Subclass of $k$-uniformly starlike functions defined by symmetric $q$-derivative operator
The theory of $q$-analogs is frequently encountered in numerous areas, including fractals and dynamical systems. The $q$-derivatives and $q$-integrals play an important role in the study of $q$-deformed quantum-mechanical simple harmonic oscillators. We define a symmetric $q$-derivative operator and...
Gespeichert in:
| Datum: | 2018 |
|---|---|
| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2018
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1653 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507477304410112 |
|---|---|
| author | Altinkaya, S. Kanas, S. Yal¸cin, S. Альтінкая, С. Канас, С. Ялцин, С. |
| author_facet | Altinkaya, S. Kanas, S. Yal¸cin, S. Альтінкая, С. Канас, С. Ялцин, С. |
| author_sort | Altinkaya, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:22:19Z |
| description | The theory of $q$-analogs is frequently encountered in numerous areas, including fractals and dynamical systems. The $q$-derivatives and $q$-integrals play an important role in the study of $q$-deformed quantum-mechanical simple harmonic oscillators. We define a symmetric $q$-derivative operator and study a new family of univalent functions defined by using this operator. We establish some new relations between the functions satisfying analytic conditions related to conical
sections. |
| first_indexed | 2026-03-24T02:09:56Z |
| format | Article |
| fulltext |
UDC 517.5
S. Kanas (Univ. Rzeszow, Poland),
Ş. Altinkaya*, S. Yalçin (Uludag Univ., Bursa, Turkey)
SUBCLASS OF \bfitk -UNIFORMLY STARLIKE FUNCTIONS
DEFINED BY SYMMETRIC \bfitq -DERIVATIVE OPERATOR
ПIДКЛАС \bfitk -РIВНОМIРНО ЗIРКОПОДIБНИХ ФУНКЦIЙ, ЩО ВИЗНАЧЕНI
ЗА ДОПОМОГОЮ СИМЕТРИЧНОГО ОПЕРАТОРА \bfitq -ПОХIДНОЇ
The theory of q-analogs is frequently encountered in numerous areas, including fractals and dynamical systems. The
q-derivatives and q-integrals play an important role in the study of q-deformed quantum-mechanical simple harmonic
oscillators. We define a symmetric q-derivative operator and study a new family of univalent functions defined by using
this operator. We establish some new relations between the functions satisfying analytic conditions related to conical
sections.
Теорiя q-аналогiв часто зустрiчається в багатьох галузях, включаючи фрактали та динамiчнi системи. Важливу
роль у вивченнi q-деформованих квантово-механiчних простих гармонiчних осциляторiв вiдiграють q-похiднi та
q-iнтеграли. Наведено визначення симетричного оператора q-похiдної та вивчено нову сiм’ю однолистих функцiй,
що визначенi за допомогою цього оператора. Встановлено також деякi новi спiввiдношення мiж функцiями, що
задовольняють аналiтичнi умови вiдносно конiчних перерiзiв.
1. Introduction, definitions and notations. The intrinsic properties of q-analogs, including the
applications in the study of quantum groups and q-deformed superalgebras, study of fractals and
multifractal measures, and in chaotic dynamical systems are known in the literature. Some integral
transforms in the classical analysis have their q-analogues in the theory of q-calculus. This has
led various researchers in the field of q-theory for extending all the important results involving the
classical analysis to their q-analogs.
For the convenience, we provide some basic definitions and concept details of q-calculus which
are used in this paper. Throughout this paper, we will assume that q satisfies the condition 0 < q < 1.
We shall follow the notation and terminology of [?]. We first recall the definitions of fractional q-
calculus operators of complex valued function f.
Definition 1.1 [?]. Let q \in (0, 1) and \lambda \in \BbbC . The q-number, denoted [\lambda ]q, we define as
[\lambda ]q =
1 - q\lambda
1 - q
.
In the case when \lambda = n \in \BbbN we obtain [\lambda ]q = 1 + q + q2 + . . . + qn - 1, and when q \rightarrow 1 - then
[n]q = n. The symmetric q-number, denoted \widetilde [n]q is defined as a number
\widetilde [n]q = qn - q - n
q - q - 1
,
that reduces to n, in the case when q \rightarrow 1 - .
We note that the symmetric q-number do not reduce to the defined above q-number, and fre-
quently occurs in the study of q-deformed quantum mechanical simple harmonic oscillator (see [?]).
Applying the above q-numbers we define q-derivative and symmetric q-derivative, below.
* Ş. Altınkaya supported by the Scientific and Technological Research Council of Turkey (TUBITAK 2214A).
c\bigcirc S. KANAS, Ş. ALTINKAYA, S. YALÇIN, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11 1499
1500 S. KANAS, Ş. ALTINKAYA, S. YALÇIN
Definition 1.2 [?]. The q-derivative of a function f, defined on a subset of \BbbC , is given by
(Dqf)(z) =
\left\{
f(z) - f(qz)
(1 - q)z
for z \not = 0,
f \prime (0) for z = 0.
We note that \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1 - (Dqf)(z) = f \prime (z) if f is differentiable at z. Additionally, if f(z) = z +
+ a2z
2 + . . . , then
(Dqf)(z) = 1 +
\infty \sum
n=2
[n]qanz
n - 1.
Definition 1.3 [?]. The symmetric q-derivative \widetilde Dqf of a function f is defined as follows:
( \widetilde Dqf)(z) =
\left\{
f(qz) - f(q - 1z)
(q - q - 1)z
for z \not = 0,
f \prime (0) for z = 0.
(1.1)
From (1.1), we deduce that \widetilde Dqz
n = \widetilde [n]qzn - 1, and a power series of \widetilde Dqf, when f(z) =
= z + a2z
2 + . . . , is
( \widetilde Dqf)(z) = 1 +
\infty \sum
n=2
\widetilde [n]qanzn - 1.
