Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order
UDC 517.5 We determine coefficient bounds for functions from  subclasses of $p$-valent starlike and $p$-valent convex functions defined by higher-order derivatives and complex order introduced with the help of a certain nonhomogeneous Cauchy – Euler differential equation for highe...
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| author | Aouf, M. K. Mostafa, A. O. Bulboacă, T. Aouf, Mohamed K. Aouf, M. K. Mostafa, A. O. Bulboacă, T. |
| author_facet | Aouf, M. K. Mostafa, A. O. Bulboacă, T. Aouf, Mohamed K. Aouf, M. K. Mostafa, A. O. Bulboacă, T. |
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UDC 517.5
We determine coefficient bounds for functions from  subclasses of $p$-valent starlike and $p$-valent convex functions defined by higher-order derivatives and complex order introduced with the help of a certain nonhomogeneous Cauchy – Euler differential equation for higher-order derivatives.  Relevant connections of some of our results with the results obtained earlier  are provided. |
| doi_str_mv | 10.37863/umzh.v74i10.6258 |
| first_indexed | 2026-03-24T03:26:46Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i10.6258
UDC 517.5
M. K. Aouf1, A. O. Mostafa (Mansoura Univ., Egypt),
T. Bulboacă (Babeş-Bolyai Univ., Cluj-Napoca, Romania)
COEFFICIENT BOUNDS FOR MULTIVALENT CLASSES OF STARLIKE
AND CONVEX FUNCTIONS DEFINED BY HIGHER-ORDER DERIVATIVES
AND COMPLEX ORDER
КОЕФIЦIЄНТНI ОЦIНКИ ДЛЯ БАГАТОЗНАЧНИХ КЛАСIВ ЗIРКОПОДIБНИХ
ТА ОПУКЛИХ ФУНКЦIЙ, ВИЗНАЧЕНИХ ПОХIДНИМИ ВИЩОГО ПОРЯДКУ
ТА КОМПЛЕКСНИМ ПОРЯДКОМ
We determine coefficient bounds for functions from subclasses of p-valent starlike and p-valent convex functions defined
by higher-order derivatives and complex order introduced with the help of a certain nonhomogeneous Cauchy – Euler
differential equation for higher-order derivatives. Relevant connections of some of our results with the results obtained
earlier are provided.
Знайдено оцiнки для коефiцiєнтiв функцiй, що належать до пiдкласiв p-значних зiркоподiбних i p-значних опуклих
функцiй, якi визначаються похiдними вищого порядку та комплексним порядком i вводяться за допомогою пев-
ного неоднорiдного диференцiального рiвняння Кошi – Ейлера для похiдних вищого порядку. Наведено вiдповiднi
спiввiдношення мiж деякими нашими результатами та результатами, що були отриманi ранiше.
1. Introduction. Denote by \scrA n(p) the class of multivalent analytic functions of the form
f(z) = zp +
\infty \sum
k=p+n
akz
k, z \in \BbbU := \{ z \in \BbbC : | z| < 1\} , p, n \in \BbbN := \{ 1, 2, . . .\} , (1.1)
and let \scrA (p) := \scrA 1(p), \scrA (n) := \scrA n(1), and \scrA := \scrA 1(1).
Definition 1.1. For p > q, p \in \BbbN , q \in \BbbN 0 := \BbbN \cup \{ 0\} , 0 \leq \gamma < p - q, we say that the function
f \in \scrA n(p) belongs to the class \BbbS \ast p(q, n, \gamma ) of (p, q)-valent starlike functions of order \gamma , if it satisfies
the inequality
\mathrm{R}\mathrm{e}
zf (1+q)(z)
f (q)(z)
> \gamma , z \in \BbbU ,
and belongs to the class \BbbK p(q, n, \gamma ) of (p, q)-valent convex functions of order \gamma , if it satisfies
\mathrm{R}\mathrm{e}
\Biggl(
1 +
zf (2+q)(z)
f (1+q)(z)
\Biggr)
> \gamma , z \in \BbbU .
