Classes of harmonic functions defined by extended Sălăgean operator
UDC 517.57 The object of the present paper is to investigate classes of harmonic functions defined by the extended Sălăgea operator. By using the extreme points theory we obtain coefficients estimates and distortion theorems for these classes of functions. Some integral mean inequalities are also po...
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2021
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| author | Dziok, J. Dziok, J. |
| author_facet | Dziok, J. Dziok, J. |
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The object of the present paper is to investigate classes of harmonic functions defined by the extended Sălăgea operator. By using the extreme points theory we obtain coefficients estimates and distortion theorems for these classes of functions. Some integral mean inequalities are also pointed out.
 
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DOI: 10.37863/umzh.v73i1.78
UDC 517.57
J. Dziok (Univ. Rzeszów, Poland)
CLASSES OF HARMONIC FUNCTIONS
DEFINED BY EXTENDED SĂLĂGEAN OPERATOR*
КЛАСИ ГАРМОНIЧНИХ ФУНКЦIЙ, ЯКI ВИЗНАЧЕНI
РОЗШИРЕНИМ ОПЕРАТОРОМ САЛАДЖАНА
The object of the present paper is to investigate classes of harmonic functions defined by the extended Sălăgean operator.
By using the extreme points theory we obtain coefficients estimates and distortion theorems for these classes of functions.
Some integral mean inequalities are also pointed out.
Дослiджуються класи гармонiчних функцiй, якi визначенi розширеним оператором Саладжана. За допомогою теорiї
екстремальних точок отримано оцiнки для коефiцiєнтiв та теореми деформацiї для класiв функцiй. Також наведено
деякi нерiвностi для iнтегральних середнiх.
1. Introduction. A complex-valued harmonic mapping f in the open unit disk \BbbU := \BbbU (1), where
\BbbU (r) := \{ z \in \BbbC : | z| < r\} , has a canonical decomposition
f = h+ g, (1)
where h and g are analytic functions in \BbbU . We call h the analytic part and g the coanalytic part of
f, respectively. Throughout this paper, we will discuss harmonic mappings that are sense-preserving
in \BbbU . By a theorem of Lewy [16], necessary and sufficient condition for f to be locally univalent
and sense-preserving in \BbbU is \bigm| \bigm| h\prime (z)\bigm| \bigm| > \bigm| \bigm| g\prime (z)\bigm| \bigm| , z \in \BbbU . (2)
Let \scrH denote the class of sense-preserving harmonic functions in the unit disc \BbbU . Any function
f \in \scrH can be written in the form
f(z) =
\infty \sum
n=0
anz
n +
\infty \sum
n=1
bnzn, z \in \BbbU . (3)
Let \BbbN l := \{ l, l + 1, . . .\} , \BbbN := \BbbN 1, k \in \BbbN 2. We denote by \scrS \scrH (k) the class of function f \in \scrH of
the form
f(z) = z +
\infty \sum
n=k
\bigl(
anz
n + bnzn
\bigr)
, z \in \BbbU , (4)
which are univalent in \BbbU .
We say that a function f \in \scrS \scrH (k) is harmonic starlike in \BbbU (r) if
\partial
\partial t
\bigl(
\mathrm{a}\mathrm{r}\mathrm{g} f
\bigl(
reit
\bigr) \bigr)
> 0,
0 \leq t \leq 2\pi , i.e., f maps the circle \partial \BbbU (r) onto a closed curve that is starlike with respect to the
origin. It is easy to verify, that the condition (5) is equivalent to the following:
* This paper was supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge,
University of Rzeszów.
c\bigcirc J. DZIOK, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1 33
34 J. DZIOK
\mathrm{R}\mathrm{e}
D\scrH f(z)
f(z)
> 0, | z| = r,
where
D\scrH f(z) := zh\prime (z) - zg\prime (z), z \in \BbbU .
For \lambda , \tau \in \BbbC , | \tau | = 1 and f = h+ g \in \scrH of the form (1), we consider the linear operator J\lambda ,\tau
\scrH :
\scrH \rightarrow \scrH defined by
J\lambda ,\tau
\scrH f(z) := z +
\infty \sum
n=2
n\lambda anz
n + \tau
\infty \sum
n=2
n\lambda bnz
n, z \in \BbbU .
For the analytic definition of the above case, see Sălăgean operator [20]. The operator J\lambda ,\tau
\scrH for
\lambda = n \in \BbbN , \tau = ( - 1)n was investigated in [17] (see also [5, 9, 11, 22]).
