Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators
UDC 517.5 The purpose of the present paper is to establish several general results concerning the partial sums of meromorphically starlike functions defined here by means of a certain class of $q$-derivative (or $q$-difference) operators. The familiar concept of neighborhood for meromorphic function...
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2021
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| author | Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. B. N. |
| author_facet | Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. B. N. |
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| description | UDC 517.5
The purpose of the present paper is to establish several general results concerning the partial sums of meromorphically starlike functions defined here by means of a certain class of $q$-derivative (or $q$-difference) operators. The familiar concept of neighborhood for meromorphic functions are also considered. Moreover, by using a Ruscheweyh-type $q$-derivative operator, we define and study another new class of functions emerging from the class of normalized meromorphic functions.
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| doi_str_mv | 10.37863/umzh.v73i9.814 |
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DOI: 10.37863/umzh.v73i9.814
UDC 517.5
H. M. Srivastava (Univ. Victoria, British Columbia, Canada and China Medical Univ., Taiwan),
M. Tahir (Abbottabad Univ. Sci. and Technology, Pakistan),
B. Khan (School Math. Sci. and Shanghai Key Laboratory PMMP, East China Normal Univ., Shanghai, China),
M. Darus (School Math. Sci., Univ. Kebangsaan Malaysia, Bangi, Selangor, Malaysia),
N. Khan (Abbottabad Univ. Sci. and Technology, Pakistan),
Q. Z. Ahmad (Covernment Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur, Pakistan)
CERTAIN SUBCLASSES OF MEROMORPHICALLY \bfitq -STARLIKE FUNCTIONS
ASSOCIATED WITH THE \bfitq -DERIVATIVE OPERATORS*
ДЕЯКI ПIДКЛАСИ МЕРОМОРФНИХ \bfitq -ЗIРКОВИХ ФУНКЦIЙ,
ПОВ’ЯЗАНI З \bfitq -ПОХIДНИМИ ОПЕРАТОРАМИ
The purpose of the present paper is to establish several general results concerning the partial sums of meromorphically
starlike functions defined here by means of a certain class of q-derivative (or q-difference) operators. The familiar concept of
neighborhood for meromorphic functions are also considered. Moreover, by using a Ruscheweyh-type q-derivative operator,
we define and study another new class of functions emerging from the class of normalized meromorphic functions.
Метою цiєї статтi є отримання кiлькох загальних результатiв, що пов’язанi з частковими сумами мероморфних
зiркових функцiй, якi визначаються за допомогою деякого класу q-похiдних (або q-рiзницевих) операторiв. Також
розглянуто вiдоме поняття околу для мероморфних функцiй. Крiм того, за допомогою q-похiдного оператора типу
Рушевая визначається та вивчається новий клас функцiй, який виводиться з класу нормалiзованих мероморфних
функцiй.
1. Introduction and definition. Let the class of functions f which are analytic in the open unit disk
\BbbU =
\bigl\{
z : z \in \BbbC and | z| < 1
\bigr\}
be denoted by \scrH (\BbbU ). Also, by \scrA we denote the subclass of the analytic functions f in \scrH (\BbbU )
satisfying the following normalization condition:
f(0) = f \prime (0) - 1 = 0.
Equivalently, the function f \in \scrA has the Taylor – Maclaurin series expansion given by
f(z) = z +
\infty \sum
n=2
anz
n \forall z \in \BbbU . (1.1)
Let \scrS be the subclass of analytic function class \scrA , consisting of all univalent functions in \BbbU .
A function f \in \scrA is said to be starlike in \BbbU , if it satisfies the following inequality:
\Re
\biggl(
zf \prime (z)
f(z)
\biggr)
> 0 \forall z \in \BbbU ,
where, for example, \Re (z) denotes the real part of z \in \BbbC .We denote by \scrS \ast all such starlike functions
in the open unit disk \BbbU .
* This paper was partially supported by UKM Grant GUP-2017-064.
c\bigcirc H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD, 2021
1260 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
CERTAIN SUBCLASSES OF MEROMORPHICALLY q-STARLIKE FUNCTIONS . . . 1261
For any two functions f and g, which are analytic in \BbbU , we say that the function f is subordinate
to g, written as follows:
f \prec g or f(z) \prec g(z),
if there is a Schwarz function w, which is analytic in \BbbU with
w(0) = 0 and
\bigm| \bigm| w(z)\bigm| \bigm| < 1,
such that
f(z) = g
\bigl(
w(z)
\bigr)
.
Furthermore, for the function g, which is univalent in \BbbU , it follows that
f(z) \prec g(z) (z \in \BbbU ) \Leftarrow \Rightarrow f(0) = g(0) and f(\BbbU ) \subset g(\BbbU ).