The classes \BbbS \ast p(q, 1, \gamma ) =: \BbbS \ast p(q, \gamma ) and \BbbK p(q, 1, \gamma ) =: \BbbK \ast
p(q, \gamma ) were introduced and studied by
Aouf [7 – 9], and note that \BbbS \ast p(0, \gamma ) =: \BbbS \ast p(\gamma ) and \BbbK p(0, \gamma ) =: \BbbK p(\gamma ) are, respectively, the classes of
p-valent starlike and convex functions of order \gamma with 0 \leq \gamma < p (see Owa [18] and Aouf [3, 5, 6]).
Also, we mention that \BbbS \ast 1(\gamma ) =: \BbbS \ast (\gamma ) and \BbbK 1(\gamma ) =: \BbbK (\gamma ), with 0 \leq \gamma < 1, were introduced and
studied by Robertson [21] (see also [24, 25]), if \BbbS \ast 1(0, n, \gamma ) =: \BbbS \ast n(\gamma ), and \BbbK 1(0, n, \gamma ) =: \BbbC n(\gamma ), if
0 \leq \gamma < 1, were introduced and studied by Srivastava et al. [26].
1 Corresponding author, e-mail: mkaouf127@yahoo.com.
c\bigcirc M. K. AOUF, A. O. MOSTAFA, T. BULBOACĂ, 2022
1308 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
COEFFICIENT BOUNDS FOR MULTIVALENT CLASSES OF STARLIKE . . . 1309
Definition 1.2. For b \in \BbbC \ast := \BbbC \setminus \{ 0\} , 0 \leq \lambda \leq 1, 0 < \beta \leq 1, p \in \BbbN , q \in \BbbN 0, and p > q, we
say that f \in \scrA n(p) belongs to the class \BbbH n(p, q, \lambda , \beta , b) if it satisfies
\mathrm{R}\mathrm{e}
\Biggl\{
1 +
1
b
\Biggl(
1
p - q
zf (1+q)(z) + \lambda z2f (2+q)(z)
(1 - \lambda )f (q)(z) + \lambda zf (1+q)(z)
- 1
\Biggr) \Biggr\}
> \beta , z \in \BbbU . (1.2)
Remark 1.1. For different values of p, q, n, b, \lambda we obtain the following subclasses:
(i) \BbbH n(p, q, 0, \beta , b) =: \BbbS n(p, q, \beta , b) =
=
\Biggl\{
f \in \scrA n(p) : \mathrm{R}\mathrm{e}
\Biggl[
1 +
1
b
\Biggl(
1
p - q
zf (1+q)(z)
f (q)(z)
- 1
\Biggr) \Biggr]
> \beta , z \in \BbbU
\Biggr\}
.
(ii) \BbbH n(p, q, 1, \beta , b) =: Cn(p, q, \beta , b) =
=
\Biggl\{
f \in \scrA n(p) : \mathrm{R}\mathrm{e}
\Biggl[
1 +
1
b
\Biggl(
1
p - q
\Biggl(
1 +
zf (2+q)(z)
f (1+q)(z)
\Biggr)
- 1
\Biggr) \Biggr]
> \beta , z \in \BbbU
\Biggr\}
.
(iii) \BbbH n(p, 0, \lambda , \beta , b) =: \BbbG n(p, \lambda , \beta , b) =
=
\biggl\{
f \in \scrA n(p) : \mathrm{R}\mathrm{e}
\biggl[
1 +
1
b
\biggl(
1
p
zf \prime (z) + \lambda z2f \prime \prime (z)
(1 - \lambda )f(z) + \lambda zf \prime (z)
- 1
\biggr) \biggr]
> \beta , z \in \BbbU
\biggr\}
.
