Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives
UDC 517.5 We investigate some generalizations of the classes of harmonic functions defined by the Sălăgean and Ruscheweyh derivatives. By using the extreme-points theory, we obtain the coefficient-estimates distortion theorems and mean integral  inequalities for these...
Gespeichert in:
| Datum: | 2022 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6157 |
| 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_ | 1860512278261006336 |
|---|---|
| author | Páll-Szabo, Á. O. Páll-Szabo, Á. O. |
| author_facet | Páll-Szabo, Á. O. Páll-Szabo, Á. O. |
| author_sort | Páll-Szabo, Á. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-01-07T13:45:37Z |
| description |
UDC 517.5
We investigate some generalizations of the classes of harmonic functions defined by the Sălăgean and Ruscheweyh derivatives. By using the extreme-points theory, we obtain the coefficient-estimates distortion theorems and mean integral  inequalities for these classes of functions. |
| doi_str_mv | 10.37863/umzh.v74i10.6157 |
| first_indexed | 2026-03-24T03:26:15Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i10.6157
UDC 517.5
Á. O. Páll-Szabo1 (Babeş-Bolyai Univ., Cluj-Napoca, Romania)
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS
DEFINED BY SĂLĂGEAN AND RUSCHEWEYH DERIVATIVES
УЗАГАЛЬНЕННЯ ЗIРКОПОДIБНИХ ГАРМОНIЧНИХ ФУНКЦIЙ,
ЩО ВИЗНАЧЕНI ПОХIДНИМИ САЛАГЕНА ТА РУШЕВЕЯ
We investigate some generalizations of the classes of harmonic functions defined by the Sălăgean and Ruscheweyh
derivatives. By using the extreme-points theory, we obtain the coefficient-estimates distortion theorems and mean integral
inequalities for these classes of functions.
Дослiджено деякi узагальнення класiв гармонiчних функцiй, що визначенi похiдними Салагена та Рушевея. З
використанням теорiї екстремальних точок отримaно теореми про спотворення оцiнок коефiцiєнтiв та нерiвностi
для iнтегральних середнiх для цих класiв функцiй.
1. Preliminaries. Let \scrA denote the class of functions of the form
f(z) = z +
\infty \sum
k=2
akz
k, (1)
which are analytic in the open unit disk U = \{ z \in \BbbC : | z| < 1\} .
A continuous function f = u+ iv is a complex-valued harmonic function in a complex domain
\scrG if both u and v are real and harmonic in \scrG . In any simply-connected domain D \subset \scrG , we can write
f = h+ g, where h and g are analytic in D. We call h the analytic part and g the co-analytic part
of f. A necessary and sufficient condition for f to be locally univalent and orientation preserving in
D is that | h\prime (z)| > | g\prime (z)| in D (see [2]).
Let \scrH denote the family of continuous complex-valued functions that are harmonic in U. Denote
by S\scrH the family of functions f \in \scrH of the form
f = h+ g, h(z) = z +
\infty \sum
k=2
akz
k, g(z) =
\infty \sum
k=2
bkz
k, (2)
which are univalent and orientation preserving in the open unit disc U. Thus, f(z) is then given by
f(z) = z +
\infty \sum
k=2
akz
k +
\infty \sum
k=2
bkzk. (3)
A function f of the form (3) is said to be in \scrS \ast
\scrH (\alpha ) if and only if (see [2, 4, 5])
\partial
\partial \theta
\Bigl(
\mathrm{a}\mathrm{r}\mathrm{g} f
\Bigl(
rei\theta
\Bigr) \Bigr)
> \alpha , 0 \leq \theta < 2\pi , | z| = r < 1, 0 \leq \alpha < 1. (4)
Similarly, a function f of the form (3) is said to be in \scrS c
\scrH (\alpha ) if and only if
1 e-mails: pallszaboagnes@math.ubbcluj.ro, agnes.pallszabo@econ.ubbcluj.ro.
c\bigcirc Á. O. PÁLL-SZABO, 2022
1388 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS . . . 1389
\partial
\partial \theta
\biggl(
\mathrm{a}\mathrm{r}\mathrm{g}
\partial
\partial \theta
\Bigl(
f
\Bigl(
rei\theta
\Bigr) \Bigr) \biggr)
> \alpha , 0 \leq \theta < 2\pi , | z| = r < 1. (5)
We note that (see [7]) a harmonic function f \in \scrS \ast
\scrH (\alpha ) if and only if
\Re J\scrH f(z)
f(z)
> \alpha , | z| = r < 1, where J\scrH f(z) = zh\prime (z) - zg\prime (z).
Definition 1 [1]. For f \in \scrA , \lambda \geq 0 and n \in \BbbN , the operator Dn
\lambda is defined by Dn
\lambda : \scrA \rightarrow \scrA ,
D0
\lambda f(z) = f(z),
Dn+1
\lambda f(z) = (1 - \lambda )Dn
\lambda f(z) + \lambda z(Dn
\lambda f(z))
\prime = D\lambda (D
n
\lambda f(z)), z \in U.
Remark 1. If f \in \scrA , then
Dn
\lambda f(z) = z +
\infty \sum
k=2
[1 + (k - 1)\lambda ]nakz
k, z \in U.
Remark 2. For \lambda = 1 in the above definition we obtain the Sălăgean differential operator [13].
