B, k, β, λ) is closed under convex linear
combinations.
Proof. Suppose that
(i)
p
f (z) = z −
∞
X
(i)
(i)
ap+n z p+n ; i = 1, 2 ; ap+n ≥ 0
n=k
are in the class Tp∗ (α, A, B, k, β, λ). Let f (z) = (1 − ς)f (1) (z) + ςf (2) (z)
with 0 ≤ ς ≤ 1. It is easy to satisfy, by Theorem 2.1, that f (z) is in
Tp∗ (α, A, B, k, β, λ).
Theorem 5.2. The class Cp (α, A, B, k, β, λ) is closed under convex linear
combinations.
16
6
H. Özlem Güney
Radius of Convexity
Theorem 6.1. If f ∈ Tp∗ (α, A, B, k, β, λ), then f is p-valent convex function | z |< R1 , where
(·
(16) R1 = inf
n≥k≥2
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ−2
(A − B)pλ−2 β(p − α)
¸ n1 )
The result is sharp with the extremal function f given by (3).
Proof. It is sufficient to show that
¯
¯
00
¯
¯
¯[1 + zf (z) ] − p¯ ≤ p for | z |< R1 .
¯
¯
0
f (z)
We have
¯
¯
¯ ¯ 0
00
¯
¯ ¯ f (z) + zf 00 (z) − pf 0 (z) ¯
zf
(z)
¯=
¯[1 +
] − p¯¯ = ¯¯
¯
¯
f 0 (z)
f 0 (z)
¯ ∞
¯
∞
P
¯ P
¯
n
¯−
n(p + n)ap+n z ¯
n(p + n)ap+n | z |n
¯ n=k
¯
n=k
¯≤
= ¯¯
.
∞
∞
¯
P
P
n
n
¯p −
¯
(p + n)ap+n z ¯ p −
(p + n)ap+n | z |
¯
n=k
Therefore
if
n=k
¯
¯
00
¯
¯
zf
(z)
¯[1 +
] − p¯¯ ≤ p
¯
0
f (z)
∞
X
(p + n)2 ap+n |z|n ≤ p2
n=k
or
∞
X
p+n 2
) ap+n |z|n ≤ 1.
(
p
n=k
By Theorem 2.1, we have
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
n=k
(A − B)pλ β(p − α)
ap+n ≤ 1.
.
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 17
Hence f is p-valently convex if
µ
¶2
p+n
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
|z|n ≤
p
(A − B)pλ β(p − α)
or if
·
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ−2
|z| ≤
(A − B)pλ−2 β(p − α)
¸ n1
; n ≥ k ≥ 2.
The extreme points of Tp∗(α, A, B, k, β, λ)
7
and Cp(α, A, B, k, β, λ)
Theorem 7.1. Let fp (z) = z p and
fp+n = z p −
(A − B)β(p − α)pλ
z p+n ; n ≥ k ≥ 2.
λ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
Then f ∈ Tp∗ (α, A, B, k, β, λ) if and only if it can be expressed in the form
∞
∞
P
P
f (z) = ξp fp (z) +
ξn fp+n (z), z ∈ U , where ξn ≥ 0 and ξp = 1 −
ξn .
n=k
n=k
Proof. Let us assume that
f (z) = ξp fp (z) +
∞
X
ξn fp+n (z) =
n=k
= [1−
∞
X
½
∞
X
ξn z p −
ξn ]z +
p
n=k
n=k
p
=z −
∞
X
n=k
(A − B)β(p − α)pλ
z p+n
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
(A − B)β(p − α)pλ
ξn z p+n .
λ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
Then from Theorem 2.1, we have
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ ·
n=k
¾
=
18
H. Özlem Güney
·
(A − B)β(p − α)pλ
ξn ≤ (A − B)pλ β(p − α)
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
Hence f ∈ Tp∗ (α, A, B, k, β, λ).