Moreover, \BbbG 1(p, 0, \beta , b) =: \BbbS p(\gamma , b), \gamma = p\beta , 0 \leq \beta < 1 (see [13] with A = 1 and B = - 1
and [11] with m = 0), that is,
\BbbS p(\gamma , b) =
\biggl\{
f \in \scrA (p) : \mathrm{R}\mathrm{e}
\biggl[
p+
1
b
\biggl(
zf \prime (z)
f(z)
- p
\biggr) \biggr]
> \gamma , z \in \BbbU , 0 \leq \gamma < p
\biggr\}
,
and \BbbG 1(p, 1, \beta , b) =: Cp(\gamma , b), \gamma = p\beta , 0 \leq \beta < 1
\Bigl(
see [10] with B = - 1, A = 1 - 2\gamma
p
, 0 \leq \gamma < p
and [11] with m = 0
\Bigr)
, that is,
Cp(\gamma , b) =
\biggl\{
f \in \scrA (p) : \mathrm{R}\mathrm{e}
\biggl[
p+
1
b
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
- p
\biggr) \biggr]
> \gamma , z \in \BbbU , 0 \leq \gamma < p
\biggr\}
.
(iv) \BbbH n(1, 0, \lambda , \beta , b) =: \BbbS \BbbC n(\lambda , \beta , b) (see [1]), that is,
\BbbS \BbbC n(\lambda , \beta , b) =
\biggl\{
f \in \scrA (n) : \mathrm{R}\mathrm{e}
\biggl[
1 +
1
b
\biggl(
zf \prime (z) + \lambda z2f \prime \prime (z)
(1 - \lambda )f(z) + \lambda zf \prime (z)
- 1
\biggr) \biggr]
> \beta , z \in \BbbU
\biggr\}
,
and, also, \BbbH 1(1, 0, \lambda , \beta , b) =: \BbbS (\lambda , \beta , b), 0 \leq \beta < 1 (see [22]).
(v) \BbbS \ast 1(p, q, \beta , \mathrm{c}\mathrm{o}\mathrm{s}\alpha e - i\alpha ) =: \BbbS \alpha p (q, \gamma ), \gamma = (p - q)\beta , 0 \leq \beta < 1, that is,
\BbbS \alpha p (q, \gamma ) =
\Biggl\{
f \in \scrA (p) : \mathrm{R}\mathrm{e}
\Biggl[
ei\alpha
zf (1+q)(z)
f (q)(z)
\Biggr]
> \gamma \mathrm{c}\mathrm{o}\mathrm{s}\alpha , | \alpha | < \pi
2
, z \in \BbbU , 0 \leq \gamma < p - q
\Biggr\}
.
Also, \BbbS \alpha p (0, \gamma ) = \BbbS \alpha p (\gamma ) (see [4, 19, 23]).
(vi) C1(p, q, \beta , \mathrm{c}\mathrm{o}\mathrm{s}\alpha e
- i\alpha ) =: C\alpha
p (q, \gamma ), \gamma = (p - q)\beta , 0 \leq \beta < 1, that is,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1310 M. K. AOUF, A. O. MOSTAFA, T. BULBOACĂ
C\alpha
p (q, \gamma ) =
=
\Biggl\{
f \in \scrA (p) : \mathrm{R}\mathrm{e}
\Biggl[
ei\alpha
\Biggl(
1 +
zf (2+q)(z)
f (1+q)(z)
\Biggr) \Biggr]
> \gamma \mathrm{c}\mathrm{o}\mathrm{s}\alpha , z \in \BbbU , | \alpha | < \pi
2
, 0 \leq \gamma < p - q
\Biggr\}
,
and C\alpha
p (0, \gamma ) =: C\alpha
p (\gamma ) (see [10, 23]).
(vii) \BbbH 1(1, 0, 0, 0, b) =: S(b) and \BbbH 1(1, 0, 1, 0, b) =: C(b) (see [15 – 17]).