Next, for a function f \in \scrA given by (1.1) and another function g \in \scrA given by
g(z) = z +
\infty \sum
n=2
bnz
n \forall z \in \BbbU ,
the convolution (or the Hadamard product) of the functions f and g is defined by
(f \ast g)(z) = z +
\infty \sum
n=2
anbnz
n = (g \ast f)(z) \forall z \in \BbbU .
Let \scrP denote the class of analytic functions p normalized by
p(z) = 1 +
\infty \sum
n=1
cnz
n
such that
\Re \{ p(z)\} > 0 \forall z \in \BbbU .
We now recall some essential definitions and concept details of the q-calculus, which are used in
this paper. We suppose throughout this paper that 0 < q < 1 and that
\BbbN = \{ 1, 2, 3, . . .\} = \BbbN \setminus \{ 0\} , \BbbN 0 = \{ 0, 1, 2, . . .\} .
Definition 1. Let q \in (0, 1) and define the q-number [\lambda ]q by
[\lambda ]q =
\left\{
1 - q\lambda
1 - q
, \lambda \in \BbbC ,\sum n - 1
k=0
qk = 1 + q + q2 + . . .+ qn - 1, \lambda = n \in \BbbN .
Definition 2. Let q \in (0, 1) and define the q-factorial [n]q! by
[n]q! =
\left\{ 1, n = 0,\prod n - 1
k=1
[k]q, n \in \BbbN .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1262 H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD
Definition 3. Let q \in (0, 1) and define q-Pochhammer symbol [t]q,n, t \in \BbbC , n \in \BbbN 0, by
[t]q,n =
(qt; q)n
(1 - q)n
=
\left\{ 1, n = 0,
[t]q[t+ 1]q[t+ 2]q . . . [t+ n - 1]q, n \in \BbbN .
Moreover, the q-gamma-function \Gamma q(z) may be defined here by the following recurrence relation:
\Gamma q(z + 1) = [z]q\Gamma q(z) and \Gamma q(1) = 1.
Definition 4 [20, 21]. The q-derivative (or the q-difference) (Dqf) of a function f is defined,
in a given subset of \BbbC , by
(Dqf)(z) =
\left\{
f(z) - f(qz)
(1 - q)z
, z \not = 0,
f \prime (0), z = 0,
(1.2)
provided that f \prime (0) exists.
We note from Definition 4 that
\mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
(Dqf)(z) = \mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
f(z) - f(qz)
(1 - q)z
= f \prime (z)
for a differentiable function f in a given subset of \BbbC . It is readily seen from (1.1) and (1.2) that
(Dqf)(z) = 1 +
\infty \sum
n=2
[n]qanz
n - 1.
A number of subclasses of the normalized analytic function class \scrA in Geometric Function
Theory have been studied already from different viewpoints (see, for example, [7, 8, 11, 12, 15]).
The above-defined q-calculus provides an important tool in order to investigate several subclasses of
the class \scrA . The q-derivative (or the q-difference) operator Dq was first used in Geometrc Function
Theory by Ismail et al. [19] in order to study the q-analogue of the class \scrS \ast of starlike functions in
\BbbU (see Definition 5 below). However, initial usages of the q-calculus in the context of Geometric
Function Theory were presented systematically, and the basic (or q-) hypergeometric functions were
first used in Geometric Function Theory, in a book chapter by Srivastava (see, for details, [31, p. 347]
and also [1, 2, 13, 14, 18, 24, 26, 33, 34, 36 – 38]).
Definition 5 [19]. A function f \in \scrA is said to belong to the class \scrS \ast
q of q-starlike functions if
f(0) = f \prime (0) - 1 = 0
and \bigm| \bigm| \bigm| \bigm| z(Dqf)(z)
f(z)
- 1
1 - q
\bigm| \bigm| \bigm| \bigm| \leq 1
1 - q
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
CERTAIN SUBCLASSES OF MEROMORPHICALLY q-STARLIKE FUNCTIONS . . . 1263
It is readily observed that, as q \rightarrow 1 - , the closed disk given by\bigm| \bigm| \bigm| \bigm| w - 1
1 - q
\bigm| \bigm| \bigm| \bigm| \leq 1
1 - q
becomes the right-half plane and the class \scrS \ast
q of q-starlike functions reduces to the familiar class \scrS \ast
of starlike functions in \BbbU .
We next let \scrM denote the class of functions f of the form
f(z) =
1
z
+
\infty \sum
n=0
anz
n, (1.3)
which are analytic in the punctured open unit disk
\BbbU \ast =
\bigl\{
z : z \in \BbbC and 0 < | z| < 1
\bigr\}
= \BbbU \setminus \{ 0\} .