It is easy to check that the following properties hold:
\widetilde Dq(f(z) + g(z)) = ( \widetilde Dqf)(z) + ( \widetilde Dqg)(z),\widetilde Dq
\bigl(
f(z)g(z)
\bigr)
= g(q - 1z)( \widetilde Dqf)(z) + f(qz)( \widetilde Dqg)(z) = g(qz)( \widetilde Dqf)(z) + f(q - 1z)( \widetilde Dqg)(z),\widetilde Dqf(z) = Dq2f(q
- 1z).
The defined above fractional q-calculus are the important tools used in a study of various families
of analytic functions, and in the context of univalent functions was first used in a book chapter by
Srivastava [23]. In contrast to the Leibnitz notation, being a ratio of two infinitisemals, the notions
of q-derivatives are plain ratios. Therefore, it appeared soon a generalization of q-calculus in many
subjects, such as hypergeometric series, complex analysis, and particle physics. It is also widely
applied in an approximation theory, especially on various operators, which includes convergence
of operators to functions in real and complex domain. In the last twenty years q-calculus served
as a bridge between mathematics and physics. The field has expanded explosively, due to the fact
that applications of basic hypergeometric series to the diverse subjects of combinatorics, quantum
theory, number theory, statistical mechanics, are constantly being uncovered. Specially, the theory of
univalent functions can be newly described by using the theory of the q-calculus. In recent years,
such q-calculus operators as the fractional q-integral and fractional q-derivative operators were used
to construct several subclasses of analytic functions (see, for example, [?, ?, ?, ?]). In the present
paper we study the symmetric q-operator, and related problems involving univalent functions.
Let \scrA denote the class of functions of the form:
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
SUBCLASS OF k-UNIFORMLY STARLIKE FUNCTIONS . . . 1501
f(z) = z +
\infty \sum
n=2
anz
n, (1.2)
which are analytic in the open unit disk \BbbD =
\bigl\{
z \in \BbbC : | z| < 1
\bigr\}
. Also, let \scrS , \bfT be the subclasses of
\scrA consisting of functions which are univalent in \BbbD , and with negative coefficients, respectively. We
denote by \scrS \scrT (\alpha ) (0 \leq \alpha < 1) a subset of \scrS consisting of all functions starlike of order \alpha , i.e., such
that \Re (zf \prime (z)/f(z)) > \alpha , z \in \BbbD . When \alpha = 0 the class \scrS \scrT (\alpha ) becomes the class \scrS \scrT of functions
f that maps \BbbD onto a starlike domain with respect to the origin. By k-\scrS \scrT (\alpha ) we denote the class
of k-starlike functions of order \alpha , 0 \leq \alpha < 1, that is a class of function f, which satisfy a condition
\Re
\biggl(
zf \prime (z)
f(z)
\biggr)
> k
\bigm| \bigm| \bigm| \bigm| zf \prime (z)
f(z)
- 1
\bigm| \bigm| \bigm| \bigm| + \alpha , k \geq 0 (1.3)
(for details see [?] and [?]).
We remark here that the class of k-starlike functions of order \alpha is an extension of the relatively
more familiar class of k-starlike functions investigated earlier by Kanas et al. [? – ?, ?] (see also
[?, ?]). For the case k = 1 that class was studied by Rønning [?], and called there “a parabolic
class”. We mention here that the name k-uniformly starlike was incorrectly attributed to the class
of k-starlike functions defined by (1.3) (for \alpha = 0), and related to the class of k-uniformly convex
functions by the well known Alexander relation. A class of uniformly starlike functions is due to
Goodman [?] and was defined by the condition
\Re
\biggl(
(z - \zeta )f \prime (z)
f(z) - f(\zeta )
\biggr)
> 0, z, \zeta \in \BbbD ,
and is completely different that the class k-stalike functions.
Definition 1.4. Let 0 \leq k < \infty and 0 \leq \alpha < 1. By k-\widetilde \scrS \scrT q(\alpha ) we denote the class of functions
f \in \scrA satisfying the condition
\Re
\Biggl(
z( \widetilde Dqf)(z)
f(z)
\Biggr)
> k
\bigm| \bigm| \bigm| \bigm| \bigm| z( \widetilde Dqf)(z)
f(z)
- 1
\bigm| \bigm| \bigm| \bigm| \bigm| + \alpha , z \in \BbbD . (1.4)
We also set k-\widetilde \scrS \scrT -
q (\alpha ) = k-\widetilde \scrS \scrT q(\alpha ) \cap \bfT . We note that \mathrm{l}\mathrm{i}\mathrm{m}q\rightarrow 1 - k-\widetilde \scrS \scrT q(\alpha ) = k-\scrS \scrT (\alpha ).
Let \scrP be the Carathèodory class of functions with positive real part consisting of all functions p
analytic in \BbbD satisfying p(0) = 1, and \Re (p(z)) > 0. Making use of a properties of the Carathèodory
functions we may rewrite a definition of k-\widetilde \scrS \scrT -
q (\alpha ). Setting p(z) =
z( \widetilde Dqf)(z)
f(z)
we may rewrite a
condition (1.4) in a form \Re p(z) > k| p(z) - 1| + \alpha (z \in \BbbD ), or p \prec pk,\alpha , where pk,\alpha is a function
with a positive real part, that maps the unit disk onto a domain \Omega k,\alpha , described by the inequality
\Re w > k| w - 1| +\alpha (here \prec denotes a symbol of a subordination of the analytic functions). We note
that \Omega k,\alpha is a domain bounded by a conic section, symmetric about real axis and contained in a right
half plane. It is also known that pk,\alpha has the real and positive coefficients (see [?, 13]). We will use
the notation pk,\alpha = 1 + P1z + P2z
2 + . . . .