Conversely, let f ∈ Tp∗ (α, A, B, k, β, λ). Using Corollary 2.1 , setting
ξn =
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
ap+n ; n = k, k + 1, · · · ; k ≥ 2,
(A − B)pλ β(p − α)
and letting ξp = 1 −
∞
P
ξn , we have
n=k
p
f (z) = z −
∞
X
ap+n z
p+n
p
=z −
n=k
−
∞
X
ξn
n=k
= [1−
∞
X
p
ξn ]z +
p
ξn z +
n=k
∞
X
ξn z p −
n=k
(A − B)β(p − α)pλ
z p+n =
λ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
∞
X
n=k
n=k
∞
X
ξn [z p −
(A − B)β(p − α)pλ
z p+n ] =
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
= ξp fp (z) +
∞
X
ξn fp+n (z).
n=k
This completes the proof of Theorem 7.1.
Corollary 7.1. The extreme points of Tp∗ (α, A, B, k, β, λ) are the functions
fp (z) = z p and
fp+n = z p −
(A − B)β(p − α)pλ
z p+n ; n ≥ k ≥ 2.
λ
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
Theorem 7.2. Let fp (z) = z p and
fp+n
(A − B)β(p − α)pλ+1
z p+n ; n ≥ k ≥ 2.
=z −
λ+1
[(1 − Bβ)n + (A − B)β(p − α)](p + n)
p
Then f ∈ Cp (α, A, B, k, β, λ) if and only if it can be expressed in the form
∞
∞
P
P
f (z) = ξp fp (z) +
ξn fp+n (z), z ∈ U , where ξn ≥ 0 and ξp = 1 −
ξn .
n=k
n=k
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 19
Corollary 7.2. The extreme points of Cp (α, A, B, k, β, λ) are the functions
fp (z) = z p and
fp+n = z p −
8
(A − B)β(p − α)pλ+1
z p+n ; n ≥ k ≥ 2.
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ+1
Modified Hadamard Products
Let the functions fj (z)(j = 1, 2) be defined by
(17)
p
fj (z) = z −
∞
X
ap+n,j z p+n
(ap+n,j ≥ 0).
n=k
The Modified Hadamard product f1 ∗ f2 of f1 and f2 is denoted by
(18)
p
(f1 ∗ f2 )(z) = z −
∞
X
ap+n,1 ap+n,2 z p+n
n=k
Theorem 8.1. Let the functions fj (z)(j = 1, 2) defined by be in the class
Tp∗ (α, A, B, k, β, λ).
Then (f1 ∗f2 )(z) belongs to the class Tp∗ (γ(p, α, A, B, k, β, λ), A, B, k, β, λ)
where
(19)
γ = γ(p, α, A, B, k, β, λ) =
(A − B)β(p − α)pλ (p − α)2 (1 − Bβ)k
=p−
.
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ − (A − B)2 β 2 pλ (p − α)2
The result is sharp for the functions fj (z)given by
fj (z) = z p −
(A − B)β(p − α)pλ (p − α)
z p+k ;
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ
k ≥ 2.
20
H. Özlem Güney
Proof. Employing the tecnique used earlier by Schild and Silverman [14],
we need to find the largest γ = γ(p, α, A, B, k, β, λ) such that
∞
X
[(1 − Bβ)n + (A − B)β(p − γ)](p + n)λ
(20)
(A − B)pλ β(p − γ)
n=k
ap+n,1 ap+n,2 ≤ 1.
Since
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
(21)
(A − B)pλ β(p − α)
n=k
ap+n,1 ≤ 1.
and
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
(22)
(A − B)pλ β(p − α)
n=k
ap+n,2 ≤ 1.
by means of Cauchy-Schwarz inequality, we have
(23)
∞
X
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ √
ap+n,1 ap+n,2 ≤ 1.