A function f \in \scrM is said to be in the class \scrM \scrS (\alpha ) of meromorphically starlike functions of order
\alpha if it satisfies the following inequality:
- \Re
\biggl(
zf \prime (z)
f(z)
\biggr)
> \alpha \forall z \in \BbbU , 0 \leq \alpha < 1.
Next, analogous to Definition 4, we extend the notion of the q-derivative (or the q-difference)
operator Dq to a function f given by (1.3) from the above-defined class \scrM and also introduce the
class \scrM \scrS q(\alpha ). Indeed, for f \in \scrM given by (1.3), the q-derivative (or the q-difference) Dqf is
given by
(Dqf)(z) =
f(z) - f(qz)
(1 - q)z
= - 1
qz2
+
\infty \sum
n=0
[n]qanz
n - 1 \forall z \in \BbbU \ast .
Definition 6. A function f \in \scrM is said to be in the class \scrM \scrS q(\alpha ) 0 \leq \alpha < 1 if it satisfies the
following condition: \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\biggl(
- z(Dqf)(z)
f(z)
\biggr)
- \alpha
1 - \alpha
- 1
1 - q
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
1
1 - q
. (1.4)
Throughout this paper, we use the notation \scrM \scrS q(\alpha ) for the class of meromorphically q-starlike
functions of order \alpha .
Remark 1. It is easily seen that
\mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
\scrM \scrS q(\alpha ) =: \scrM \scrS (\alpha ) and \mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
\scrM \scrS q(0) =: \scrM \scrS ,
where \scrM \scrS is the function class which was introduced and studied by Clunie (see [10]).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1264 H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD
Since the work in the meromorphically univalent case has parallel to that of the analytically
univalent case, one is tempted to search for results analogous to those of Silverman [30] for mero-
morphically univalent functions in \BbbU \ast . Thus, in this paper, we are motivated essentially by the works
[4, 9, 16, 29, 30] (see also [23, 25, 27, 35]). We propose to investigate the ratio of a function of the
form (1.3) to its sequence of partial sums given by
fk(z) =
1
z
+
k\sum
n=0
anz
n, k \in \BbbN , (1.5)
when the coefficients are sufficiently small. We will determine sharp lower bounds for
\Re
\biggl(
f(z)
fk(z)
\biggr)
, \Re
\biggl(
fk(z)
f(z)
\biggr)
, \Re
\biggl(
(Dqf)(z)
(Dqfk)(z)
\biggr)
and \Re
\biggl(
(Dqfk)(z)
(Dqf)(z)
\biggr)
.
Furthermore, in this paper, we introduce the (\xi , q)-neighborhood of a function f \in \scrM of the
form (1.3) by means of the following definition.
Definition 7. For \xi \geq 0, 0 \leq \alpha < 1 and f \in \scrM given by (1.3), we define the (\xi , q)-
neighborhood of the function f by
N(\xi ,q)(f) =
\Biggl\{
g : g \in \scrM , g(z) =
1
z
+
\infty \sum
n=0
bnz
n and
\infty \sum
n=0
\scrL (n, q, \alpha )| ak - bk| \leq \xi
\Biggr\}
, (1.6)
where
\scrL (n, q, \alpha ) =
\bigl(
2[n]q + (1 + q)\alpha
\bigr)
q
q - 1 + (1 + q)(1 - \alpha q)
, n \in \BbbN 0. (1.7)
2. Main results and their demonstration. First of all, we give a sufficient condition for a
function f \in \scrM of the form (1.3) to be in the class \scrM \scrS q(\alpha ).
Theorem 1. Let
1
q
- \alpha > 0.
Suppose also that the function f \in \scrM is given by (1.3). If
\infty \sum
n=0
\bigl(
[n]q + \alpha
\bigr)
| an| \leq
1
q
- \alpha , (2.1)
then f \in \scrM \scrS q(\alpha ).
Proof. Let f \in \scrM . Then, from (1.4) we have\bigm| \bigm| \bigm| \bigm| z(Dqf)(z)
f(z)
+
1 - \alpha q
1 - q
\bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
1
qz
+
\sum \infty
n=0
[n]qanz
n +
1 - \alpha q
1 - q
\biggl(
1
z
+
\sum \infty
n=0
anz
n
\biggr)
1
z
+
\sum \infty
n=0
anz
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
CERTAIN SUBCLASSES OF MEROMORPHICALLY q-STARLIKE FUNCTIONS . . . 1265
\leq
1 - \alpha q
1 - q
- 1
q
+
\sum \infty
n=0
\biggl(
[n]q +
1 - \alpha q
1 - q
\biggr)
| an|
1 +
\sum \infty
n=0
| an|
. (2.2)
This last expression in (2.2) is bounded above by
1 - \alpha
1 - q
if the condition (2.1) is satisfied.
