Open Access Library Journal
2020, Volume 7, e5324
ISSN Online: 23339721
ISSN Print: 23339705
Bounded Turning of an mth Partial Sum of
Modiﬁed Caputo’s Fractional Calculus
Derivative Operator
Ajai P. Terwase1,2*, S. Longwap3, N. M. Choji2
1Department of Mathematics, University of the Gambia, Banjul, the Gambia
2Department of Mathematics, Plateau state University, Bokkos, Plateau, Nigeria
3Department of Mathematics, University of Jos, Jos, Plateau, Nigeria
Abstract
In this article, we consider subclasses of functions with bounded turning for
normalized analytic functions in the unit disk, we investigate certain condi
tions under which the partial sums of the modiﬁed Caputo’s fractional deriv
ative operators of analytic univalent functions of bounded turning are also of
bounded turning.
Subject Areas
Mathematical Analysis
Keywords
Analytic Functions, ClosetoConvex, Bounded Turning, Univalent
1. Introduction and Deﬁnitions
Let
denote a class of all analytic functions of the form
( )
2
k
k
k
f z z az
∞
=
= +
∑
(1.1)
which are analytic in the open unit disk
{ }
:1U zz= <
and normalized by
( ) ( )
0 0 10ff
′
= −=
Deﬁnition 1.
Let
( )
,0 1B
µµ
≤<
denote the class of functions of the Form (1.1) then if
{ }
f
µ
′
ℜ>
, that is the real part of its first derivative map the unit disk onto the
right half plane, then the class of functions in
( )
B
µ
are called functions of
bounded turning.
How to cite this paper:
Terwase, A.P.,
Longwap, S.
and Choji, N.M. (2020)
Bounded
Turning of an
m
th Partial Sum of Modiﬁed
Caputo’s Fractional Calculus Derivative
Operator
.
Open Access Library Journal
,
7
:
e5324
.
https://doi.org/10.4236/oalib.1105324
Received:
October 18, 2019
Accepted:
March 24, 2020
Published:
March 27, 2020
Copyright © 20
20 by author(s) and Open
Access Library Inc
.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
DOI: 10.4236/oalib.1105324 Mar. 27, 2020 1 Open Access Library Journal
A. P. Terwase et al.
By Nashiro Warschowski, see [1], it is proved that the functions in
( )
B
µ
are
univalent and also close to convex in
U
. In [2], it was also shown that the partial
sums of the Libera integral operator of functions of bounded turning are also of
bounded turning. For more works on bounded turning see [3] [4].
Deﬁnition 2.
If
( )
0
n
n
n
f z az
∞
=
=∑
and
( )
0
n
n
n
g z bz
∞
=
=∑
are analytic in
U
, then their Hadamard
product
*fg
defined by the power series is given by:
( )( )
0
*.
n
nn
n
f g z abz
∞
=
=∑
(1.2)
Note that the convolution so defined is also analytic in
U
.
For ƒ of the Form (1.1) several interesting derivatives operators in their dif
ferent forms have been studied, here we consider (1.1) using the modified Ca
puto’s derivative operator
( )
,
J fz
ηλ
, see [5] [6], stated as follow:
For
f∈
,
()( )
( )
,1
0
2d
z
f
J fz z z
η
λη
ηλ λη
ξ
ηλ ξ
ηλ ξ
−
+−
Ω
+−
=−−
∫
(1.3)
where
η
is a real number and
12
η λη
−< ≤ <
. Notice that (1.3) can also be
express as:
( ) ( ) ( )( )
( )( )
2
,2
12 2
11
n
n
n
n
J f z z az
nn
ηλ
ηλ η
ηλ η
∞
=
+ +− −
= + +−+ −+
∑
(1.4)
and its partial sum given as:
( ) ( ) ( )( )
( )( )
2
2
12 2
11
Mn
Mn
n
n
P z z az
nn
ηλ η
ηλ η
=
+ +− −
= + +−+ −+
∑
(1.5)
We determine conditions under which the partial sums of the operator given
in (1.4) are of bounded turning. We shall use the following lemmas in the sequel
to establish our result.