Definition 2 [12]. For f \in \scrA , n \in \BbbN , the operator Rn is defined by Rn : \scrA \rightarrow \scrA ,
R0f(z) = f(z),
(n+ 1)Rn+1f(z) = z(Rnf(z))\prime + nRnf(z), z \in U.
Remark 3. If f \in \scrA , then
Rnf(z) = z +
\infty \sum
k=2
(n+ k - 1)!
n!(k - 1)!
akz
k, z \in U,
which is the Ruscheweyh differential operator [12].
Definition 3. Let \gamma , \lambda \geq 0, n \in \BbbN . Denote by Ln the operator given by Ln : \scrA \rightarrow \scrA ,
Lnf(z) = (1 - \gamma )Rnf(z) + \gamma Dn
\lambda f(z), z \in U.
Remark 4. If f \in \scrA , then
Lnf(z) = z +
\infty \sum
k=2
\biggl\{
\gamma [1 + (k - 1)\lambda ]n + (1 - \gamma )
(n+ k - 1)!
n!(k - 1)!
\biggr\}
akz
k, z \in U.
We consider the linear operator Ln
\scrH : \scrH \rightarrow \scrH defined for a function f = h+ g \in \scrH by
Ln
\scrH f := Lnh+ ( - 1)nLng.
For a function f \in \scrH of the form (3), we have
Ln
\scrH f(z) = z +
\infty \sum
k=2
[\gamma \eta (k, n, \lambda ) + (1 - \gamma )\mu (k, n)]akz
k+
+( - 1)n
\infty \sum
k=2
[\gamma \eta (k, n, \lambda ) + (1 - \gamma )\mu (k, n)]bkz
k, z \in U,
where \eta (k, n, \lambda ) = [1 + (k - 1)\lambda ]n and \mu (k, n) =
(n+ k - 1)!
n!(k - 1)!
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1390 Á. O. PÁLL-SZABO
Definition 4. For - B \leq A < B \leq 1 and n \in \BbbN , let \widetilde \scrS n
\scrH (A,B) denote the class of functions
f \in \scrH of the form (3) such that\bigm| \bigm| \bigm| \bigm| \bigm| Ln+1
\scrH f(z) - Ln
\scrH f(z)
BLn+1
\scrH f(z) - ALn
\scrH f(z)
\bigm| \bigm| \bigm| \bigm| \bigm| < 1, z \in U. (6)
Remark 5. Dziok et al. studied the case \gamma = 0 in [3], while the case \gamma = 1 and \lambda = 1 was
studied in [4].
Note that the classes \widetilde \scrS 0
\scrH (A,B) for the analytic case, i.e., g \equiv 0, were introduced by Janowski
[8]. Jahangiri [6, 7] and Silverman [14] studied the classes \scrS \ast
\scrH (\alpha ) =
\widetilde \scrS 0
\scrH (2\alpha - 1, 1) and \scrS c
\scrH (\alpha ) =
= \widetilde \scrS 1
\scrH (2\alpha - 1, 1) for the harmonic case.
2. Coefficient estimates.
Theorem 1. A function f \in \scrH of the form (3) belongs to the class \widetilde \scrS n
\scrH (A,B) if it satisfies the
condition
\infty \sum
k=2
(\alpha k| ak| + \beta k| bk| ) \leq B - A, (7)
where
\alpha k = \sigma (A,B, n, \gamma , \lambda , k) + \sigma (1, 1, n, \gamma , \lambda , k),
\beta k = \delta (A,B, n, \gamma , \lambda , k) + \delta (1, 1, n, \gamma , \lambda , k),
\sigma (A,B, n, \gamma , \lambda , k) = \gamma \eta (k, n, \lambda )[(k - 1)\lambda B +B - A]+
+(1 - \gamma )\mu (k, n)
(B - A)n+Bk - A
n+ 1
,
\delta (A,B, n, \gamma , \lambda , k) = \gamma \eta (k, n, \lambda )[(k - 1)\lambda B +B +A]+
+(1 - \gamma )\mu (k, n)
(B +A)n+Bk +A
n+ 1
.
Proof. We know from Definition 4 that f \in \widetilde \scrS n
\scrH (A,B) if and only if\bigm| \bigm| \bigm| \bigm| \bigm| Ln+1
\scrH f(z) - Ln
\scrH f(z)
BLn+1
\scrH f(z) - ALn
\scrH f(z)
\bigm| \bigm| \bigm| \bigm| \bigm| < 1, z \in U.
It is sufficient to prove that\bigm| \bigm| Ln+1
\scrH f(z) - Ln
\scrH f(z)
\bigm| \bigm| - \bigm| \bigm| BLn+1
\scrH f(z) - ALn
\scrH f(z)
\bigm| \bigm| < 0, z \in U \setminus \{ 0\} .