Theorem 1 is proved.
Our next result is based upon Definition 7.
Theorem 2. Let \xi \geq 0 and f \in \scrM given by (1.3) satisfy the following condition:
f(z) + \varepsilon z - 1
1 + \varepsilon
\in \scrM \scrS q(\alpha ) (2.3)
for any complex number \varepsilon such that | \varepsilon | < \xi . Then
N(\xi ,q)(f) \subset \scrM \scrS q(\alpha ). (2.4)
Proof. By noting that the condition (1.4) can be written as follows:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
z(Dq)f(z)
f(z)
+ 1
z(Dqf)(z)
f(z)
+ (1 + q)\alpha - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < 1, (2.5)
it is easy to see from the condition (2.5) that g(z) \in \scrM \scrS q(\alpha ) if and only if
z(Dqg)(z) + g(z)
z(Dqg)(z) +
\bigl(
(1 + q)\alpha - 1
\bigr)
g(z)
\not = \sigma \forall z \in \BbbU , \sigma \in \BbbC , | \sigma | = 1,
which is equivalent to
(g \ast h)(z)
z - 1
\not = 0 \forall z \in \BbbU . (2.6)
The function h(z), which is involved in (2.6), is given by
h(z) =
1
z
+
\infty \sum
n=0
\Upsilon nz
n
and
\Upsilon n :=
\bigl(
[n]q + 1 -
\bigl(
[n]q + (1 + q)\alpha - 1
\bigr)
\sigma
\bigr)
q
q - 1 + (1 + q)(1 - \alpha q)\sigma
. (2.7)
It follows from (2.7) that
| \Upsilon n| =
\bigm| \bigm| \bigm| \bigm| \bigm| [n]q + 1 -
\bigl( \bigl(
[n]q + (1 + q)\alpha - 1
\bigr)
\sigma
\bigr)
q
q - 1 + (1 + q)(1 - \alpha q)\sigma
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigl(
[n]q + 1 +
\bigl(
[n]q + (1 + q)\alpha - 1
\bigr)
| \sigma |
\bigr)
q
q - 1 + (1 + q)(1 - \alpha q)| \sigma |
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1266 H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD
=
(2[n]q + (1 + q)\alpha )q
q - 1 + (1 + q)(1 - \alpha q)
=: \scrL (n, q, \alpha ), | \sigma | = 1, n \in \BbbN 0.
Now, if f \in \scrM given by (1.3) satisfies the condition (2.3), we deduce from (2.6) that
(g \ast h)(z)
z - 1
\not = \varepsilon | \varepsilon | < \xi , \xi > 0, (2.8)
or, equivalently, the condition in (2.8) can be written as follows:\bigm| \bigm| \bigm| \bigm| (g \ast h)(z)z - 1
\bigm| \bigm| \bigm| \bigm| \geq \xi , z \in \BbbU , \xi > 0. (2.9)
Next, if we suppose that
q(z) =
1
z
+
\infty \sum
n=0
dnz
n \in N(\xi ,q)(f),
it follows from (1.6) that \bigm| \bigm| \bigm| \bigm| \bigm|
\bigl(
(q - f) \ast h
\bigr)
(z)
z - 1
\bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
n=0
(dn - an)\Upsilon nz
n+1
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq | z|
\infty \sum
n=0
2[n]q + (1 + q)\alpha
q - 1 + (1 + q) (1 - \alpha q)
| dn - an| < \xi . (2.10)
Upon combining (2.9) and (2.10), we easily see that\bigm| \bigm| \bigm| \bigm| (q \ast h)(z)z - 1
\bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm|
\bigl( \bigl[
f + (q - f
\bigr]
\ast h
\bigr)
)(z)
z - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq
\bigm| \bigm| \bigm| \bigm| (f \ast h)(z)
z - 1
\bigm| \bigm| \bigm| \bigm| -
\bigm| \bigm| \bigm| \bigm| \bigm|
\bigl(
(q - f) \ast h
\bigr)
(z)
z - 1
\bigm| \bigm| \bigm| \bigm| \bigm| > 0. (2.11)
The inequality in (2.11) now implies that\bigm| \bigm| \bigm| \bigm| (q \ast h)(z)z - 1
\bigm| \bigm| \bigm| \bigm| \not = 0.
Consequently, we have
q(z) \in \scrM \scrS q(\alpha ),
which completes the proof of Theorem 2.