It is known, that if p \in \scrP has a Taylor series expansion p(z) = 1 +B1z +B2z
2 +B3z
3 + . . . ,
then | Bn| \leq 2 for n \in \BbbN [?].
More refinement result was obtained by Grenander and Szegö [6].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1502 S. KANAS, Ş. ALTINKAYA, S. YALÇIN
Lemma 1.1 [6]. If the function p \in \scrP , then
2B2 = B2
1 + x(4 - B2
1),
4B3 = B3
1 + 2(4 - B2
1)B1x - B1(4 - B2
1)x
2 + 2(4 - B2
1)(1 - | x| 2)z
for some x, z with | x| \leq 1 and | z| \leq 1.
2. Fundamental properties. Several new subclasses of the families of k-starlike and k-uni-
formly convex functions making use of linear operators and fractional calculus were studied (see, for
example, [?, ?]), and various interesting properties were obtained. In light of this, it is of interest to
consider the behaviour of the classes k-\widetilde \scrS \scrT q(\alpha ) and k-\widetilde \scrS \scrT -
q (\alpha ) defined by symmetric q-derivative
operator. We provide necessary and sufficient coefficient conditions, distortion bounds, and extreme
points. In the first theorems we provide a necessary and a necessary and sufficient conditions to be a
member of k-\widetilde \scrS \scrT q(\alpha ) and k-\widetilde \scrS \scrT -
q (\alpha ), respectively.
Theorem 2.1. Let 0 < q < 1, and f \in \scrS be given by (1.2). If the inequality
\infty \sum
n=2
\Bigl[ \widetilde [n]q(k + 1) - (k + \alpha )
\Bigr]
| an| \leq 1 - \alpha (2.1)
holds true for some k, 0 \leq k < \infty and \alpha , 0 \leq \alpha < 1, then f \in k-\widetilde \scrS \scrT q(\alpha ).
Proof. By a Definition 1.4, it suffices to prove that
k
\bigm| \bigm| \bigm| \bigm| \bigm| z( \widetilde Dqf)(z)
f(z)
- 1
\bigm| \bigm| \bigm| \bigm| \bigm| - \Re
\Biggl(
z( \widetilde Dqf)(z)
f(z)
- 1
\Biggr)
< 1 - \alpha .
Observe that
k
\bigm| \bigm| \bigm| \bigm| \bigm| z( \widetilde Dqf)(z)
f(z)
- 1
\bigm| \bigm| \bigm| \bigm| \bigm| - \Re
\Biggl(
z( \widetilde Dqf)(z)
f(z)
- 1
\Biggr)
\leq (k + 1)
\bigm| \bigm| \bigm| \bigm| \bigm| z( \widetilde Dqf)(z)
f(z)
- 1
\bigm| \bigm| \bigm| \bigm| \bigm| =
= (k + 1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum \infty
n=2
\Bigl( \widetilde [n]q - 1
\Bigr)
anz
n - 1
1 +
\sum \infty
n=2
anz
n - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (k + 1)
\sum \infty
n=2
\Bigl( \widetilde [n]q - 1
\Bigr)
| an|
1 -
\sum \infty
n=2
| an|
.
The last expression is bounded by 1 - \alpha , if the inequality (2.1) holds.
Theorem 2.1 is proved.
The inequality (2.1) gives a tool to obtain some special members k-\widetilde \scrS \scrT q(\alpha ). For example, we
have the following corollary.
Corollary 2.1. Let 0 \leq k < \infty , 0 < q < 1 and 0 \leq \alpha < 1. If , for f(z) = z + anz
n, the
inequality
| an| \leq
1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
, n \geq 2,
holds, then f \in k-\widetilde \scrS \scrT q(\alpha ). Specially f(z) = z +
(1 - \alpha )q
(q2 + 1)(k + 1) - q(k + \alpha )
z2 \in k-\widetilde \scrS \scrT q(\alpha ).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
SUBCLASS OF k-UNIFORMLY STARLIKE FUNCTIONS . . . 1503
Theorem 2.2. Let 0 \leq k < \infty , 0 < q < 1, and 0 \leq \alpha < 1. A necessary and sufficient condition
for f of the form f(z) = z - a2z
2 - . . . , an \geq 0, to be in the class k-\widetilde \scrS \scrT -
q (\alpha ) is that
\infty \sum
n=2
\Bigl[ \widetilde [n]q(k + 1) - (k + \alpha )
\Bigr]
an \leq 1 - \alpha . (2.2)
The result is sharp, equality holds for the function f given by
f(z) = z - 1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn.
Proof. In view of Theorem 2.1, we need only to prove the necessity. If f \in k-\widetilde \scrS \scrT -
q (\alpha ), then
by | \Re (z)| \leq | z| for any z we get\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1 -
\sum \infty
n=2
\widetilde [n]qanzn - 1
1 -
\sum \infty
n=2
anz
n - 1
- \alpha
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq k
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum \infty
n=2
\Bigl( \widetilde [n]q - 1
\Bigr)
anz
n - 1
1 +
\sum \infty
n=2
anz
n - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| . (2.3)
Choose values of z on the real axis so that \widetilde Dqf(z) is real. Upon clearing the dominator of (2.3) and
letting z \rightarrow 1 - through the real values, we obtain (2.2).
Theorem 2.2 is proved.