Letting | z| = r, 0 < r < 1, we have\bigm| \bigm| Ln+1
\scrH f(z) - Ln
\scrH f(z)
\bigm| \bigm| - \bigm| \bigm| BLn+1
\scrH f(z) - ALn
\scrH f(z)
\bigm| \bigm| \leq
\leq
\infty \sum
k=2
\biggl[
\gamma \eta (k, n, \lambda )(k - 1)\lambda + (1 - \gamma )\mu (k, n)
k - 1
n+ 1
\biggr]
| ak| rk+
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS . . . 1391
+
\infty \sum
k=2
\biggl[
\gamma \eta (k, n, \lambda )[2 + (k - 1)\lambda ] + (1 - \gamma )\mu (k, n)
2n+ k + 1
n+ 1
\biggr]
| bk| rk - (B - A)r+
+
\infty \sum
k=2
\biggl[
\gamma \eta (k, n, \lambda )[(k - 1)\lambda B +B - A] + (1 - \gamma )\mu (k, n)
\biggl(
B
n+ k
n+ 1
- A
\biggr) \biggr]
| ak| rk+
+
\infty \sum
k=2
\biggl[
\gamma \eta (k, n, \lambda )[(k - 1)\lambda B +B +A] + (1 - \gamma )\mu (k, n)
\biggl(
B
n+ k
n+ 1
+A
\biggr) \biggr]
| bk| rk \leq
\leq r
\Biggl\{ \infty \sum
k=2
(\alpha k| ak| + \beta k| bk| )rk - 1 - (B - A)
\Biggr\}
< 0,
whence f \in \widetilde \scrS n
\scrH (A,B).
Theorem 1 is proved.
Lemma 1. If \lambda \geq 1, \gamma \in [0, 1], n \geq 0, - B \leq A < B \leq 1, k \in \BbbN , k \geq 2, then
\alpha k \geq k(B - A), \beta k \geq k(B - A),
where \alpha k, \beta k is defined in (7).
Proof. It is known that
\eta (k, n, \lambda ) = [1 + (k - 1)\lambda ]n \geq kn. (8)
First we prove that
\mu (k, n) =
(n+ k - 1)!
n!(k - 1)!
\geq n+ 1. (9)
For the proof we use the mathematical induction method.
1. Let k \geq 2 be fixed and n = 0, then \mu (k, 0) =
(k - 1)!
0!(k - 1)!
= 1 is true.
Let k \geq 2 be fixed and n = 1, then \mu (k, 1) =
k!
1!(k - 1)!
\geq 2 \leftrightarrow k! \geq 2(k - 1)! \leftrightarrow k \geq 2 is true.
2. Assume, for n = l, that the formula displayed below holds:
\mu (k, l) =
(l + k - 1)!
l!(k - 1)!
\geq l + 1 \leftrightarrow (l + k - 1)! \geq l!(k - 1)!(l + 1) = (l + 1)!(k - 1)! .
3. Let n = l + 1, so we have to prove that
\mu (k, l + 1) =
(l + k)!
(l + 1)!(k - 1)!
\geq l + 2 \leftrightarrow (l + k)! \geq (l + 1)!(k - 1)!(l + 2).
This holds using the previous item
(l + k)! = (l + k)(l + k - 1)! \geq (l + k)(l + 1)!(k - 1)! \geq (l + 2)(l + 1)!(k - 1)!.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1392 Á. O. PÁLL-SZABO
Now, using (8) and (9), we prove that \alpha k \geq k(B - A):
\alpha k = \sigma (A,B, n, \gamma , \lambda , k) + \sigma (1, 1, n, \gamma , \lambda , k) \geq
\geq \gamma kn[(k - 1)\lambda B +B - A]+
+(1 - \gamma )[(B - A)n+Bk - A] + \gamma kn(k - 1)\lambda + (1 - \gamma )(k - 1).
But
kn[(k - 1)\lambda B +B - A] + kn(k - 1)\lambda = kn[(B - A) + (k - 1)\lambda (B + 1)\underbrace{} \underbrace{}
>0
] >
> kn(B - A) > k(B - A)
and
(B - A)n+Bk - A+ (k - 1) \geq B(k - 1) +B - A+ k - 1 =
= (k - 1)(B + 1) +B - A \geq (k - 1)(B - A) +B - A = k(B - A).
So, \alpha k \geq \gamma (B - A)k + (1 - \gamma )(B - A)k = k(B - A).
Now we prove that \beta k \geq k(B - A):
\beta k = \delta (A,B, n, \gamma , \lambda , k) + \delta (1, 1, n, \gamma , \lambda , k) \geq
\geq \gamma kn[(k - 1)\lambda B +B +A] + (1 - \gamma )[(B +A)n+Bk +A]+
+\gamma kn[(k - 1)\lambda + 2] + (1 - \gamma )[2n+ k + 1] >
> \gamma kn[(k - 1)(B + 1) +B +A+ 2] + (1 - \gamma )[(B +A)n+ 2n+Bk + k +A+ 1].
But
(k - 1)(B + 1) +B +A+ 2 = kB + k + 1 +A \geq
\geq k(B - A), B \geq - 1, A \geq - 1,
k + 1 +A \geq - kA \leftrightarrow k(A+ 1) +A+ 1 \geq 0 \leftrightarrow (k + 1)(A+ 1) \geq 0
and
(B +A)n+ 2n+Bk + k +A+ 1 \geq Bk + k +A+ 1 \geq Bk - Ak,
because
k +A+ 1 \geq - Ak \leftrightarrow k(A+ 1) +A+ 1 \geq 0 \leftrightarrow (k + 1)(A+ 1) \geq 0.
So, \beta k \geq \gamma (B - A)k + (1 - \gamma )(B - A)k = k(B - A).
Lemma 1 is proved.