λ β(p − α)
(A
−
B)p
n=k
Therefore, it is sufficient to show that
[(1 − Bβ)n + (A − B)β(p − γ)](p + n)λ
ap+n,1 ap+n,2 ≤
(A − B)pλ β(p − γ)
≤
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ √
ap+n,1 ap+n,2
(A − B)pλ β(p − α)
that is, that
(24)
√
ap+n,1 ap+n,2 ≤
(p − γ)[(1 − Bβ)n + (A − B)β(p − α)]
; n ≥ k ≥ 2.
(p − α)[(1 − Bβ)n + (A − B)β(p − γ)]
Note that
(25)
√
ap+n,1 ap+n,2 ≤
(A − B)pλ β(p − α)
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
;
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 21
n ≥ k ≥ 2.
Consequently, we need only to prove that
(A − B)pλ β(p − α)
≤
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ
≤
(p − γ)[(1 − Bβ)n + (A − B)β(p − α)]
(p − α)[(1 − Bβ)n + (A − B)β(p − γ)]
; n ≥ k ≥ 2.
or , equivalently, if
(26)
=p−
γ = γ(p, α, A, B, k, β, λ) =
(A − B)β(p − α)pλ (p − α)2 (1 − Bβ)n
.
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ − (A − B)2 β 2 pλ (p − α)2
Since Φ(n)by
(27)
=p−
Φ(n) =
(A − B)β(p − α)pλ (p − α)2 (1 − Bβ)n
,
[(1 − Bβ)n + (A − B)β(p − α)](p + n)λ − (A − B)2 β 2 pλ (p − α)2
is an increasing function of n, n ≥ k ≥ 2, letting n = k in (8.11) we obtain,
therefore,
(28)
=p−
γ ≤ Φ(k) =
(A − B)β(p − α)pλ (p − α)2 (1 − Bβ)k
.
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ − (A − B)2 β 2 pλ (p − α)2
which completes the assertion of theorem.
Corollary 8.1. For f1 (z)and f2 (z) as in Theorem 8.1, the function
p
h(z) = z −
∞
X
√
ap+n,1 ap+n,2 z p+n
n=k
belongs to the class Tp∗ (α, A, B, k, β, λ).
Proof. It follows from the Cauchy-Shwarz inequality (8.7). It is sharp for
the same functions as in Theorem 8.1.
22
H. Özlem Güney
Theorem 8.2. Let the functions fj (z)(j = 1, 2) defined by be in the class
Cp (α, A, B, k, β, λ).
Then (f1 ∗f2 )(z) belongs to the class Cp (ψ(p, α, A, B, k, β, λ), A, B, k, β, λ)
(29)
= p−
ψ = ψ(p, α, A, B, k, β, λ) =
(A − B)β(p − α)pλ+1 (p − α)2 (1 − Bβ)k
.
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1 − (A − B)2 β 2 pλ+1 (p − α)2
The result is sharp for the functions fj (z)given by
fj (z) = z p −
9
(A − B)β(p − α)pλ+1 (p − α)
z p+k ;
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1
k ≥ 2.
Definitions and Applications of The Fractional Calculus
In this section, we shall prove several distortion theorems for functions to
general classes Tp∗ (α, A, B, k, β, λ) and Cp (α, A, B, k, β, λ).
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 [10] (and more recently, by Owa and
Srivastava [11], and Srivastava and Owa [12], ; see also Srivastava et all.
[13] )
Definition 9.1. The fractional integral of order µ is defined, for a function
f , by
(30)
1
Dz−µ f (z) =
Γ(µ)
Zz
0
f (ζ)
dζ, (µ > 0)
(z − ζ)1−µ
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 23
where f is an analytic function in a simply – connected region of the 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 9.2.
The fractional derivative of order µ is defined, for a
function f , by
(31)
Dzµ f (z)
1
d
=
Γ(1 − µ) dz
Zz
f (ζ)
dζ, (0 ≤ µ < 1)
(z − ζ)µ
0
where f is constrained, and the multiplicity of (z − ζ)−µ is removed, as in
Definition 9.1.