Theorem 2.3. Let 0 \leq k < \infty , 0 < q < 1 and 0 \leq \alpha < 1. Let the function f defined by
f(z) = z - a2z
2 - . . . , an \geq 0, be in the class k-\widetilde \scrS \scrT -
q (\alpha ). Then for | z| = r < 1 it holds
r -
q
\bigl(
1 - \alpha
\bigr) \bigl(
q2 + 1
\bigr)
(k + 1) - q(k + \alpha )
r2 \leq
\bigm| \bigm| f(z)\bigm| \bigm| \leq r +
q(1 - \alpha )\bigl(
q2 + 1
\bigr)
(k + 1) - q(k + \alpha )
r2. (2.4)
Equality in (2.4) holds true for the function f given by
f(z) = z +
q(1 - \alpha )
(q2 + 1)(k + 1) - q(k + \alpha )
z2. (2.5)
Proof. Since f \in k-\widetilde \scrS \scrT -
q (\alpha ), then in view of Theorem 2.2, we have\Bigl[ \widetilde [2]q(k + 1) - (k + \alpha )
\Bigr] \infty \sum
n=2
an \leq
\infty \sum
n=2
\Bigl[ \widetilde [n]q(k + 1) - (k + \alpha )
\Bigr]
| an| \leq 1 - \alpha ,
which gives
\infty \sum
n=2
an \leq 1 - \alpha \widetilde [2]q(k + 1) - (k + \alpha )
. (2.6)
Therefore
| f(z)| \leq | z| +
\infty \sum
n=2
an | z| n \leq r +
q (1 - \alpha )
(q2 + 1) (k + 1) - q(k + \alpha )
r2
and
| f(z)| \geq | z| -
\infty \sum
n=2
an | z| n \geq r - q (1 - \alpha )
(q2 + 1) (k + 1) - q(k + \alpha )
r2.
The results follows by letting r \rightarrow 1 - .
Theorem 2.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1504 S. KANAS, Ş. ALTINKAYA, S. YALÇIN
Theorem 2.4. Let 0 \leq k < \infty , 0 < q < 1 and 0 \leq \alpha < 1. Let the function f with the Taylor
series f(z) = z - a2z
2 - . . . , an \geq 0, be a member of the class k-\widetilde \scrS \scrT -
q (\alpha ). Then for | z| = r < 1
1 - 2q (1 - \alpha )
(q2 + 1) (k + 1) - q(k + \alpha )
r \leq
\bigm| \bigm| f \prime (z)
\bigm| \bigm| \leq 1 +
2q (1 - \alpha )
(q2 + 1) (k + 1) - q(k + \alpha )
r. (2.7)
Proof. Differentiating f and using triangle inequality for the modulus, we obtain
\bigm| \bigm| f \prime (z)
\bigm| \bigm| \leq 1 +
\infty \sum
n=2
nan | z| n - 1 \leq 1 + r
\infty \sum
n=2
nan (2.8)
and \bigm| \bigm| f \prime (z)
\bigm| \bigm| \geq 1 -
\infty \sum
n=2
nan| z| n - 1 \geq 1 - r
\infty \sum
n=2
nan. (2.9)
The assertion (2.7) now follows from (2.8) and (2.9) by means of a rather simple consequence of (2.6)
given by
\infty \sum
n=2
nan \leq 2(1 - \alpha )\widetilde [2]q(k + 1) - (k + \alpha )
.
Theorem 2.4 is proved.
Theorem 2.5. Let 0 \leq k < \infty , 0 < q < 1, 0 \leq \alpha < 1, and set
f1(z) = z, fn(z) = z - 1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn, n = 2, 3, . . . .
Then f \in k-\widetilde \scrS \scrT -
q (\alpha ) if and only if f can be expressed in the form
f(z) =
\infty \sum
n=1
\lambda nfn(z), \lambda n > 0,
\infty \sum
n=1
\lambda n = 1.
Proof. Suppose that
f(z) =
\infty \sum
n=1
\lambda nfn(z) = \lambda 1f1(z) +
\infty \sum
n=2
\lambda nfn(z) =
= \lambda 1f1(z) +
\infty \sum
n=2
\lambda n
\Biggl[
z - 1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn
\Biggr]
=
= \lambda 1z +
\infty \sum
n=2
\lambda nz -
\infty \sum
n=2
\lambda n
1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn =
=
\Biggl( \infty \sum
n=1
\lambda n
\Biggr)
z -
\infty \sum
n=2
\lambda n
1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn =
= z -
\infty \sum
n=2
\lambda n
1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
SUBCLASS OF k-UNIFORMLY STARLIKE FUNCTIONS . . . 1505
Then
\infty \sum
n=2
\lambda n
1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
\widetilde [n]q(k + 1) - (k + \alpha )
1 - \alpha
=
\infty \sum
n=2
\lambda n =
\infty \sum
n=1
\lambda n - \lambda 1 = 1 - \lambda 1 \leq 1,
and we have f \in k-\widetilde \scrS \scrT -
q (\alpha ).
Conversely, suppose that f \in k-\widetilde \scrS \scrT -
q (\alpha ). Since | an| \leq (1 - \alpha )/
\bigl[ \widetilde [n]q(k + 1) - (k + \alpha )
\bigr]
, we
may set
\lambda n =
\widetilde [n]q(k + 1) - (k + \alpha )
1 - \alpha
| an| and \lambda 1 = 1 -
\infty \sum
n=2
\lambda n.
Then
f(z) = z +
\infty \sum
n=2
anz
n = z +
\infty \sum
n=2
\lambda n
1 - \alpha \widetilde [n]q(k + 1) - (k + \alpha )
zn =
= z +
\infty \sum
n=2
\lambda n(z + fn(z)) = z +
\infty \sum
n=2
\lambda nz +
\infty \sum
n=2
\lambda nfn(z) =
=
\Biggl(
1 -
\infty \sum
n=2
\lambda n
\Biggr)
z +
\infty \sum
n=2
\lambda nfn(z) = \lambda 1z +
\infty \sum
n=2
\lambda nfn(z) =
\infty \sum
n=1
\lambda nfn(z).
Theorem 2.5 is proved.