We now derive the partial sums for the function class \scrM \scrS q(\alpha ).
Theorem 3. Let f \in \scrM given by (1.3) and define the partial sum fk(z) of the function f
by (1.5), where an empty sum is interpreted (as usual) to be nil. If
\infty \sum
n=0
\scrL (n, q, \alpha )| an| \leq 1, (2.12)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
CERTAIN SUBCLASSES OF MEROMORPHICALLY q-STARLIKE FUNCTIONS . . . 1267
then
f(z) \in \scrM \scrS q(\alpha ), (2.13)
\Re
\biggl(
f(z)
fk(z)
\biggr)
\geq 1 - 1
\scrL (k + 1, q, \alpha )
\forall z \in \BbbU , k \in \BbbN , (2.14)
and
\Re
\biggl(
fk(z)
f(z)
\biggr)
\geq \scrL (k + 1, q, \alpha )
1 + \scrL (k + 1, q, \alpha )
\forall z \in \BbbU , k \in \BbbN , (2.15)
where \scrL (n, q, \alpha ), n \in \BbbN 0, is defined by (1.7). The bound in (2.14) and (2.15) are sharp.
Proof. First of all, we set
f1(z) =
1
z
and we know that
f1(z) + \varepsilon z - 1
1 + \varepsilon
=
1
z
\in \scrM \scrS q(\alpha ).
Also, from (2.12), we can easily see that
\infty \sum
n=0
\scrL (n, q, \alpha )| an - 0| \leq 1, (2.16)
where \scrL (n, q, \alpha ), n \in \BbbN 0, is given by (1.7). Inequality in (2.16) now implies that f \in N(1,q)(z
- 1).
From Theorem 2, we conclude that
f(z) \in N(1,q)(z
- 1) \subset \scrM \scrS q(\alpha ).
We deduce that the assertion (2.13) holds true.
Next, it is easy to verify that
\scrL (k + 1, q, \alpha ) > \scrL (k, q, \alpha ) > 1.
Thus, we find
k\sum
n=0
| an| + \scrL (k + 1, q, \alpha )
\infty \sum
n=0
| an| \leq
\infty \sum
n=k+1
\scrL (n+ 1, q, \alpha ) | an| \leq 1. (2.17)
If we set
h1(z) = \scrL (k + 1, q, \alpha )
\biggl\{
f(z)
fk(z)
-
\biggl(
1 - 1
\scrL (k + 1, q, \alpha )
\biggr) \biggr\}
=
= 1 +
\scrL (k + 1, q, \alpha )
\sum \infty
n=k+1
anz
n+1
1 +
\sum k
n=0
anz
n+1
. (2.18)
It follows from (2.17) and (2.18) that
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1268 H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD
\bigm| \bigm| \bigm| \bigm| h1(z) - 1
h1(z) + 1
\bigm| \bigm| \bigm| \bigm| \leq \scrL (k + 1, q, \alpha )
\sum \infty
n=k+1
| an|
2 - 2
\sum k
n=0
| an| - \scrL (k + 1, q, \alpha )
\sum \infty
n=k+1
| an|
\leq 1 \forall z \in \BbbU . (2.19)
Clearly, the inequality in (2.19) now shows that
\Re
\bigl(
h1(z)
\bigr)
\geq 0 \forall z \in \BbbU . (2.20)
Thus, by combining (2.17) and (2.20), we deduce that the assertion (2.14) holds true.
Next, by taking
f(z) =
1
z
- zn+1
\scrL (k + 1, q, \alpha )
, (2.21)
we easily observe that
f(z)
fk(z)
= 1 - zn+2
\scrL (k + 1, q, \alpha )
\rightarrow 1 - 1
\scrL (k + 1, q, \alpha )
, z \rightarrow 1 - ,
which shows that the bound in (2.14) is best possible for each k \in \BbbN .
Just as above, we set that
h2(z) = (1 + \scrL (k + 1, q, \alpha ))
\biggl\{
fk(z)
f(z)
- \scrL (k + 1, q, \alpha )
1 + \scrL (k + 1, q, \alpha )
\biggr\}
=
= 1 -
(1 + \scrL (k + 1, q, \alpha ))
\sum \infty
n=k+1
anz
n+1
1 +
\sum \infty
n=0
anz
n+1
. (2.22)
By the virtue of (2.17) and (2.22), we conclude that
\bigm| \bigm| \bigm| \bigm| h2(z) - 1
h2(z) + 1
\bigm| \bigm| \bigm| \bigm| \leq
\bigl(
1 + \scrL (k + 1, q, \alpha )
\bigr) \sum \infty
n=k+1
| an|
2 - 2
\sum k
n=0
| an| +
\bigl(
1 - \scrL (k + 1, q, \alpha )
\bigr) \sum \infty
n=k+1
| an|
\leq 1 \forall z \in \BbbU ,
which shows that
\Re
\bigl(
h2(z)
\bigr)
\geq 0 \forall z \in \BbbU . (2.23)
Finally, upon combining (2.22) and (2.23), we readily get the assertion (2.15) of Theorem 3. The
bound in (2.15) is sharp with the extremal function f(z) given by (2.21).