Definition 1.3. A function f \in \scrA n(p) belongs to the class \BbbB n(p, q, \lambda , \beta , b) if w = f(z) satisfies
the following nonhomogeneous Cauchy – Euler differential equation (see [14])
z2
d(2+q)w
dz(2+q)
+ 2(1 + \mu )z
d(1+q)w
dz(1+q)
+ \mu (1 + \mu )
d(q)w
dz(q)
= (p - q + \mu )(p - q + \mu + 1)
d(q)g
dz(q)
, (1.3)
where g \in \BbbH n(p, q, \lambda , \beta , b), \mu \in \BbbR with \mu > q - p, and p \in \BbbN , q \in \BbbN 0 .
Note that \BbbB 1(1, 0, \lambda , \beta , b) =: \scrH (\lambda , \beta , \mu , b) (see [1, 12]) and \BbbB n(p, 0, \lambda , \beta , \mu , b) =: \scrG n(p, \lambda , \beta , \mu , b)
(see [11] with m = 0).
2. Coefficient estimates for the function class \BbbH \bfitn (\bfitp , \bfitq , \bfitlambda , \bfitbeta , \bfitb ). Unless otherwise stated we
assume that b \in \BbbC \ast , 0 \leq \lambda \leq 1, 0 < \beta \leq 1, p \in \BbbN , q \in \BbbN 0, p > q, \mu > q - p, and \mu \in \BbbR . Let \Gamma
denotes the well-known Euler integral of the second kind, that is,
\Gamma (z) :=
\infty \int
0
tz - 1e - t dt,
that converges absolutely on \scrD := \{ z \in \BbbC : \mathrm{R}\mathrm{e} z > 0\} , therefore \Gamma (1) = 1 and \Gamma (m+ 1) = m! for
all m \in \BbbN .
Theorem 2.1. Let the function f \in \scrA n(p) defined by (1.1) belongs to the class \BbbH n(p, q, \lambda , \beta , b).
Then
| ap+n| \leq
2[1 + \lambda (p - q - 1)]\delta (p, q + 1)(1 - \beta )| b|
n[1 + \lambda (p+ n - q - 1)]\delta (p+ n, q)
and
| ak| \leq
2\Gamma (n)[1 + \lambda (p - q - 1)]\delta (p, q + 1)(1 - \beta )| b|
\Gamma (k - p+ 1)[1 + \lambda (k - q - 1)]\delta (k, q)
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2p(1 - \beta )| b|
\bigr]
,
k \geq p+ n+ 1,
where \delta (p, q) :=
p!
(p - q)!
, p > q .
Proof. For f \in \scrA n(p) given by (1.1) we define the function F\lambda ,p,q by
F\lambda ,p,q(z) :=
(1 - \lambda )f (q)(z) + \lambda zf (1+q)(z)\bigl[
1 + \lambda (p - q - 1)
\bigr]
\delta (p, q)
, z \in \BbbU , (2.1)
that is,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
COEFFICIENT BOUNDS FOR MULTIVALENT CLASSES OF STARLIKE . . . 1311
F\lambda ,p,q(z) = zp - q +
\infty \sum
k=n+p
Ak,qz
k - q, z \in \BbbU ,
where
Ak,q =
\bigl[
1 + \lambda (k - q - 1)
\bigr]
\delta (k, q)\bigl[
1 + \lambda (p - q - 1)
\bigr]
\delta (p, q)
ak, k \geq n+ p. (2.2)
From (1.2) and (2.1) we have
\mathrm{R}\mathrm{e}
\Biggl[
1 +
1
b
\Biggl(
1
p - q
zF \prime
\lambda ,p,q(z)
F\lambda ,p,q(z)
- 1
\Biggr) \Biggr]
> \beta , z \in \BbbU .