3. Hankel determinant. Let n and s be the natural numbers, such that n \geq 0 and s \geq 1.
In 1976 Noonan and Thomas [?] defined the sth Hankel determinant of f as
Hs(n) =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
an an+1 \cdot \cdot \cdot an+s - 1
an+1 an+2 \cdot \cdot \cdot an+s
...
...
...
...
an+s - 1 an+s \cdot \cdot \cdot an+2s - 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
(a1 = 1). (3.1)
This determinant has been considered by several authors. For example, Noor [?] determined the rate
of growth of Hs(n) as n \rightarrow \infty for functions f given by (1.2) with bounded boundary. In particular,
sharp upper bounds on H2(2), known as a second Hankel determinant, were obtained in [?, ?] for
different classes of functions.
Note that
H2(1) =
\bigm| \bigm| \bigm| \bigm| \bigm| a1 a2
a2 a3
\bigm| \bigm| \bigm| \bigm| \bigm| = a3 - a22, H2(2) =
\bigm| \bigm| \bigm| \bigm| \bigm| a2 a3
a3 a4
\bigm| \bigm| \bigm| \bigm| \bigm| = a2a4 - a23,
and the first Hankel determinant H2(1) = a3 - a22 is known as a special case of the Fekete –
Szegö functional.
In this section will look more closely at the behaviour of the first and second Hankel determinant
in the class k-\widetilde \scrS \scrT q(\alpha ), additionally we find a bound of the Fekete – Szegö functional and, as a special
case, we obtain a bound of | H2(1)| . For convenience, in the sequel we use the abbreviations
q2 = \widetilde [2]q - 1, q3 = \widetilde [3]q - 1, q4 = \widetilde [4]q - 1, where 0 < q < 1.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1506 S. KANAS, Ş. ALTINKAYA, S. YALÇIN
Theorem 3.1. Let 0 \leq k < \infty , 0 < q < 1, 0 \leq \alpha < 1, and let f \in k-\widetilde \scrS \scrT q(\alpha ).
1. If
U - P1q2(q2q4 - 1) \leq 0, V - P 2
1 q
2
2q4 \leq 0,
then the second Hankel determinant satisfies
\bigm| \bigm| a2a4 - a23
\bigm| \bigm| \leq P 2
1
q23
.
2. If
U - P1q2(q2q4 - 1) \geq 0, 2S - U - P 2
1 q2(1 + q2q4) \geq 0,
or
U - P1q2(q2q4 - 1) \leq 0, V - P 2
1 q
2
2q4 \geq 0,
then the second Hankel determinant satisfies\bigm| \bigm| a2a4 - a23
\bigm| \bigm| \leq V
q22q
2
3q4
.
3. If
U - P1q2(q2q4 - 1) > 0, 2V - U - P 2
1 q2(1 + q2q4) \leq 0,
then \bigm| \bigm| a2a4 - a23
\bigm| \bigm| \leq 4P 2
1 q
2
2q4V - 2P 2
1 q2(1 + q2q4)U - U2 - P 4
1 q
2
2(1 + q2q4)
2
4
\bigl(
V - U - P 2
1 q2
\bigr)
q22q
2
3q4
,
where U, V, and M, N, S are given by
U = | M + 2P 2
1 q2 + 2P1q2q4S| , V = | M +N + P 2
1 q2 - q4S
2 + 2P1q2q4S| ,
N = P1q3
\bigl[
P 3
1 + (P3 - 2P2)q2q3 + P1(P2 - P1)(q2 + q3) + P1q2q3
\bigr]
, (3.2)
M = P1q3
\bigl[
2q2q3(P2 - P1) + P 2
1 (q2 + q3)
\bigr]
, S = P 2
1 + q2(P2 - P1).
Proof. Let f \in k-\widetilde \scrS \scrT q(\alpha ). Then, there exists a Schwarz function w, w(0) = 1, | w(z)| < 1 for
z \in \BbbD , such that
z( \widetilde Dqf)(z)
f(z)
= pk,\alpha (w(z)).
Let
p0(z) =
1 + w(z)
1 - w(z)
= 1 +B1z +B2z
2 + . . . , (3.3)
or, equivalently,
w(z) =
p0(z) - 1
p0(z) + 1
=
1
2
\biggl(
B1z +
\biggl(
B2 -
B2
1
2
\biggr)
z2 + . . .
\biggr)
.
Such function p0 is analytic in the unit disk, and has a positive real part there. By using the Taylor
expansion of pk,\alpha and w, we obtain
pk,\alpha (w(z)) = 1 +
P1B1
2
z +
\biggl(
P1B2
2
+
B2
1(P2 - P1)
4
\biggr)
z2+
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
SUBCLASS OF k-UNIFORMLY STARLIKE FUNCTIONS . . . 1507
+
\biggl(
P1B3 + (P2 - P1)B1B2
2
+
B3
1(P3 + P1)
8
- P2B
3
1
4
\biggr)
z3 + . . . . (3.4)
Since
z( \widetilde Dqf)(z)
f(z)
= 1 + q2a2z +
\bigl[
q3a3 - q2a
2
2
\bigr]
z2 +
\bigl[
q4a4 - (q2 + q3)a2a3 + q2a
3
2
\bigr]
z3 + . . . ,
then, combining (3.3) with (3.4), we have
a2 =
P1B1
2q2
, a3 =
1
4q2q3
\bigl[
P 2
1B
2
1 - P1B
2
1q2 + P2B
2
1q2 + 2P1B2q2
\bigr]
,
(3.5)
a4 =
B3
1
\bigl(
P 3
1 + (P3 - 2P2 + P1)q2q3 + P1(P2 - P1)(q2 + q3)
\bigr)
8q2q3q4
+
+
2B1B2
\bigl(
P 2
1 (q2 + q3) + 2q2q3(P2 - P1)
\bigr)
+ 4B3P1q2q3
8q2q3q4
.