Theorem 3 is proved.
In its special case when q \rightarrow 1 - , Theorem 3 yields the following known result proved by Cho
and Owa ([9], see also Remark 1).
Corollary 1 [9]. If the function f of the form (1.3) satisfies the following condition:
\infty \sum
n=0
(n+ \alpha ) | an| \leq 1 - \alpha ,
then
f \in \scrM \scrS (\alpha ),
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CERTAIN SUBCLASSES OF MEROMORPHICALLY q-STARLIKE FUNCTIONS . . . 1269
\Re
\biggl(
f(z)
fk(z)
\biggr)
\geq k + 2\alpha
k + 1 + \alpha
\forall z \in \BbbU , k \in \BbbN ,
and
\Re
\biggl(
fk(z)
f(z)
\biggr)
\geq k + 1 + \alpha
k + 2
\forall z \in \BbbU , k \in \BbbN .
The proof of Theorem 4 below is similar to that of Theorem 3, so we have chosen to omit the
analogous details.
Theorem 4. Let f \in \scrM given by (1.3) and define the partial sum fk(z) of f by (1.5). If the
condition (2.12) holds true, then
\Re
\biggl(
(Dqf)(z)
(Dqfk)(z)
\biggr)
\geq 1 - [k + 1]q
\scrL (k + 1, q, \alpha )
(2.24)
and
\Re
\biggl(
(Dqfk)(z)
(Dqf)(z)
\biggr)
\geq \scrL (k + 1, q, \alpha )
[k + 1]q + \scrL (k + 1, q, \alpha )
, (2.25)
where \scrL (n, q, \alpha ), n \in \BbbN 0, is given in (1.7) and the bounds in (2.24) and (2.25) are sharp with the
extremal function given by (2.21).
As an application of Theorem 4 (with \alpha = 0), we immediately deduce Corollary 2 below.
Corollary 2. If the function f \in \scrM given by (1.3) satisfies the condition (2.12) with \alpha = 0, then
\Re
\biggl(
(Dqf)(z)
(Dqfk)(z)
\biggr)
\geq 1 - [k + 1]q
\scrL (k + 1, q, 0)
and
\Re
\biggl(
(Dqfk)(z)
(Dqf)(z)
\biggr)
\geq \scrL (k + 1, q, 0)
[k + 1]q + \scrL (k + 1, q, 0)
,
where \scrL (n, q, \alpha ), n \in \BbbN 0, is given by (1.7).
In the limit case when q \rightarrow 1 - , Theorem 4 yields the following known result.
Corollary 3 [9]. If the function f of the form (1.3) satisfies the following condition:
\infty \sum
n=0
(n+ \alpha )| an| \leq 1 - \alpha ,
then
\Re
\biggl(
f \prime (z)
f \prime k(z)
\biggr)
\geq 1 - (k + 1)(1 - \alpha )
k + 1 + \alpha
\forall z \in \BbbU , k \in \BbbN ,
and
\Re
\biggl(
f \prime k(z)
f \prime (z)
\biggr)
\geq k + 1 + \alpha
2(k + 1) - k\alpha
\forall z \in \BbbU , k \in \BbbN .
3. Ruscheweyh-type \bfitq -derivative operator for meromorphic functions. In this section,
by using a Ruscheweyh-type q-derivative operator, we define and study a new class of functions
emerging from the class \scrM of normalized meromorphic functions. We also investigate the results
analogous to those that have been proved in the preceding section.
Analogues of the Ruscheweyh derivative for analytic functions (see, for details, [28]), Al-
Amiri [3] studied what he called the m-order Ruscheweyh-type derivative. Subsequently, Ganigi
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1270 H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD
and Uralegaddi [17] introduced the meromorphic analogue of this derivative. More recently, Kanas
and Răducanu [22] introduced the Ruscheweyh derivative operator for analytic functions by using
the q-derivative operator. We propose to define a q-extension of the meromorphic analogue of the
Ruscheweyh derivative by using the q-derivative operator.