Lemma 2. If \lambda \geq 1, \gamma > 1, n \geq 0, - B \leq A < B \leq 1, k \in \BbbN , k \geq 2, then
\alpha k \geq k(B - A), \beta k \geq k(B - A),
where \alpha k, \beta k is defined in (7).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS . . . 1393
Proof. First we note that
\mu (k, n) =
(n+ k - 1)!
n!(k - 1)!
\leq kn, k, n \in \BbbN , k \geq 2. (10)
Let k be fixed. If n = 0 then (10) holds true.
Suppose that, for n, (10) is true, then, for n+ 1, we have
(n+ k)! = (n+ k)(n+ k - 1)! \leq (n+ k)knn!(k - 1)! \leq
\leq (n+ 1)kknn!(k - 1)! = kn(n+ 1)!(k - 1)!.
Now
\alpha k \geq \gamma kn[(k - 1)(B + 1) +B - A] - (\gamma - 1)kn
(B - A)n+Bk - A
n+ 1
by (8) and (10).
But
(B - A)n+Bk - A+ k - 1
n+ 1
< (B - A) + (k - 1)(B + 1)
and so
\alpha k \geq [\gamma - (\gamma - 1)][B - A+ (k - 1)(B + 1)]kn \geq k(B - A),
\beta k \geq \gamma kn[(k - 1)(B + 1) +B +A+ 2]+
+(1 - \gamma )kn
(B +A)n+ 2n+Bk + k +A+ 1
n+ 1
\geq
\geq kn[(k - 1)(B + 1) +B +A+ 2] \geq k(B - A),
because (B +A)n+ 2n+Bk + k +A+ 1 < (n+ 1)[(k - 1)(B + 1) +B +A+ 2].
Lemma 2 is proved.
Theorem 2. If f \in \scrH of the form (3) and f satisfies the condition (7), then f \in \scrS \scrH .
Proof. The theorem is true for the function f(z) \equiv z. Let f \in \scrH be a function of the form
(3) and let us assume that exists k \in \{ 2, 3, . . .\} such that ak \not = 0 or bk \not = 0. Since
\alpha k
B - A
\geq k,
\beta k
B - A
\geq k, k = 2, 3, . . . , proved in Lemma 1 and 2, then by (7) we have
\infty \sum
k=2
(k| ak| + k| bk| ) \leq 1 (11)
and
\bigm| \bigm| h\prime (z)\bigm| \bigm| - \bigm| \bigm| g\prime (z)\bigm| \bigm| \geq 1 -
\infty \sum
k=2
k| ak| | z| k -
\infty \sum
k=2
k| bk| | z| k \geq 1 - | z|
\infty \sum
k=2
(k| ak| + k| bk| ) \geq
\geq 1 - | z|
B - A
\infty \sum
k=2
(\alpha k| ak| + \beta k| bk| ) \geq 1 - | z| > 0, z \in U.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1394 Á. O. PÁLL-SZABO
In this case the function f is locally univalent and sense-preserving in U. Moreover, if z1, z2 \in U,
z1 \not = z2, then \bigm| \bigm| \bigm| \bigm| zk1 - zk2
z1 - z2
\bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm|
k\sum
l=1
zl - 1
1 zk - l
2
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
k\sum
l=1
| z1| l - 1| z2| k - 1 < k, k = 2, 3, . . . .
Therefore, by (11), we have
| f(z1) - f(z2)| \geq | h(z1) - h(z2)| - | g(z1) - g(z2)| \geq
\geq
\bigm| \bigm| \bigm| \bigm| \bigm| z1 - z2 -
\infty \sum
k=2
ak
\Bigl(
zk1 - zk2
\Bigr) \bigm| \bigm| \bigm| \bigm| \bigm| -
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k=2
bk
\bigl(
zk1 - zk2
\bigr) \bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq | z1 - z2|
\Biggl(
1 -
\infty \sum
k=2
| ak|
\bigm| \bigm| \bigm| \bigm| zk1 - zk2
z1 - z2
\bigm| \bigm| \bigm| \bigm| - \infty \sum
k=2
| bk|
\bigm| \bigm| \bigm| \bigm| zk1 - zk2
z1 - z2
\bigm| \bigm| \bigm| \bigm|
\Biggr)
>
> | z1 - z2|
\Biggl(
1 -
\infty \sum
k=2
k| ak| -
\infty \sum
k=2
k| bk|
\Biggr)
\geq 0.
This leads to the univalence of f, so f \in \scrS \scrH .
Theorem 2 is proved.
Let \scrN denote the class of functions f = h+ g \in \scrH of the form (see [14])
f(z) = z -
\infty \sum
k=2
| ak| zk + ( - 1)n
\infty \sum
k=2
| bk| zk, (12)
and denote by \widetilde \scrS n
\scrH \scrN (A,B) the class \scrN \cap \widetilde \scrS n
\scrH (A,B).
Theorem 3. Let f = h+g be defined by (12). Then f \in \widetilde \scrS n
\scrH \scrN (A,B) if and only if the condition
(7) holds true.
