Research Article
OnthePartialSumsoftheq-GeneralizedDiniFunction
AlaaH.El-Qadeem ,
1
MohamedA.Mamon ,
2
andIbrahimS.Elshazly
3
1
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
2
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
3
Department of Basic Sciences, Common First Year, King Saud University, Alriyad 11451, Saudi Arabia
CorrespondenceshouldbeaddressedtoAlaaH.El-Qadeem;ahhassan@science.zu.edu.eg
Received 29 December 2021; Accepted 29 January 2022; Published 21 February 2022
AcademicEditor:V.Ravichandran
Copyright©2022AlaaH.El-Qadeemetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Ourobjectiveinthispaperistointroduceaq-analogofthegeneralizedDinifunction.Also,weinvestigatethelowerboundforthe
ratio of the q-generalized Dini function to its sequences of partial sums.
1.IntroductionandBasicConcepts
Let A denotetheclassoffunctions f thathavethefollowing
Maclaurin’s form,
f(z)� z +
∞
n�2
a
n
z
n
, (1)
which is analytic and univalent in the open unit disc U �
z: |z| < 1 { } and satisfies the normalization conditions
f(0)� f
′
(0)− 1 � 0.
Special functions play an inspired role in applied
mathematics and physics. e widespread use of these
functions has attracted many researchers to work in many
directions. Lately, many authors studied the geometric
properties of some special functions such as starlikness,
univalence,andconvexity,see[1–6].ereareseveralresults
related to partial sums of analytic univalent functions that
were developed by the authors in [7–9]. Specifically, the
authors in [10] investigated the partial sums of the gener-
alized Bessel function, and then, a lot of authors followed
them in studying the same problem for different special
functions such as Bessel [11, 12], Struve [13], Lommel [14],
Wright [15], and Mittag-Leffler [16], see also [17].
Our aim in this study is to develop a q-analog of the
generalizedDinifunction,whichisinspiredbyearlystudies
on analytic and special functions. We also provide lower
bounds for the ratio of q-generalized Dini function to its
sequences of partial sums, for m ∈ N � 1, 2, 3, ... { },
(ψ
a
],b,c
(z; q))
m
� z +
m
n�1
ζ
n
z
n+1
. We will investigate the
following:
R
ψ
a
],b,c
(z; q)
ψ
a
],b,c
(z; q)
m
⎧ ⎨
⎩
⎫ ⎬
⎭
, R
ψ
a
],b,c
(z; q)
m
ψ
a
],b,c
(z; q)
⎧ ⎨
⎩
⎫ ⎬
⎭
, R
ψ
a
],b,c
(z; q)
′
ψ
a
],b,c
(z; q)
m
′
⎧ ⎨
⎩
⎫ ⎬
⎭
, and R
ψ
a
],b,c
(z; q)
m
′
ψ
a
],b,c
(z; q)
′
⎧ ⎨
⎩
⎫ ⎬
⎭
. (2)
To introduce the main results, we would like to recall
some fundamentals and concepts related to geometric
function theory and the definition of q-generalized Bessel
function.Atfirst,letusconsiderthefollowingsecond-order
linear homogenous differential equation (for more details,
see [18–20]):
Hindawi
Journal of Mathematics
Volume 2022, Article ID 8496249, 7 pages
https://doi.org/10.1155/2022/8496249