Definition 8. For f \in \scrM , the meromorphic analogue of the Ruscheweyh-type q-derivative
operator is defined by
\scrM \scrR \delta
qf(z) = f(z) \ast \phi (q, \delta + 1; z) =
1
z
+
\infty \sum
n=1
\psi n(\delta )anz
n, z \in \BbbU \ast , \delta > - 1, (3.1)
where
\phi (q, \delta + 1; z) =
1
z
+
\infty \sum
n=1
\psi n(\delta )z
n
and
\psi n(\delta ) =
[\delta + n+ 1]q!
[n+ 1]q![\delta ]q!
, n \in \BbbN . (3.2)
It is easily seen from (3.1) that
\scrM \scrR 0
qf(z) = f(z), \scrM \scrR 1
qf(z) - [2]q\scrM \scrR 0
qf(qz) = zDqf(z)
and
\scrM \scrR m
q f(z) =
z - 1Dq
\bigl(
zm+1f(z)
\bigr)
[m]q!
, m \in \BbbN .
We also note that
\mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
\phi (q, \delta + 1; z) =
1
z(1 - z)\delta +1
and
\mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
\scrM \scrR \delta
qf(z) = f(z) \ast 1
z(1 - z)\delta +1
,
which is the familiar Ruscheweyh derivative operator for meromorphic functions introduced and
studied in [5, 6].
Definition 9. A function f \in \scrM is said to be in the class \scrM \scrS \delta
q(\alpha ), 0 \leq \alpha < 1, if it satisfies
the following condition: \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl(
-
zDq
\bigl(
\scrM \scrR \delta
qf(z)
\bigr)
\scrM \scrR \delta
qf(z)
\Biggr)
- \alpha
1 - \alpha
- 1
1 - q
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\leq 1
1 - q
.
Remark 2. First of all, we see that
\scrM \scrS 0
q(\alpha ) = \scrM \scrS q(\alpha ),
where \scrM \scrS q(\alpha ) is the function class in Definition 6. Secondly, we have
\mathrm{l}\mathrm{i}\mathrm{m}
q\rightarrow 1 -
\scrM \scrS 0
q(0) = \scrM \scrS ,
where \scrM \scrS is the function class, which was introduced and studied by Clunie (see [10]).
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CERTAIN SUBCLASSES OF MEROMORPHICALLY q-STARLIKE FUNCTIONS . . . 1271
The following results can be proved by using the arguments similar to those that were already
use in Section 2, so we choose to omit the details of our proof of Theorems 5 – 8 below.
Theorem 5. Let
1
q
- \alpha > 0.
Suppose also that the function f \in \scrM is given by (1.3). If
\infty \sum
n=0
\bigl(
[n]q + \alpha
\bigr)
\psi n| an| \leq
1
q
- \alpha ,
then f \in \scrM \scrS \delta
q(\alpha ).
Remark 3. Upon letting \delta = 0 in Theorem 5, we are led to Theorem 1 of the preceding section.
Theorem 6. For \xi \geq 0, and let the function f \in \scrM given by (1.3) satisfy the following
condition:
f(z) + \varepsilon z - 1
1 + \varepsilon
\in \scrM \scrS \delta
q(\alpha )
for any complex number \varepsilon such that | \varepsilon | < \xi . Then
N(\xi ,q)(f) \subset \scrM \scrS \delta
q(\alpha ).
Theorem 7. Let the function f \in \scrM be given by (1.3) and define the partial sum fk(z) of the
function f by (1.5), where an empty sum is interpreted (as usual) to be nil. If
\infty \sum
n=0
\kappa n(\alpha )| an| \leq 1, (3.3)
where
\kappa n(\alpha ) =
\bigl(
2[n]q + (1 + q)\alpha
\bigr)
q\psi n(\delta )
q - 1 + (1 + q)(1 - \alpha q)
= \scrL (n, q, \alpha )q\psi n(\delta ), n \in \BbbN , (3.4)
in terms of \scrL (n, q, \alpha ) and \psi n(\delta ) given by (1.7) and (3.2), respectively, then
f(z) \in \scrM \scrS \delta
q(\alpha ),
\Re
\biggl(
f(z)
fk(z)
\biggr)
\geq 1 - 1
\kappa k+1(\alpha )
, z \in \BbbU , k \in \BbbN , (3.5)
and
\Re
\biggl(
fk(z)
f(z)
\biggr)
\geq \kappa k+1(\alpha )
1 + \kappa k+1(\alpha )
, z \in \BbbU , k \in \BbbN . (3.6)
The bounds in (3.5) and (3.6) are sharp.