If we define the function g by
g(z) :=
1 +
1
b
\Biggl(
1
p - q
zF \prime
\lambda ,p,q(z)
F\lambda ,p,q(z)
- 1
\Biggr)
- \beta
1 - \beta
, z \in \BbbU ,
then g is analytic in \BbbU with g(0) = 1 and \mathrm{R}\mathrm{e} g(z) > 0, z \in \BbbU . Since the above relation is equivalent
to
1
b
\Biggl(
1
p - q
zF \prime
\lambda ,p,q(z)
F\lambda ,p,q(z)
- 1
\Biggr)
= (1 - \beta )
\bigl[
g(z) - 1
\bigr]
, z \in \BbbU ,
and F\lambda ,p,q \in \scrA n(p - q), it follows that
g(z) = 1 + cnz
n + cn+1z
n+1 + . . . , z \in \BbbU .
Therefore, we obtain
1
b
\Biggl(
1
p - q
zF \prime
\lambda ,p,q(z)
F\lambda ,p,q(z)
- 1
\Biggr)
= (1 - \beta )(cnz
n + cn+1z
n+1 + . . .), z \in \BbbU ,
or, equivalently,
zF \prime
\lambda ,p,q(z) - (p - q)F\lambda ,p,q(z) = (p - q)b(1 - \beta )(cnz
n + cn+1z
n+1 + . . .)F\lambda ,p,q(z), z \in \BbbU .
The last equality implies that
(k - p)Ak,q = (p - q)b(1 - \beta )
\bigl(
ck - p + ck - p - nAp+n,q + . . .+ cnAk - n,q
\bigr)
,
and putting k = p+ n+ r, r \in \BbbN 0, we have
(n+ r)Ap+n+r,q = (p - q)b(1 - \beta )
\bigl(
cn+r + crAp+n,q + . . .+ cnAp+r,q
\bigr)
.
Applying the coefficient estimates | ck| \leq 2, k \geq 1, for the Carathéodory functions (see [20]), we
obtain
| Ap+n+r,q| \leq
2(p - q)(1 - \beta )| b|
n+ r
\bigl(
1 + | Ap+n,q| + . . .+ | Ap+r,q|
\bigr)
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1312 M. K. AOUF, A. O. MOSTAFA, T. BULBOACĂ
But, for r = 0, 1, 2, we get
| Ap+n,q| \leq
2(p - q)(1 - \beta )| b|
n
,
| Ap+n+1,q| \leq
2(p - q)(1 - \beta )| b|
n+ 1
\bigl(
1 + | Ap+n,q|
\bigr)
\leq
\leq
2(p - q)(1 - \beta )| b|
\bigl[
n+ 2(p - q)(1 - \beta )| b|
\bigr]
n(n+ 1)
, (2.3)
| Ap+n+2,q| \leq
2(p - q)(1 - \beta )| b|
n+ 2
\bigl(
1 + | Ap+n,q| + | Ap+n+1,q|
\bigr)
\leq
\leq
2(p - q)(1 - \beta )| b|
\bigl[
n+ 2(p - q)(1 - \beta )| b|
\bigr] \bigl[
n+ 1 + 2(p - q)(1 - \beta )| b|
\bigr]
n(n+ 1)(n+ 2)
,
respectively. By mathematical induction we have
| Ap+n+r,q| \leq
2(p - q)(1 - \beta )| b|
n(n+ 1) . . . (n+ r)
r - 1\prod
j=0
\bigl[
n+ j + 2(p - q)(1 - \beta )| b|
\bigr]
=
=
2(p - q)(1 - \beta )| b| \Gamma (n)
\Gamma (n+ r + 1)
r - 1\prod
j=0
\bigl[
n+ j + 2(p - q)(1 - \beta )| b|
\bigr]
, r \geq 1. (2.4)
Then, from (2.3), (2.4) and k = p+ n+ r, we get
| Ak,q| \leq
2(p - q)(1 - \beta )| b| \Gamma (n)
\Gamma (k - p+ 1)
k - p - n - 1\prod
j=0
\bigl[
n+ j + 2(p - q)(1 - \beta )| b|
\bigr]
, k \geq p+ n+ 1.