From the above we find that
H2(2) = a2a4 - a23 =
B4N + (2B2)B
2M + (4B3)BP 2
1 q2q
2
3 -
\bigl[
(2B2)P1q2 +B2S
\bigr] 2
q4
16q22q
2
3q4
,
where, without loss of generality, we set B := B1 > 0, and N, M, S are given by (3.2). Applying
Lemma 1.1 and performing the necessary computations, we obtain
H2(2) =
B4
\bigl[
N +M + P 2
1 q2 - q4S
2 + 2P1q2q4S
\bigr]
+ xB2(4 - B2)
\bigl[
M + 2P 2
1 q2 - 2P1q2q4S
\bigr]
16q22q
2
3q4
+
+
- x2(4 - B2)
\bigl[
B2P 2
1 q2 + 4P 2
1 q
2
2q4
\bigr]
+ 2B(4 - B2)(1 - | x| 2)zP 2
1 q2q
2
3
16q22q
2
3q4
.
Set now \rho = | x| , where \leq \rho \leq 1, and take an absolute value of H2(2). Applying additionally
| z| \leq 1, we have | H2(2)| \leq \Phi (\rho ,B) = W (\alpha \rho 2 + \beta \rho + \gamma ), where
\alpha = (4 - B2)
\bigl[
B2P 2
1 q2 + 4P 2
1 q
2
2q4
\bigr]
- 2B(4 - B2)2P 2
1 q2q
2
3,
\beta = B2(4 - B2)
\bigm| \bigm| M + 2P 2
1 q2 + 2P1q2q4S
\bigm| \bigm| ,
\gamma = 2B(4 - B2)P 2
1 q2q
2
3 +B4
\bigm| \bigm| N +M + P 2
1 q2 - q4S
2 + 2P1q2q4S
\bigm| \bigm| ,
and W = 1/(16q22q
2
3q4). We note that \alpha \geq 0, \beta \geq 0. Indeed, an inequality \beta \geq 0 is obvious, and
we get \alpha = (4 - B2)P 2
1 q2
\bigl[
B2 - 2Bq23 +4q2q4
\bigr]
. The expression in a square brackets \Psi (B) = B2 -
- 2Bq23+4q2q4 is a quadratic function of B (0 \leq B \leq 2) with roots at B = 2, and B = 2(q23 - 1) >
> 2. Since \Psi (0) = 4q2q4 > 0, then \Psi (B) > 0 for 0 \leq B \leq 2. Hence \partial \Phi /\partial \rho = W (2\alpha \rho + \beta ) \geq 0,
and from this fact we conclude that \Phi is increasing function of \rho . Therefore, for fixed B \in [0, 2],
the maximum of \Phi (\rho ,B) is attained at \rho = 1, that is \mathrm{m}\mathrm{a}\mathrm{x}\Phi (\rho ,B) = \Phi (1, B) =: G(B). We note
that
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1508 S. KANAS, Ş. ALTINKAYA, S. YALÇIN
G(B) =
1
16q22q
2
3q4
\Biggl(
B4
\Bigl[ \bigm| \bigm| M +N + P 2
1 q2 - q4S
2 + 2P1q2q4S
\bigm| \bigm| -
-
\bigm| \bigm| M + 2P 2
1 q2 + 2P1q2q4S
\bigm| \bigm| - P 2
1 q2
\Bigr]
+B2
\Bigl[
4
\bigm| \bigm| M + 2P 2
1 q2 + 2P1q2q4S
\bigm| \bigm| +
+4P 2
1 q2(1 - q2q4)
\Bigr]
+ 16P 2
1 q
2
2q4
\Biggr)
.
Let
P =
\bigm| \bigm| M +N + P 2
1 q2 - q4S
2 + 2P1q2q4S
\bigm| \bigm| - \bigm| \bigm| M + 2P 2
1 q2 + 2P1q2q4S
\bigm| \bigm| - P 2
1 q2,
Q = 4
\bigm| \bigm| M + 2P 2
1 q2 + 2P1q2q4S
\bigm| \bigm| + 4P 2
1 q2(1 - q2q4), (3.6)
R = 16P 2
1 q
2
2q4.
Now, analyzing the maximum of a Pt2 +Qt+R, over 0 \leq t \leq 4, we conclude that
| H2(2)| \leq
1
16q22q
2
3q4
\left\{
R for Q \leq 0, P \leq - Q/4,
16P + 4Q+R for Q \geq 0, P \geq - Q/8 or Q \leq 0, P \geq - Q/4,
R - Q2/(4P ) for Q > 0, P \leq - Q/8,
where P, Q, R are given by (3.6).
Theorem 3.1 is proved.
Corollary 3.1. Let q \rightarrow 1 - . Then k-\widetilde \scrS \scrT q(\alpha ) \rightarrow k-\scrS \scrT (\alpha ), for which P1 =
8
\pi 2
. Then we get
\bigm| \bigm| a2a4 - a23
\bigm| \bigm| \leq 16
\pi 2
.
Theorem 3.2. Let 0 \leq k < \infty , 0 < q < 1, 0 \leq \alpha < 1, and let f \in k-\widetilde \scrS \scrT q(\alpha ). Then for
complex \mu it holds \bigm| \bigm| a3 - \mu a22
\bigm| \bigm| \leq P 2
1 | q2 - \mu q3| + P2q
2
2
q22q3
.
In the case, when \mu is real, then
\bigm| \bigm| a3 - \mu a22
\bigm| \bigm| \leq
\left\{
P2q
2
q4 + 1
+ P 2
1 q
2 q(q
2 - q + 1) - \mu (q4 + 1)
(q4 + 1)(q2 - q + 1)2
for \mu \leq q(q2 - q + 1)
q4 + 1
,
P2q
2
q4 + 1
+ P 2
1 q
2\mu (q
4 + 1) - q(q2 - q + 1)
(q4 + 1)(q2 - q + 1)2
for \mu \geq q(q2 - q + 1)
q4 + 1
.