Theorem 8. Let the function f \in \scrM be given by (1.3) and define the partial sum fk(z) of the
function f by (1.5). If the condition (3.3) holds true, then
\Re
\biggl(
Dqf(z)
Dqfk(z)
\biggr)
\geq 1 - [k + 1]q
\kappa k+1(\alpha )
(3.7)
and
\Re
\biggl(
(Dqfk)(z)
(Dqf)(z)
\biggr)
\geq \kappa k+1(\alpha )
[k + 1]q + \kappa k+1(\alpha )
, (3.8)
where \kappa n(\alpha ), n \in \BbbN 0, is given in (3.4). The bounds in (3.7) and (3.8) are sharp with the extremal
function given by (2.21).
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1272 H. M. SRIVASTAVA, M. TAHIR, B. KHAN, M. DARUS, N. KHAN, Q. Z. AHMAD
In its special case when \alpha = 0, Theorem 8 yields the following corollary.
Corollary 4. Let the function f \in \scrM , given by (1.3), satisfy the condition (3.3) with \alpha = 0.
Suppose also that the partial sum fk(z) of the function f is defined by (1.5). Then
\Re
\biggl(
(Dqf)(z)
(Dqfk)(z)
\biggr)
\geq 1 - 1
q\psi k+1(\delta )
and
\Re
\biggl(
(Dqfk)(z)
(Dqf)(z)
\biggr)
\geq q\psi k+1(\delta )
1 + q\psi k+1(\delta )
,
where \psi n(\delta ) is given by (3.2).
4. Conclusion. For the triviality and inconsequential nature of the so-called (p, q)-variations
of our q-results, with an obviously redundant (or superfluous) parameter p, the reader is referred
to a recent survey-cum-expository review article by Srivastava [32, p. 340]. In this paper, we have
established several general results involving the partial sums of meromorphically q-starlike functions
defined here by means of a certain class of q-derivative (or q-difference) operators. We have also
investigated the familiar concept of neighborhood for meromorphic functions. Moreover, by using
a Ruscheweyh-type q-derivative operator, we have introduced and studied another new class of
functions emerging from the class of normalized meromorphic functions.
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ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
|
| id | umjimathkievua-article-814 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:08Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/04/24defb4e5e79c206d63c0d3ce7588604.pdf |
| spelling | umjimathkievua-article-8142025-03-31T08:46:40Z Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. B. N. Analytic and univalent functions Meromorphic functions Meromorphically starlike functions q-Derivative (or q-difference) operator Neighborhoods Partial sums Ruscheweyh-type q-derivative operator Analytic and univalent functions Meromorphic functions Meromorphically starlike functions q-Derivative (or q-difference) operator Neighborhoods Partial sums Ruscheweyh-type q-derivative operator UDC 517.5 The purpose of the present paper is to establish several general results concerning the partial sums of meromorphically starlike functions defined here by means of a certain class of $q$-derivative (or $q$-difference) operators. The familiar concept of neighborhood for meromorphic functions are also considered. Moreover, by using a Ruscheweyh-type $q$-derivative operator, we define and study another new class of functions emerging from the class of normalized meromorphic functions. &nbsp; УДК 517.5 Деякi пiдкласи мероморфних $q$ -зiркових функцiй, пов’язанi з $q$ -похiдними операторами Метою цієї статті є отримання кількох загальних результатів, що пов'язані з частковими сумами мероморфних зіркових функцій, які визначаються за допомогою деякого класу $q$-похідних (або $q$-різницевих) операторів. Також розглянуто відоме поняття околу для мероморфних функцій. Крім того, за допомогою $q$-похідного оператора типу Рушевая визначається та вивчається новий клас функцій, який виводиться з класу нормалізованих мероморфних функцій. Institute of Mathematics, NAS of Ukraine 2021-09-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/814 10.37863/umzh.v73i9.814 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 9 (2021); 1260 - 1273 Український математичний журнал; Том 73 № 9 (2021); 1260 - 1273 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/814/9109 Copyright (c) 2021 Qazi Zahoor Ahmad, Hari M Srivastava, Muhammad Tahir, Bilal Khan, Maslina Darus, Nazar Khan |
| spellingShingle | Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. Srivastava, H. M. Tahir, M. Khan, B. Darus, M. Khan, N. Ahmad, Q. Z. B. N. Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title | Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title_alt | Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title_full | Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title_fullStr | Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title_full_unstemmed | Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title_short | Certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| title_sort | certain subclasses of meromorphically $q$-starlike functions associated with the $q$-derivative operators |
| topic_facet | Analytic and univalent functions Meromorphic functions Meromorphically starlike functions q-Derivative (or q-difference) operator Neighborhoods Partial sums Ruscheweyh-type q-derivative operator Analytic and univalent functions Meromorphic functions Meromorphically starlike functions q-Derivative (or q-difference) operator Neighborhoods Partial sums Ruscheweyh-type q-derivative operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/814 |
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