From (2.2) we obtain
ak =
[1 + \lambda (p - q - 1)]\delta (p, q)
[1 + \lambda (k - q - 1)]\delta (k, q)
Ak,q, k \geq n+ p,
so, we have
| ap+n| \leq
2(1 - \beta )| b|
\bigl[
1 + \lambda (p - q - 1)
\bigr]
\delta (p, q + 1)
n
\bigl[
1 + \lambda (p+ n - q - 1)
\bigr]
\delta (p+ n, q)
,
| ak| \leq
2\Gamma (n)(1 - \beta )| b|
\bigl[
1 + \lambda (p - q - 1)
\bigr]
\delta (p, q + 1)
\Gamma (k - p+ 1)
\bigl[
1 + \lambda (k - q - 1)
\bigr]
\delta (k, q)
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2(p - q)(1 - \beta )| b|
\bigr]
.
Theorem 2.1 is proved.
Putting \lambda = 0 and \lambda = 1 in Theorem 2.1, we have the next two results, respectively.
Corollary 2.1. If f \in \BbbS n(p, q, \beta , b) (see Remark 1.1 (i)), then
| ap+n| \leq
2(1 - \beta )| b| \delta (p, q + 1)
n\delta (p+ n, q)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
COEFFICIENT BOUNDS FOR MULTIVALENT CLASSES OF STARLIKE . . . 1313
and
| ak| \leq
2(1 - \beta )| b| \Gamma (n)\delta (p, q + 1)
\Gamma (k - p+ 1)\delta (k, q)
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2(p - q)(1 - \beta )| b|
\bigr]
,
k \geq p+ n+ 1.
Corollary 2.2. If f \in Cn(p, q, \beta , b) (see Remark 1.1 (ii)), then
| ap+n| \leq
2(1 - \beta )| b| (p - q)\delta (p, q + 1)
n\delta (p+ n+ 1, q)
and
| ak| \leq
2(1 - \beta )| b| \Gamma (n)(p - q)\delta (p, q + 1)
\Gamma (k - p+ 1)\delta (k, q + 1)
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2(p - q)(1 - \beta )| b|
\bigr]
,
k \geq p+ n+ 1.
Considering \beta = q = 0 in Corollaries 2.1 and 2.2, we obtain the next two special cases,
respectively.
Example 2.1. If f \in \BbbS n(p, 0, 0, b), then | ap+n| \leq
2p| b|
n
and
| ak| \leq
2p| b| \Gamma (n)
\Gamma (k - p+ 1)
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2p| b|
\bigr]
, k \geq p+ n+ 1.
Example 2.2. If f \in Cn(p, 0, 0, b), then | ap+n| \leq
2p2| b|
n(p+ n)
and
| ak| \leq
2p2| b| \Gamma (n)
k\Gamma (k - p+ 1)
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2p| b|
\bigr]
, k \geq p+ n+ 1.
Putting n = 1 in Example 2.1, we get the following example.
Example 2.3. If f \in \BbbS 1(p, 0, 0, b), then
| ap+1| \leq 2p| b|
and
| ak| \leq
2p| b|
\Gamma (k - p+ 1)
k - (p+2)\prod
j=0
\bigl[
j + 1 + 2p| b|
\bigr]
, k \geq p+ 2.
Taking n = 1 in Example 2.2, we have (see [2], Corollary 1 and Theorem 4, and [10], Corollary 1
and Theorem 3 with A = 1 and B = - 1) the following example.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1314 M. K. AOUF, A. O. MOSTAFA, T. BULBOACĂ
Example 2.4. If f \in C1(p, 0, 0, b), then
| ap+1| \leq
2p2| b|
p+ 1
and
| ak| \leq
2p2| b|
k\Gamma (k - p+ 1)
k - (p+2)\prod
j=0
\bigl[
j + 1 + 2p| b|
\bigr]
, k \geq p+ 2.