Lemma 1. [7]
For
zU∈
, we have
( )
1
1,
23
n
n
zzU
n
∞
=
ℜ >− ∈
+
∑
(1.6)
Lemma 2. [1]
Let
P
(
z
) be analytic in
U
, such that
P
(0) = 1, and
()
( )
1
2
Pzℜ>
in
U
. For
function
Q
analytic in
U
the convolution function
*PQ
takes values in the
convex hull of the image
U
under
Q
.
We shall implore lemmas 1 and 2 to show conditions under which the mth
partial sum (2.1) of the modiﬁed Caputoes derivative operator of analytic univa
lent functions of bounded turning is also of bounded turning.
2. Main Theorem
Let
( )
fz∈
be of the Form (1.1), if
11
2
µ
<<
and
( ) ( )
fz B
µ
∈
, then
DOI: 10.4236/oalib.1105324 2 Open Access Library Journal
A. P. Terwase et al.
( ) ( )( )( )
( )
32 2 1
3
M
Pz B
ηλ η µ
− +− − −
∈
,
12
η λη
−< ≤ <
.
Proof.
Let
()
fz
be of the Form (1.1) and
( )
{ }
1
, 1,
2
fz zU
µµ
′
ℜ > << ∈
. This im
plies that
1
2
12
n
n
n
na z
µ
∞−
=
ℜ+ >
∑
(2.1)
Now for
11
2
µ
<<
we have
11
22
11
1
nn
nn
nn
n
a z na z
µ
∞∞
−−
= =
ℜ+ >ℜ+
−
∑∑
(2.2)
Applying the convolution properties to
( )
Pz
′
, where
( ) ( ) ( )( )
()( )
21
2
12 2
111
Mn
Mn
n
nn
P z az
nn
ηλ η
ηλ η
−
=
+ +− −
′= + +−+ −+
∑
(2.3)
( ) ( )( )
( )( ) ( )
( ) ( )
2
11
22
12 2
1 *1 1
1 11
*
M
nn
nn
nn
nn
n
a z az
nn
Pz Qz
ηλ η µ
µ ηλ η
∞−−
= =
+ +− −
++ −
− +−+ −+
=
∑∑
(2.4)
with recourse for Lemma 1 and
1Jm= −
we have
1
2
1
13
n
M
n
z
n
−
=
ℜ >−
+
∑
(2.5)
Then for
12
η λη
−< ≤ <
( )
( )
( )( )( )( )( )
1
1
21
2
1
2
1 2 2 1 11
1
n
M
n
nn
n
M
n
z
nn n n az
z
n
ηλ η ηλ η µ
−
−−
=
−
=
ℜ
+ +− − +−+ −+ −
≥ℜ
+
∑
∑
(2.6)
Hence
( )
( )
( )( )( )( )( )
1
1
21
2
1 2 2 1 11
1
3
n
M
n
nn
z
nn n n az
ηλ η ηλ η µ
−
−−
=
ℜ
+ +− − +−+ −+ −
≥−
∑
(2.7)
Relating Lemma 1 and with
( )
Qz
, a computation gives
( ) ( ) ( )( )
( )( ) ( )
( )( )( )
21
2
12 2
11
11
32 2 1
3
Mn
n
n
nn
Qz az
nn
ηλ η µ
ηλ η
ηλ η µ
−
=
+ +− −
ℜ=+ −
+−+ −+
− +− − −
>
∑
(2.8)
DOI: 10.4236/oalib.1105324 3 Open Access Library Journal
A. P. Terwase et al.
Recall the power series
()
1
2
1,
1n
n
n
n
Pz a z z U
µ
∞−
=
=+∈
−
∑
(2.9)
satisfies
( )
01p=
and
( )
( )
1
2
1
1,
12
n
n
n
n
Pz a z z U
µ
∞−
=
ℜ =ℜ+ > ∈
−
∑
. Therefore
by Lemma 2 we have
( )
( )
( )( )( )
32 2 1 ,
3
Pz zU
ηλ η µ
− +− − −
′
ℜ> ∈
(2.10)
This proves our results.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa
per.
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DOI: 10.4236/oalib.1105324 4 Open Access Library Journal