Proof. For the ‘if’ part see Theorem 1. For the ‘only if’ part, assume that f \in \widetilde \scrS n
\scrH \scrN (A,B), then,
by (6), we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum \infty
k=2
\Bigl[
\sigma (1, 1, n, \gamma , \lambda , k)| ak| zk - 1 + \delta (1, 1, n, \gamma , \lambda , k)| bk| zk - 1
\Bigr]
(B - A) -
\sum \infty
k=2
\Bigl[
\sigma (A,B, n, \gamma , \lambda , k)| ak| zk - 1 + \delta (A,B, n, \gamma , \lambda , k)| bk| zk - 1
\Bigr]
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < 1, z \in U.
For z = r < 1, we obtain\sum \infty
k=2
[\sigma (1, 1, n, \gamma , \lambda , k)| ak| + \delta (1, 1, n, \gamma , \lambda , k)| bk| ]rk - 1
(B - A) -
\sum \infty
k=2
[\sigma (A,B, n, \gamma , \lambda , k)| ak| + \delta (A,B, n, \gamma , \lambda , k)| bk| ]rk - 1
< 1.
The denominator of the left-hand side can not vanish for r \in [0, 1) and it is positive. So\sum \infty
k=2
(\alpha k| ak| + \beta k| bk| )rk - 1 \leq B - A, which, upon letting r \rightarrow 1 - , yields to assertion (7).
Theorem 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS . . . 1395
3. Extreme points.
Definition 5. We say that a class \scrF is convex if \eta f + (1 - \eta )g \in \scrF for all f and g in \scrF and
0 \leq \eta \leq 1. The closed convex hull of \scrF , denoted by \mathrm{c}\mathrm{o}\scrF , is the intersection of all closed convex
subsets of \scrH (with respect to the topology of locally uniform convergence) that contain \scrF .
Definition 6. Let \scrF be a convex set. A function f \in \scrF \subset \scrH is called an extreme point of \scrF if
f = \eta f1 + (1 - \eta )f2 implies f1 = f2 = f for all f1 and f2 in \scrF and 0 < \eta < 1. We shall use the
notation E\scrF to denote the set of all extreme points of \scrF . It is clear that E\scrF \subset \scrF .
For the extreme points we use the Krein – Milman theorem (see [3, 4, 9]) which implies.
Lemma 3 [3, 4]. Let \scrF be a non-empty compact convex subclass of the class \scrH and \scrJ : \scrH \rightarrow \BbbR
be a real-valued, continuous, and convex functional on \scrF . Then
\mathrm{m}\mathrm{a}\mathrm{x}\{ \scrJ (f) : f \in \scrF \} = \mathrm{m}\mathrm{a}\mathrm{x}\{ \scrJ (f) : f \in E\scrF \} .
Since \scrH is a complete metric space, we can use Montel’s theorem [10].
Lemma 4 [3, 4]. A class \scrF \subset \scrH is compact if and only if \scrF is closed and locally uniformly
bounded.
Theorem 4. The class \widetilde \scrS n
\scrH \scrN (A,B) is a convex and compact subset of \scrH .
Proof. For 0 \leq \eta \leq 1, let f1, f2 \in \widetilde \scrS n
\scrH \scrN (A,B) be defined by (2). Then
\eta f1(z) + (1 - \eta )f2(z) = z -
\infty \sum
k=2
(\eta | a1,k| + (1 - \eta )| a2,k| )zk+
+( - 1)n
\infty \sum
k=2
(\eta | b1,k| + (1 - \eta )| b2,k| zk)
and
\infty \sum
k=2
\Bigl\{
\alpha k| \eta | a1,k| + (1 - \eta )| a2,k| | + \beta k
\bigm| \bigm| \bigm| \eta | b1,k| + (1 - \eta )| b2,k| zk
\bigm| \bigm| \bigm| \Bigr\} =
= \eta
\infty \sum
k=2
\{ \alpha k| a1,k| + \beta k| b1,k| \} + (1 - \eta )
\infty \sum
k=2
\alpha k| a2,k| + \beta k| b2,k| \leq
\leq \eta (B - A) + (1 - \eta )(B - A).
Therefore, the function \phi = \eta f1 + (1 - \eta )f2 belongs to the class \widetilde \scrS n
\scrH \scrN (A,B), so \widetilde \scrS n
\scrH \scrN (A,B) is
convex.
On the other hand, for f \in \widetilde \scrS n
\scrH \scrN (A,B), | z| \leq r and 0 < r < 1, we have
| f(z)| \leq r +
\infty \sum
k=2
(| ak| + | bk| )rn \leq r +
\infty \sum
k=2
(\alpha k| ak| + \beta k| bk| ) \leq r + (B - A).
From this comes that \widetilde \scrS n
\scrH \scrN (A,B) is locally uniformly bounded. Let
fe(z) = z +
\infty \sum
k=2
ae,kz
k +
\infty \sum
k=1
be,kzk, z \in U, k \in \BbbN ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1396 Á. O. PÁLL-SZABO
and f \in \scrH . Using Theorem 3, we have
\infty \sum
k=2
(\alpha k| ae,k| + \beta k| be,k| ) \leq B - A, k \in \BbbN .
If fe \rightarrow f, then | ae,k| \rightarrow | ak| and | be,k| \rightarrow | bk| when k \rightarrow \infty , k \in \BbbN . This gives condition (7).
Therefore, f \in \widetilde \scrS n
\scrH \scrN (A,B) and the class \widetilde \scrS n
\scrH \scrN (A,B) is closed. We can now say, by Lemma 3, that
the class \widetilde \scrS n
\scrH \scrN (A,B) is compact subset of \scrH .