Putting p = n = 1 and q = 0 in Theorem 2.1, we have the next corollary (see [1], Theorem 1,
and Deng [12], Theorem 1 with n = 0).
Corollary 2.3. If f \in \BbbS (\lambda , \beta , b) (see Remark 1.1 (iv)), then
| ak| \leq
2(1 - \beta )| b|
\Gamma (k)[1 + \lambda (k - 1)]
k - 3\prod
j=0
\bigl[
j + 1 + 2(1 - \beta )| b|
\bigr]
=
=
1
(k - 1)![1 + \lambda (k - 1)]
k - 2\prod
j=0
\bigl[
j + 2(1 - \beta )| b|
\bigr]
, k \geq 2.
Remark 2.1. Putting \lambda = 0 in Corollary 2.3, we get the result obtained by Deng [12] (Corollary 2
with n = 0).
If we take \beta = 0 in Corollary 2.3 we get the next result (see also [1], Corollary 1).
Example 2.5. If f \in \BbbS (\lambda , 0, b), then
| ak| \leq
1
(k - 1)![1 + \lambda (k - 1)]
k - 2\prod
j=0
\bigl[
j + 2| b|
\bigr]
, k \geq 2.
Putting \lambda = 0 in Example 2.5, we obtain the result of [17] (Theorems 2 and 3).
Example 2.6. If f \in \BbbS (0, 0, b), then
| ak| \leq
1
(k - 1)!
k - 2\prod
j=0
\bigl[
j + 2| b|
\bigr]
, k \geq 2.
Remark 2.2. For the special case \lambda = 1, Corollary 2.3 reduces to the result of [15] (Theorem 2).
3. Coefficient bounds for the function class \BbbB \bfitn (\bfitp , \bfitq , \bfitlambda , \bfitbeta , \bfitmu , \bfitb ).
Theorem 3.1. Let the function f \in \scrA n(p) defined by (1.1) belongs to the class \BbbB n(p, q, \lambda , \beta , \mu , b).
Then
| ap+n| \leq
2(p - q + \mu )(p - q + \mu + 1)
\bigl[
1 + \lambda (p - q - 1)
\bigr]
\delta (p, q + 1)(1 - \beta )| b|
n(p+ n - q + \mu )(p+ n - q + \mu + 1)
\bigl[
1 + \lambda (p+ n - q - 1)
\bigr]
\delta (p+ n, q)
and
| ak| \leq
2\Gamma (n)(p - q + \mu )(p - q + \mu + 1)[1 + \lambda (p - q - 1)]\delta (p, q + 1)(1 - \beta )| b|
(k - q + \mu )(k - q + \mu + 1)\Gamma (k - p+ 1)[1 + \lambda (k - q - 1)]\delta (k, q)
\times
\times
k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2p(1 - \beta )| b|
\bigr]
,
where k \geq p+ n+ 1 and \mu \in \BbbR with \mu > q - p.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
COEFFICIENT BOUNDS FOR MULTIVALENT CLASSES OF STARLIKE . . . 1315
Proof. For f \in \scrA n(p) given by (1.1), since f \in \BbbB n(p, q, \lambda , \beta , \mu , b), there exists a function
g \in \BbbH n(p, q, \lambda , \beta , b) of the form
g(z) = zp +
\infty \sum
k=p+n
bkz
k, z \in \BbbU ,
that satisfies the Cauchy – Euler differential equation (1.3). Equating the coefficients of both sides of
this differential equation it follows that
bk =
(k - q + \mu )(k - q + \mu + 1)
(p - q + \mu )(p - q + \mu + 1)
ak, k \geq p+ n, \mu > q - p.
Thus, by using the conclusions of Theorem 2.1 for the function g we obtain the required inequalities.
Theorem 3.1 is proved.
Putting q = 0 in Theorem 3.1, we get the next special case (see [11], Theorem 3.1 with m = 0).