We remark that in Definition 9.1 and 9.2, Γ denotes the Gamma function.
Definition 9.3.
Under the hypotheses of Definition 9.2, the fractional
derivative of order (n + µ) is defined by
(32)
Dzn+µ f (z) =
dn µ
D f (z)
dz n z
where 0 ≤ µ < 1 and n ∈ N0 = N ∪ {0} .
From Definition 9.2, we have
Dz0 f (z) = f (z)
(33)
which, in view of Definition 9.3 yields,
(34)
Dzn+0 f (z) =
dn 0
Dz f (z) = f n (z).
n
dz
Thus, it follows from (33) and (34) that lim Dz−µ f (z) = f (z) and
µ→0
lim Dz1−µ f (z) = f 0 (z).
µ→0
24
H. Özlem Güney
Theorem 9.1. Let the function f defined by
p
f (z) = z −
∞
X
ap+n z p+n
; ap+n ≥ 0
n=k
be in the class Tp∗ (α, A, B, k, β, λ).
Then
¯ −µ i
¯
¯Dz (D f (z))¯ ≥ |z|p+µ
(35)
½
Γ(p + 1)
−
Γ(µ + p + 1)
(A − B)β(p − α)pλ Γ(p + k + 1)
−
|z|k
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ−i Γ(µ + p + k + 1)
¾
and
¯ −µ i
¯
¯Dz (D f (z))¯ ≤ |z|p+µ
(36)
½
Γ(p + 1)
+
Γ(µ + p + 1)
(A − B)β(p − α)pλ Γ(p + k + 1)
+
|z|k
λ−i
[(1 − Bβ)k + (A − B)β(p − α)](p + k) Γ(µ + p + k + 1)
¾
for µ > 0 , 0 ≤ i ≤ λ and z ∈ U . The equalities in (35) and (36) are
attained for the function f (z) given by
(37)
f (z) = z p −
(A − B)β(p − α)pλ
(p + k)−λ z p+k .
[(1 − Bβ)k + (A − B)β(p − α)]
Proof. We note that
Γ(p + 1 + µ) −µ −µ i
z Dz (D f (z)) =
Γ(p + 1)
p
=z −
∞
X
Γ(p + n + 1)Γ(p + µ + 1)
n=k
Γ(p + 1)Γ(p + n + µ + 1)
(p + n)i ap+n z p+n .
Defining the function ϕ(n)
ϕ(n) =
Γ(p + n + 1)Γ(p + µ + 1)
,
Γ(p + 1)Γ(p + n + µ + 1)
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 25
µ>0;n ≥k≥2,
We can see that ϕ(n) is decreasing in n , that is , that
0 < ϕ(n) ≤ ϕ(k) =
Γ(p + k + 1)Γ(p + µ + 1)
.
Γ(p + 1)Γ(p + k + µ + 1)
On the other hand, from [8 ],
∞
X
(p+n)i ap+n ≤
n=k
(A − B)β(p − α)pλ
(p+k)−(λ−i) ; 0 ≤ i ≤ λ.
[(1 − Bβ)k + (A − B)β(p − α)]
Therefore,
¯
¯
∞
X
¯ Γ(p + 1 + µ) −µ −µ i
¯
p
p+k
¯
¯
(p + n)i ap+n ≥
¯ Γ(p + 1) z Dz (D f (z))¯ ≥ |z| − ϕ(k) |z|
n=k
≥ |z|p −
Γ(p + k + 1)Γ(p + µ + 1)
(A − B)β(p − α)pλ
·
Γ(p + 1)Γ(p + k + µ + 1) [(1 − Bβ)k + (A − B)β(p − α)]
·(p + k)−(λ−i) |z|p+k
and
¯
¯
∞
X
¯ Γ(p + 1 + µ) −µ −µ i
¯
p
p+k
¯
¯
(p + n)i ap+n ≤
¯ Γ(p + 1) z Dz (D f (z))¯ ≤ |z| + ϕ(k) |z|
n=k
Γ(p + k + 1)Γ(p + µ + 1)
(A − B)β(p − α)pλ
≤ |z| +
·
Γ(p + 1)Γ(p + k + µ + 1) [(1 − Bβ)k + (A − B)β(p − α)]
p
·(p + k)−(λ−i) |z|p+k
which completes the proof of theorem.