Theorem 4 is proved.
Theorem 5. The set of extreme points of the class \widetilde \scrS n
\scrH \scrN (A,B) is E \widetilde \scrS n
\scrH \scrN (A,B) = \{ hk : k \in \BbbN \} \cup
\cup \{ gk : k \in \{ 2, 3, . . .\} \} , where
h1 = z, hk(z) = z - B - A
\alpha k
zk,
gk(z) = z + ( - 1)n
B - A
\beta k
zk, z \in U, k \in \{ 2, 3, . . .\} . (13)
Proof. If we use (7), we can see that the functions of the above form are the extreme points of the
class \widetilde \scrS n
\scrH \scrN (A,B). Supposing that f \in E \widetilde \scrS n
\scrH \scrN (A,B) and f is not of the form seen above, there exists
m \in \{ 2, 3, . . .\} such that 0 < | am| < B - A
\alpha m
or 0 < | bm| < B - A
\beta m
. If 0 < | am| < B - A
\alpha m
, then
putting \gamma =
| am| \alpha m
B - A
,\varphi =
1
1 - \eta
(f - \eta hm), we have 0 < \eta < 1, hm, \varphi \in \widetilde \scrS \ast
\scrH \scrN (A,B), hm \not = \varphi
and f = \eta hm + (1 - \eta )\varphi . Thus, f /\in E \widetilde \scrS n
\scrH \scrN (A,B). We get the same result for 0 < | bm| < B - A
\beta m
.
Theorem 5 is proved.
If the class \scrF = \{ fk \in \scrH : k \in \BbbN \} is locally uniformly bounded, then its closed convex hull is
\mathrm{c}\mathrm{o}\scrF =
\Biggl\{ \infty \sum
k=1
\eta kfk :
\infty \sum
k=1
\eta k = 1, \eta k \geq 0, k \in \BbbN
\Biggr\}
.
Corollary 1. Let hk, gk be defined by (13), then
\widetilde \scrS n
\scrH \scrN (A,B) =
\Biggl\{ \infty \sum
k=1
(\eta khk + \delta kgk) :
\infty \sum
k=1
(\eta k + \delta k) = 1, \delta 1 = 0, \eta k, \delta k \geq 0, k \in \BbbN
\Biggr\}
.
For each fixed value of k \in \BbbN , z \in U, the following real-valued functionals are continuous and
convex on \scrH :
\scrJ (f) = | ak| , \scrJ (f) = | bk| , \scrJ (f) = | f(z)| , \scrJ (f) =
\bigm| \bigm| \bigm| Lk
\scrH f(z)
\bigm| \bigm| \bigm| , f \in \scrH .
The real-valued functional
\scrJ (f) =
\left( 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| f\Bigl( rei\theta \Bigr) \bigm| \bigm| \bigm| \gamma d\theta
\right) 1/\gamma
, f \in \scrH , \gamma \geq 1, 0 < r < 1,
is continuous on \scrH . For \gamma \geq 1 it is also convex on \scrH (Minkowski’s inequality).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS . . . 1397
Corollary 2. Let f \in \widetilde \scrS n
\scrH \scrN (A,B) be a function of the form (12). Then
| ak| \leq
B - A
\alpha k
, | bk| \leq
B - A
\beta k
, k = 2, 3, . . . ,
where \alpha k, \beta k are defined by (7). The result is sharp. The extremal functions are hk, gk of the
form (13).
Theorem 6. Let f \in \widetilde \scrS n
\scrH \scrN (A,B) and | z| = r < 1. Then
r - B - A
\alpha 2
r2 \leq | f(z)| \leq r +
B - A
\alpha 2
r2,
r - (B - A)[\gamma (1 + \lambda )n + (1 - \gamma )(n+ 1)]
\alpha 2
r2 \leq | Ln
\scrH f(z)| \leq
\leq r +
(B - A)[\gamma (1 + \lambda )n + (1 - \gamma )(n+ 1)]
\alpha 2
r2.
The result is sharp. The extremal functions are h2 of the form (13).
Proof. We only prove the right-hand side inequality. The proof for the left-hand side inequality
is similar and will be omitted. We have
| f(z)| \leq r +
\infty \sum
k=2
(| ak| + | bk| )rk \leq r +
\infty \sum
k=2
(| ak| + | bk| )r2 \leq
\leq r +
\Biggl(
1
\alpha 2
\infty \sum
k=2
\alpha 2| ak| +
1
\beta 2
\infty \sum
k=2
\beta 2| bk|
\Biggr)
r2 \leq
\leq r +
1
\alpha 2
\infty \sum
k=2
(\alpha k| ak| + \beta k| bk| )r2 \leq
\leq r +
B - A
\alpha 2
r2, \alpha 2 \leq \alpha k, \alpha 2 \leq \beta 2, \beta 2 \leq \beta k for all k \geq 2.
An other proof can be made using the Lemma 3 with extreme points.
Theorem 6 is proved.
Corollary 3. If f \in \widetilde \scrS n
\scrH \scrN (A,B), then U(r) \subset f(U(r)), where
r = 1 - B - A
\alpha 2
and
U(r) := \{ z \in \BbbC : | z| < r \leq 1\} .