Proof. We apply a form of a2, a3, given by (3.5), and assume as in the proof of the first part
that B := B1 > 0. Then, for complex \mu , we have
a3 - \mu a22 =
B2
\bigl(
P 2
1 q2 + q22(P2 - P1) - \mu P 2
1 q3
\bigr)
+ (2B2)P1q
2
2
4q22q3
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
SUBCLASS OF k-UNIFORMLY STARLIKE FUNCTIONS . . . 1509
Making use of Lemma 1.1, we obtain
a3 - \mu a22 =
B2
\bigl(
P 2
1 q2 + q22(P2 - P1) - \mu P 2
1 q3
\bigr)
+ (B2 + x(4 - B2))P1q
2
2
4q22q3
,
where x is a complex number satisfying | x| \leq 1. Hence
a3 - \mu a22 =
B2
\bigl[
q2(P
2
1 + P2q2) - \mu P 2
1 q3
\bigr]
+ (4 - B2)P1q
2
2
4q22q3
.
After simplification and using B \leq 2, we get
| a3 - \mu a22| =
\bigm| \bigm| P 2
1 (q2 - \mu q3) + P2q
2
2
\bigm| \bigm|
q22q3
.
We note also that P1, P2 are nonnegative, and q2, q3 are positive real number, therefore
| a3 - \mu a22| =
P 2
1
\bigm| \bigm| q2 - \mu q3
\bigm| \bigm| + P2q
2
2
q22q3
,
that establishes our first assertion. For real \mu our claim is deduced by the observation that q2 =
= q + 1/q - 1, and q3 = q2 + 1/q2, where 0 < q < 1.
Theorem 3.2 is proved.
A trivial computation gives the bound for the first Hankel derivative, and for the third coefficient,
below.
Corollary 3.2. Let 0 \leq k < \infty , 0 < q < 1, 0 \leq \alpha < 1, and let f \in k-\widetilde \scrS \scrT q(\alpha ). Then the first
Hankel determinant satisfy
\bigm| \bigm| a3 - a22
\bigm| \bigm| \leq q2(P2 + P 2
1 q)
q4 + 1
- P 2
1 q
2
q2 - q + 1
.
Corollary 3.3. Under the assumption the same as in the Corollary 3.2 we have
| a3| \leq
q2
\bigl(
P2 + P 2
1 q
\bigr)
q4 + 1
.
References
1. Bharati R., Parvatham R., Swaminathan A. On subclasses of uniformaly convex functions and correspondding class
of starlike functions // Tamkang J. Math. – 1997. – 28. – P. 17 – 32.
2. Biedenharn L. C. The quantum group SUq(2) and a q-analogue of the boson operators // J. Phys. A. – 1984. – 22. –
P. L873 – L878.
3. Brahim K. L., Sidomou Y. On some symmetric q-special functions // La Mat. – 2013. – 68. – P. 107 – 122.
4. Gasper G., Rahman M. Basic hypergeometric series. – Cambridge, MA: Cambridge Univ. Press, 1990.
5. Goodman A. W. On uniformly starlike functions // J. Math. Anal. and Appl. – 1991. – 155. – P. 364 – 370.
6. Grenander U., Szegö G. Toeplitz forms and their applications // California Monographs Math. Sci. – Berkeley:
California Univ. Press, 1958.
7. Jackson F. H. On q-functions and a certain difference operator // Trans. Roy. Soc. Edinburgh. – 1908. – 46. –
P. 253 – 281.
8. Kanas S. Coefficient estimates in subclasses of the Caratheodory class related to conical domains // Acta Math. Univ.
Comenian. – 2005. – 74, № 2. – P. 149 – 161.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1510 S. KANAS, Ş. ALTINKAYA, S. YALÇIN
9. Kanas S., Srivastava H. M. Linear operators associated with k-uniformly convex functions // Integral Transforms
Spec. Funct. – 2000. – 9. – P. 121 – 132.
10. Kanas S., Yaguchi T. Subclasses of k-uniformly convex and starlike functions defined by generalized derivative //
Publ. Inst. Math. (Beograd) (N.S.). – 2001. – 69 (83). – P. 91 – 100.
11. Kanas S. Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions // Appl.
Math. and Comput. – 2009. – 215. – P. 2275 – 2282.
12. Kanas S., Raducanu D. Some class of analytic functions related to conic domains // Math. Slovaca. – 2014. – 64,
№ 5. – P. 1183 – 1196.
13. Kanas S., Sugawa T. Conformal representations of the interior of an ellipse // Ann. Acad. Sci. Fenn. Math. – 2006. –
31. – P. 329 – 348.
14. Kanas S., Wisniowska A. Conic regions and k-uniformly starlike functions // Rev. Roum. Math. Pures et Appl. –
2000. – 45, № 4. – P. 647 – 657.
15. Noonan J. W., Thomas D. K. On the second Hankel determinant of areally mean p-valent functions // Trans. Amer.
Math. Soc. – 1976. – 223, № 2. – P. 337 – 346.
16. Noor K. I. Hankel determinant problem for the class of functions with bounded boundary rotation // Rev. Roum.
Math. Pures et Appl. – 1983. – 28. – P. 731 – 739.
17. Hayami T., Owa S. Generalized Hankel determinant for certain classes // Int. J. Math. Anal. – 2010. – 52, № 4. –
P. 2473 – 2585.
18. Polatoğlu Y. Growth and distortion theorems for generalized q-starlike functions // Adv. Math. – 2016. – 5, № 1. –
P. 7 – 12.
19. Pommerenke C. Univalent functions. – Göttingen: Vandenhoeck & Ruprecht, 1975.
20. Purohit S. D., Raina R. K. Certain subclass of analytic functions associated with fractional q-calculus operators //
Math. Scand. – 2011. – 109. – P. 55 – 70.