Remark. By letting µ → 0, taking i = 0 and λ = 0 in Theorem 9.1 , we
have the former results by Aouf and Darwish [4].
Next, we prove,
26
H. Özlem Güney
Theorem 9.2. Let the function f defined by
p
f (z) = z −
∞
X
ap+n z p+n
; ap+n ≥ 0
n=k
be in the class Tp∗ (α, A, B, k, β, λ).
Then,
(38)
¯ µ i
¯
¯Dz (D f (z))¯ ≥ |z|p−µ
½
Γ(p + 1)
−
Γ(p − µ + 1)
(A − B)β(p − α)pλ Γ(p + k + 1)
−
|z|k
λ−i−1
[(1 − Bβ)k + (A − B)β(p − α)](p + k)
Γ(p + k − µ + 1)
(39)
¯ µ i
¯
¯Dz (D f (z))¯ ≤ |z|p−µ
½
Γ(p + 1)
+
Γ(p − µ + 1)
(A − B)β(p − α)pλ Γ(p + k + 1)
+
|z|k
λ−i−1
[(1 − Bβ)k + (A − B)β(p − α)](p + k)
Γ(p + k − µ + 1)
¾
¾
for 0 ≤ µ < 1 , 0 ≤ i ≤ λ − 1 and z ∈ U . The equalities in (38) and (39)
are attained for the function f (z) given by (37).
Proof. We can easily take
∞
X
Γ(p + k − µ) µ µ
Γ(p + n + 1)Γ(p − µ + 1)
p
z Dz f (z) = z −
(p+n)i ap+n z p+n .
Γ(p + 1)
Γ(p
+
1)Γ(p
+
n
−
µ
+
1)
n=k
Since the function
φ(n) =
Γ(p − µ + 1)Γ(p + n)
; n ≥ k ≥ 2.
Γ(p + 1)Γ(p + n − µ + 1)
In decreasing in n, we have
0 < φ(n) ≤ φ(k) =
Γ(p − µ + 1)Γ(p + k)
.
Γ(p + 1)Γ(p + k − µ + 1)
Further, we note that from [8],
∞
X
(p + n)i+1 ap+n ≤
n=k
(A − B)β(p − α)pλ
(p + k)−(λ−i−1) ,
[(1 − Bβ)k + (A − B)β(p − α)]
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 27
0 ≤ i ≤ λ − 1,
for f (z) ∈ Tp∗ (α, A, B, k, β, λ).
Then it follows that
¯
¯
∞
X
¯ Γ(p + 1 − µ) µ µ i
¯
p
p+k
¯
¯
(p + n)i+1 ap+n ≥
¯ Γ(p + 1) z Dz (D f (z))¯ ≥ |z| − φ(k) |z|
n=k
Γ(p + k)Γ(p − µ + 1)
(A − B)β(p − α)pλ
≥ |z| −
·
Γ(p + 1)Γ(p + k − µ + 1) [(1 − Bβ)k + (A − B)β(p − α)]
p
·(p + k)−(λ−i−1) |z|p+k
and
¯
¯
∞
X
¯ Γ(p + 1 − µ) µ µ i
¯
p
p+k
¯
¯
z
D
(D
f
(z))
(p + n)i+1 ap+n ≤
≤
|z|
+
φ(k)
|z|
z
¯ Γ(p + 1)
¯
n=k
≤ |z|p +
Γ(p + k)Γ(p − µ + 1)
(A − B)β(p − α)pλ
·
Γ(p + 1)Γ(p + k − µ + 1) [(1 − Bβ)k + (A − B)β(p − α)]
·(p + k)−(λ−i−1) |z|p+k
which completes the proof of theorem.