Corollary 4. Let 0 < r < 1 and \xi \geq 1. If f \in \widetilde \scrS n
\scrH \scrN (A,B), then
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| f\Bigl( rei\theta \Bigr) \bigm| \bigm| \bigm| \xi d\theta \leq 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| h2\Bigl( rei\theta \Bigr) \bigm| \bigm| \bigm| \xi d\theta ,
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| Lk
\scrH f
\Bigl(
rei\theta
\Bigr) \bigm| \bigm| \bigm| \xi d\theta \leq 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| Lk
\scrH h2
\Bigl(
rei\theta
\Bigr) \bigm| \bigm| \bigm| \xi d\theta , \xi = 1, 2, . . . .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1398 Á. O. PÁLL-SZABO
4. Radii of starlikeness and convexity. We note that a harmonic function f \in \scrS \ast
\scrH (\alpha ) if and
only if
\Re L\scrH f(z)
f(z)
> \alpha , | z| = r < 1,
where L\scrH f(z) = zh\prime (z) - zg\prime (z). For 0 \leq \alpha < 1, f \in \scrS c
\scrH (\alpha ) is equivalent with L\scrH f(z) \in \scrS \ast
\scrH (\alpha ).
Let \scrB \subseteq \scrH . We define the radius of starlikeness and the radius of convexity of the class \scrB :
R\ast
\alpha (\scrB ) := \mathrm{i}\mathrm{n}\mathrm{f}
f\in \scrB
(\mathrm{s}\mathrm{u}\mathrm{p}\{ r \in (0, 1] : f is starlike of order \alpha \in U(r)\} ),
Rc
\alpha (\scrB ) := \mathrm{i}\mathrm{n}\mathrm{f}
f\in \scrB
(\mathrm{s}\mathrm{u}\mathrm{p}\{ r \in (0, 1] : f is convex of order \alpha \in U(r)\} ).
Theorem 7. Let 0 \leq \alpha < 1 and \alpha k, \beta k be defined by (7). Then
R\ast
\alpha
\Bigl( \widetilde \scrS n
\scrH \scrN (A,B)
\Bigr)
= \mathrm{i}\mathrm{n}\mathrm{f}
k\geq 2
\biggl(
1 - \alpha
B - A
\mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
\alpha k
k - \alpha
,
\beta k
k + \alpha
\biggr\} \biggr) 1
k - 1
.
Proof. Let f \in \widetilde \scrS n
\scrH \scrN (A,B) be of the form (12).
We note that f is starlike of order \alpha in U(r) if and only if (see [7])
\infty \sum
k=2
\biggl(
k - \alpha
1 - \alpha
| ak| +
k + \alpha
1 - \alpha
| bk|
\biggr)
rk - 1 \leq 1. (14)
Also, we have, from Theorem 3, that
\infty \sum
k=2
\biggl(
\alpha k
B - A
| ak| +
\beta k
B - A
| bk|
\biggr)
\leq 1.
Since \alpha k < \beta k, k = 2, 3, . . . , the condition (14) is true if
k - \alpha
1 - \alpha
rk - 1 \leq \alpha k
B - A
and
k + \alpha
1 - \alpha
rk - 1 \leq \beta k
B - A
, k = 2, 3, . . . ,
or
r \leq
\biggl(
1 - \alpha
B - A
\mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
\alpha k
k - \alpha
,
\beta k
k + \alpha
\biggr\} \biggr) 1
k - 1
, k = 2, 3, . . . .
So, the function f is starlike of order \alpha in the disk U(r\ast ), where
r\ast := \mathrm{i}\mathrm{n}\mathrm{f}
k\geq 2
\biggl(
1 - \alpha
B - A
\mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
\alpha k
k - \alpha
,
\beta k
k + \alpha
\biggr\} \biggr) 1
k - 1
.
From the function
fk = hk(z) + gk(z) = z - B - A
\alpha k
zk + ( - 1)n
B - A
\beta k
zk
comes that the radius r\ast cannot be any larger.
Theorem 7 is proved.
Similarly, we get the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
GENERALIZATIONS OF STARLIKE HARMONIC FUNCTIONS . . . 1399
Theorem 8. Let 0 \leq \alpha < 1 and \alpha k and \beta k be defined by (7). Then
Rc
\alpha
\Bigl( \widetilde \scrS n
\scrH \scrN (A,B)
\Bigr)
= \mathrm{i}\mathrm{n}\mathrm{f}
k\geq 2
\biggl(
1 - \alpha
B - A
\mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
\alpha k
k(k - \alpha )
,
\beta k
k(k + \alpha )
\biggr\} \biggr) 1
k - 1
.
Now, we will examine the closure properties of the class \widetilde \scrS n
\scrH (A,B) under the generalized
Bernardi – Libera – Livingston integral operator \scrL c(f), c > - 1, which is defined by \scrL c(f) =
= \scrL c(h) + \scrL c(g), where
\scrL c(h)(z) =
c+ 1
zc
z\int
0
tc - 1h(t)dt \mathrm{a}\mathrm{n}\mathrm{d} \scrL c(g)(z) =
c+ 1
zc
z\int
0
tc - 1g(t)dt.
Theorem 9. Let f \in \widetilde \scrS n
\scrH (A,B). Then \scrL c(f) \in \widetilde \scrS n
\scrH (A,B).