Corollary 3.1. If f \in \scrG n(p, \lambda , \beta , \mu , b) := \BbbB n(p, 0, \lambda , \beta , \mu , b), then
| ap+n| \leq
2(p+ \mu )(p+ \mu + 1)[1 + \lambda (p - 1)](1 - \beta )| b|
n(p+ n+ \mu )(p+ n+ \mu + 1)
\bigl[
1 + \lambda (p+ n - 1)
\bigr]
and
| ak| \leq
2\Gamma (n)(p+ \mu )(p+ \mu + 1)
\bigl[
1 + \lambda (p - 1)
\bigr]
(1 - \beta )| b|
(k + \mu )(k + \mu + 1)\Gamma (k - p+ 1)
\bigl[
1 + \lambda (k - 1)
\bigr] k - (p+n+1)\prod
j=0
\bigl[
n+ j + 2p(1 - \beta )| b|
\bigr]
,
where k \geq p+ n+ 1 and \mu \in \BbbR with \mu > q - p.
Putting p = n = 1 in the last corollary, we have the following result (see also [1], Theorem 2,
[26], Corollary 4, and [12], Theorem 2 with n = 0).
Corollary 3.2. If f \in \scrH (\lambda , \beta , \mu , b) := \BbbB 1(1, 0, \lambda , \beta , b), then
| ak| \leq
(1 + \mu )(2 + \mu )
\prod k - 2
j=0
\bigl[
j + 2(1 - \beta )| b|
\bigr]
(k + \mu )(k + \mu + 1)[1 + \lambda (k - 1)](k - 1)!
, k \geq 2, \mu > - 1.
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ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
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| id | umjimathkievua-article-6258 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:46Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/15/664135bd3b4d45ec8d3d4076ca965615.pdf |
| spelling | umjimathkievua-article-62582023-01-07T13:45:38Z Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order Aouf, M. K. Mostafa, A. O. Bulboacă, T. Aouf, Mohamed K. Aouf, M. K. Mostafa, A. O. Bulboacă, T. CLASSES OF STARLIKE UDC 517.5 We determine coefficient bounds for functions from&nbsp; subclasses of $p$-valent starlike and $p$-valent convex functions defined by higher-order derivatives and complex order introduced with the help of a certain nonhomogeneous Cauchy – Euler differential equation for higher-order derivatives.&nbsp;&nbsp;Relevant connections of some of our results with the results obtained earlier&nbsp; are provided. УДК 517.5 Коефіцієнтні оцінки для &nbsp; багатозначних класів зіркоподібних та опуклих функцій, визначених похідними вищого порядку та комплексним порядком Знайдено оцінки для коефіцієнтів функцій, що належать до підкласів $p$-значних зіркоподібних і $p$-значних опуклих функцій, які визначаються похідними вищого порядку та комплексним порядком&nbsp; і вводяться&nbsp; за допомогою певного неоднорідного диференціального рівняння Коші – Ейлера для похідних вищого порядку.&nbsp;&nbsp;Наведено відповідні співвідношення між деякими нашими&nbsp; результатами та результатами, що були отримані раніше. Institute of Mathematics, NAS of Ukraine 2022-11-27 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6258 10.37863/umzh.v74i10.6258 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 10 (2022); 1308 - 1316 Український математичний журнал; Том 74 № 10 (2022); 1308 - 1316 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6258/9306 Copyright (c) 2022 Mohamed K. Aouf |
| spellingShingle | Aouf, M. K. Mostafa, A. O. Bulboacă, T. Aouf, Mohamed K. Aouf, M. K. Mostafa, A. O. Bulboacă, T. Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title | Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title_alt | Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title_full | Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title_fullStr | Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title_full_unstemmed | Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title_short | Coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| title_sort | coefficient bounds for multivalent classes of starlike and convex functions defined by higher-order derivatives and complex order |
| topic_facet | CLASSES OF STARLIKE |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6258 |
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