Remark. By taking i = 0 and λ = 0 and letting µ → 1 in Theorem 9.2 ,
we have the former results by Aouf and Darwish [4].
Theorem 9.3. Let the function f defined by
p
f (z) = z −
∞
X
ap+n z +np
; ap+n ≥ 0
n=k
be in the class Cp (α, A, B, k, β, λ).
Then
(40)
¯ −µ i
¯
¯Dz (D f (z))¯ ≥ |z|p+µ
½
Γ(p + 1)
−
Γ(µ + p + 1)
(A − B)β(p − α)pλ+1 Γ(p + k + 1)
|z|k
−
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1−i Γ(µ + p + k + 1)
¾
28
H. Özlem Güney
¯ −µ i
¯
¯Dz (D f (z))¯ ≤ |z|p+µ
(41)
½
Γ(p + 1)
+
Γ(µ + p + 1)
(A − B)β(p − α)pλ+1 Γ(p + k + 1)
+
|z|k
[(1 − Bβ)k + (A − B)β(p − α)](p + k)λ+1−i Γ(µ + p + k + 1)
¾
for µ > 0 , 0 ≤ i ≤ λ and z ∈ U, and
¯ µ i
¯
¯Dz (D f (z))¯ ≥ |z|p−µ
(42)
½
Γ(p + 1)
−
Γ(p − µ + 1)
(A − B)β(p − α)pλ+1 Γ(p + k)
−
|z|k
λ−i
[(1 − Bβ)k + (A − B)β(p − α)](p + k) Γ(p + k − µ + 1)
¾
and
¯ µ i
¯
¯Dz (D f (z))¯ ≤ |z|p−µ
(43)
½
Γ(p + 1)
+
Γ(p − µ + 1)
(A − B)β(p − α)pλ+1 Γ(p + k)
+
|z|k
λ−i
[(1 − Bβ)k + (A − B)β(p − α)](p + k) Γ(p + k − µ + 1)
¾
for
0 ≤ µ < 1 , 0 ≤ i ≤ λ − 1andz ∈ U.
All inequalities in above are attained for the function f (z) given by (37).
Remark. By letting µ → 0, taking i = 0 and λ = 0 in (40) - (41) and by
taking i = 0 and λ = 0 and letting µ → 1in (42) - (43) of Theorem 9.3, we
have the former results by Aouf [6].
References
[1] O. P. Ahuja, P. K. Jain, On starlike and convex functions with missing
and negative coefficients, Bull.Malaysian Math.Soc.(2) 3 (1980), 95101.
On New Classes of Sălăgean-type p-valent Functions with Negative and ... 29
[2] V. P. Gupta, P. K. Jain, Certain classes of univalent functions with
negative coefficients, Bull.Aust.Math.Soc., 14 (1976), 406-416.
[3] H.
Silverman,
Univalent
functions
with
negative
coefficients,
Proc.Amer.Math.Soc., 51 (1975), 109-116.
[4] M. K. Aouf, H. E. Darwish, On p-valent functions with negative and
missing coefficients II, Commun.Fac.Sci.Univ.Ank.Series A1 , Vol.44,
(1995) pp.1-14.
[5] S. M. Srangi, V. J. Patil, On multivalent functions with negative and
missing coefficients, J.Math.Res.Exposition 10 (1990), no.3, 341-348.
[6] M. K. Aouf, On subclasses of starlike and convex functions with negative and missing coefficients, Bolletino U.M.I.(7), 4-B (1990), 711-720.
[7] R. M. Goel, N. S. Sohi, Multivalent functions with negative coefficients,
Indian J.Pure and Appl.Math. 12(7), July 1981, 844-853.
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University of Dicle,
Faculty of Science & Arts,
Department of Mathematics,
21280-Diyarbakır- TURKEY,
E-mail address: ozlemg@dicle.edu.tr