Proof. From the representation of \scrL c(f(z)), it follows that
\scrL c(f)(z) =
c+ 1
zc
z\int
0
tc - 1
\Bigl[
h(t) + g(t)
\Bigr]
dt =
=
c+ 1
zc
\left[ z\int
0
tc - 1
\Biggl(
t -
\infty \sum
k=2
akt
k
\Biggr)
dt+
z\int
0
tc - 1
\Biggl(
t+ ( - 1)n
\infty \sum
k=2
bktk
\Biggr)
dt
\right] =
= z -
\infty \sum
k=2
Akz
k + ( - 1)n
\infty \sum
k=2
Bkz
k,
where
Ak =
c+ 1
c+ k
ak, Bk =
c+ 1
c+ k
bk.
Therefore,
\infty \sum
k=2
(\alpha k| Ak| + \beta k| Bk| ) \leq
\infty \sum
k=2
\biggl(
\alpha k
c+ 1
c+ k
| ak| + \beta k
c+ 1
c+ k
| bk|
\biggr)
\leq
\leq
\infty \sum
k=2
(\alpha k| ak| + \beta k| bk| ) \leq B - A.
Since f \in \widetilde \scrS n
\scrH (A,B), therefore by Theorem 1, \scrL c(f) \in \widetilde \scrS n
\scrH (A,B).
Theorem 9 is proved.
References
1. F. M. Al-Oboudi, On univalent functions defined by a generalized Sălăgean operator, Int. J. Math. and Math. Sci.,
27, 1429 – 1436 (2004).
2. J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Math., 9, 3 – 25 (1984).
3. J. Dziok, M. Darus, J. Sokol, T. Bulboacă, Generalizations of starlike harmonic functions, C. R. Acad. Sci. Paris,
Ser. I, 354, 13 – 18 (2016).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1400 Á. O. PÁLL-SZABO
4. J. Dziok, J. Jahangiri, H. Silverman, Harmonic functions with varying coefficients, J. Inequal. and Appl., 139 (2016);
DOI 10.1186/s13660-016-1079-z.
5. P. L. Duren, Harmonic mappings in the plane, Cambridge Tracts in Math., 156 (2004).
6. J. M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann.
Univ. Mariae Curie-Sklodowska Sect. A, 52, № 2, 57 – 66 (1998).
7. J. M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. and Appl., 235, 470 – 477 (1999).
8. W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math., 28, 297 – 326
(1973).
9. M. Krein, D. Milman, On the extreme points of regularly convex sets, Stud. Math., 9, 133 – 138 (1940).
10. P. Montel, Sur les families de functions analytiques qui admettent des valeurs exceptionelles dans un domaine, Ann.
Sci. Éc. Norm. Super., 23, 487 – 535 (1912).
11. G. Murugusundaramoorthy, K. Vijaya, R. K. Raina, A subclass of harmonic functions with varying arguments defined
by Dziok – Srivastava operator, Arch. Math., 45, № 1, 37 – 46 (2009).
12. St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49, 109 – 115 (1975).
13. G. S. Sălăgean, Subclasses of univalent functions, Lect. Notes in Math., 1013, 362 – 372 (1983).
14. H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. and Appl., 220, 283 – 289
(1998).
Received 07.06.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
|
| id | umjimathkievua-article-6157 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:15Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c7/408fcd7355c76394422ec883c2d174c7.pdf |
| spelling | umjimathkievua-article-61572023-01-07T13:45:37Z Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives Páll-Szabo, Á. O. Páll-Szabo, Á. O. harmonic functions Salagean and Ruscheweyh derivative negative coefficients UDC 517.5 We investigate some generalizations of the classes of harmonic functions defined by the Sălăgean and Ruscheweyh derivatives.&nbsp;By using the extreme-points theory, we obtain the coefficient-estimates distortion theorems and mean integral&nbsp; inequalities for these classes of functions. УДК 517.5 Узагальнення зіркоподібних гармонічних функцій, що визначені похідними Салагена та Рушевея Досліджено деякі узагальнення класів гармонічних функцій, що визначені похідними&nbsp; Салагена та&nbsp; Рушевея.&nbsp;З використанням теорії екстремальних точок отримaно теореми про спотворення оцінок коефіцієнтів та нерівності для інтегральних середніх&nbsp; для цих класів функцій. Institute of Mathematics, NAS of Ukraine 2022-11-27 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6157 10.37863/umzh.v74i10.6157 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 10 (2022); 1388 - 1400 Український математичний журнал; Том 74 № 10 (2022); 1388 - 1400 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6157/9314 Copyright (c) 2022 Agnes-Orsolya Pall-Szabo |
| spellingShingle | Páll-Szabo, Á. O. Páll-Szabo, Á. O. Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title | Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title_alt | Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title_full | Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title_fullStr | Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title_full_unstemmed | Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title_short | Generalizations of starlike harmonic functions defined by Sălăgean and Ruscheweyh derivatives |
| title_sort | generalizations of starlike harmonic functions defined by sălăgean and ruscheweyh derivatives |
| topic_facet | harmonic functions Salagean and Ruscheweyh derivative negative coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6157 |
| work_keys_str_mv | AT pallszaboao generalizationsofstarlikeharmonicfunctionsdefinedbysalageanandruscheweyhderivatives AT pallszaboao generalizationsofstarlikeharmonicfunctionsdefinedbysalageanandruscheweyhderivatives |