21. Rønning F. A survey on uniformly convex and uniformly starlike functions // Ann. Univ. Mariae Curie-Skłodowska
Sect. A. – 1993. – 47, № 13. – P. 123 – 134.
22. Şeker B., Acu M., Sumer Eker S. Subclasses of k-uniformly convex and k-starlike functions defined by Salagean
operator // Bull. Korean Math. Soc. – 2011. – 48, № 1. – P. 169 – 182.
23. Srivastava H. M. Univalent functions, fractional calculus, and associated generalized hypergeometric functions //
Univalent Functions, Fractional Calculus and Their Applications / Eds H. M. Srivastava, S. Owa. – New York etc.:
John Wiley and Sons, 1989.
24. Srivastava H. M., Mishra A. K. Applications of fractional calculus to parabolic starlike and uniformly convex
functions // Comput. Math. Appl. – 2000. – 39, № 3-4. – P. 57 – 69.
25. Özkan Uçar H. E. Coefficient inequalties for q-starlike functions // Appl. Math. and Comput. – 2016. – 276. –
P. 122 – 126.
Received 16.09.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
|
| id | umjimathkievua-article-1653 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:56Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/66/0488b9177d9d284f06ea3da3054e5766.pdf |
| spelling | umjimathkievua-article-16532019-12-05T09:22:19Z Subclass of $k$-uniformly starlike functions defined by symmetric $q$-derivative operator Пiдклас $k$ -рiвномiрно зiркоподiбних функцiй, що визначенi за допомогою симетричного оператора $q$ -похiдної Altinkaya, S. Kanas, S. Yal¸cin, S. Альтінкая, С. Канас, С. Ялцин, С. The theory of $q$-analogs is frequently encountered in numerous areas, including fractals and dynamical systems. The $q$-derivatives and $q$-integrals play an important role in the study of $q$-deformed quantum-mechanical simple harmonic oscillators. We define a symmetric $q$-derivative operator and study a new family of univalent functions defined by using this operator. We establish some new relations between the functions satisfying analytic conditions related to conical sections. Теорiя $q$-аналогiв часто зустрiчається в багатьох галузях, включаючи фрактали та динамiчнi системи. Важливу роль у вивченнi $q$-деформованих квантово-механiчних простих гармонiчних осциляторiв вiдiграють $q$-похiднi та $q$-iнтеграли. Наведено визначення симетричного оператора $q$-похiдної та вивчено нову сiм’ю однолистих функцiй, що визначенi за допомогою цього оператора. Встановлено також деякi новi спiввiдношення мiж функцiями, що задовольняють аналiтичнi умови вiдносно конiчних перерiзiв. Institute of Mathematics, NAS of Ukraine 2018-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1653 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 11 (2018); 1499-1510 Український математичний журнал; Том 70 № 11 (2018); 1499-1510 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1653/635 Copyright (c) 2018 Altinkaya S.; Kanas S.; Yal¸cin S. |
| spellingShingle | Altinkaya, S. Kanas, S. Yal¸cin, S. Альтінкая, С. Канас, С. Ялцин, С. Subclass of $k$-uniformly starlike functions defined by symmetric $q$-derivative operator |
| title | Subclass of $k$-uniformly starlike functions defined by symmetric
$q$-derivative operator |
| title_alt | Пiдклас $k$ -рiвномiрно зiркоподiбних функцiй, що визначенi за допомогою симетричного оператора $q$ -похiдної |
| title_full | Subclass of $k$-uniformly starlike functions defined by symmetric
$q$-derivative operator |
| title_fullStr | Subclass of $k$-uniformly starlike functions defined by symmetric
$q$-derivative operator |
| title_full_unstemmed | Subclass of $k$-uniformly starlike functions defined by symmetric
$q$-derivative operator |
| title_short | Subclass of $k$-uniformly starlike functions defined by symmetric
$q$-derivative operator |
| title_sort | subclass of $k$-uniformly starlike functions defined by symmetric
$q$-derivative operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1653 |
| work_keys_str_mv | AT altinkayas subclassofkuniformlystarlikefunctionsdefinedbysymmetricqderivativeoperator AT kanass subclassofkuniformlystarlikefunctionsdefinedbysymmetricqderivativeoperator AT yalcins subclassofkuniformlystarlikefunctionsdefinedbysymmetricqderivativeoperator AT alʹtínkaâs subclassofkuniformlystarlikefunctionsdefinedbysymmetricqderivativeoperator AT kanass subclassofkuniformlystarlikefunctionsdefinedbysymmetricqderivativeoperator AT âlcins subclassofkuniformlystarlikefunctionsdefinedbysymmetricqderivativeoperator AT altinkayas pidklaskrivnomirnozirkopodibnihfunkcijŝoviznačenizadopomogoûsimetričnogooperatoraqpohidnoí AT kanass pidklaskrivnomirnozirkopodibnihfunkcijŝoviznačenizadopomogoûsimetričnogooperatoraqpohidnoí AT yalcins pidklaskrivnomirnozirkopodibnihfunkcijŝoviznačenizadopomogoûsimetričnogooperatoraqpohidnoí AT alʹtínkaâs pidklaskrivnomirnozirkopodibnihfunkcijŝoviznačenizadopomogoûsimetričnogooperatoraqpohidnoí AT kanass pidklaskrivnomirnozirkopodibnihfunkcijŝoviznačenizadopomogoûsimetričnogooperatoraqpohidnoí AT âlcins pidklaskrivnomirnozirkopodibnihfunkcijŝoviznačenizadopomogoûsimetričnogooperatoraqpohidnoí |