Open Access Library Journal
2020, Volume 7, e5324
ISSN Online: 2333-9721
ISSN Print: 2333-9705
Bounded Turning of an m-th 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, Close-to-Convex, 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 m-th
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.
References
[1] Goodman, A.W. (1983) Univalent Functions. Vol. I & II. Polygonal Publishing House,
Washinton, New Jersey.
[2] Jahangiri, J.M. and Farahmand, K. (2004) Partial Sums of Functions of Bounded
Turning.
International Journal of Mathematics and Mathematical Sciences
, 2004,
Article ID: 451028. https://doi.org/10.1155/S0161171204305284
[3] Liu, J.L. and Owa, S. (1997) On Partial Sums of the Libera Integral Operator.
Jour-
nal of Mathematical Analysis and Applications
, 213, 444-454.
https://doi.org/10.1006/jmaa.1997.5549
[4] Darus, M. and Ibrahim, R.W. (2010) Partial Sums of Analytic Functions of Bounded
Turning with Applications.
Computational & Applied Mathematics
, 29, 81-88.
https://doi.org/10.1590/S1807-03022010000100006
[5] Jamal, S. (2016) A Note on the Modified Caputos Fractional Calculus Derivative
Operator.
Far East Journal of Mathematical Sciences
(
FJMS
), 100, 609-615.
https://doi.org/10.17654/MS100040609
[6] Salah, J. and Darus, M. (2010) A Subclass of Uniformly Convex Functions Associated
with a Fractional Calculus Operator Involving Caputos Fractional Differentiation.
Acta
Universitatis Apulensis
.
Mathematics
-
Informatics
, 24, 295-306.
[7] Gasper, G. (1969) Nonnegative Sums of Cosines, Ultraspherical and Jacobi Poly-
nomials.
Journal of Mathematical Analysis and Applications
, 26, 60-68.
https://doi.org/10.1016/0022-247X(69)90176-0
DOI: 10.4236/oalib.1105324 4 Open Access Library Journal