We say that a function f \in \scrH is subordinate to a function F \in \scrH , and write f(z) \prec F (z)
(or simply f \prec F ) if there exists a complex-valued function \omega which maps \BbbU into oneself with
\omega (0) = 0 such that f(z) = F (\omega (z)), z \in \BbbU .
Let A,B be complex parameters, A \not = B. We denote by \scrS \lambda ,\tau
\scrH (k;A,B) the class of functions
f \in \scrS \scrH (k) such that
J\lambda +1, - \tau
\scrH f(z)
J\lambda ,\tau
\scrH f(z)
\prec 1 +Az
1 +Bz
. (5)
Also, by \scrR \lambda ,\tau
\scrH (k;A,B) we denote the class of functions f \in \scrS \scrH (k) such that
J\lambda ,\tau
\scrH f(z)
z
\prec 1 +Az
1 +Bz
.
In particular, if we put \lambda = n \in \BbbN 0, \tau = ( - 1)n , then we obtain the classes
\scrS n
\scrH (k;A,B) := \scrS n,( - 1)n
\scrH (k;A,B), \scrR n
\scrH (k;A,B) := \scrR n,( - 1)n
\scrH (k;A,B)
studied in [8]. The classes \scrS \scrH (k;A,B) := \scrS 0
\scrH (k;A,B), \scrK \scrH (k;A,B) := \scrS 1
\scrH (k;A,B) and
\scrR \scrH (k;A,B) := \scrR 1
\scrH (k;A,B) are defined in [5] (see also) with restrictions - B \leq A < B \leq 1,
k = 2.
In this paper, we obtain some necessary and sufficient conditions for defined classes of functions.
Some topological properties and extreme points of the classes are also considered. By using extreme
points theory we obtain coefficients estimates, distortion theorems, integral mean inequalities for
these classes of functions.
2. Dual sets. For functions f1, f2 \in \scrH of the form
fl(z) = z +
\infty \sum
n=k
al,nz
n +
\infty \sum
n=k
bl,nzn, z \in \BbbD , l \in \BbbN , (6)
we define the Hadamard product or convolution of f1 and f2 by
(f1 \ast f2) (z) =
\infty \sum
k=0
\Bigl(
a1,ka2,kz
k + b1,kb2,kzk
\Bigr)
, z \in \BbbU .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
CLASSES OF HARMONIC FUNCTIONS DEFINED BY EXTENDED SĂLĂGEAN OPERATOR 35
Let \scrV \subset \scrH , \BbbU 0 := \BbbU \smallsetminus \{ 0\} . Motivated by Ruscheweyh [19] we define the dual set of \scrV by
\scrV \ast :=
\left\{ f \in \scrS \scrH (k) :
\bigwedge
q\in \scrV
(f \ast q) (z) \not = 0, z \in \BbbU 0
\right\} .
The object of the section is to show that the defined classes of functions can be presented as dual
sets.
Theorem 1.
\scrS \lambda ,\tau
\scrH (k;A,B) =
\Bigl\{
J\lambda ,\tau
\scrH (\psi \xi ) : | \xi | = 1
\Bigr\} \ast
,
where
\psi \xi (z) := z
(B - A) \xi + (1 +A\xi ) z
(1 - z)2
- z
2 + (A+B) \xi - (1 +A\xi ) z
(1 - z)2
, z \in \BbbU . (7)
Proof. Let f \in \scrH . Then f \in \scrH \lambda (A,B) if and only if and the condition (5) holds or equivalently
J\lambda +1
\scrH f(z)
J\lambda
\scrH f(z)
\not = 1 +A\zeta
1 +B\zeta
, \zeta \in \BbbC , | \zeta | = 1. (8)
Now for J\lambda +1
\scrH h(z) = J\lambda
\scrH h(z) \ast z/(1 - z)2 and J\lambda
\scrH h(z) = J\lambda
\scrH h(z) \ast z/(1 - z), the above inequality
(8) yields
(1 +B\zeta ) J\lambda +1
\scrH f(z) - (1 +A\zeta ) J\lambda
\scrH f(z) =
= (1 +B\zeta ) J\lambda +1
\scrH h(z) - (1 +A\zeta ) J\lambda
\scrH h(z) -
- ( - 1)\lambda
\Bigl[
(1 +B\zeta ) J\lambda +1
\scrH g(z) + (1 +A\zeta ) J\lambda
\scrH g(z)
\Bigr]
=
= J\lambda
\scrH h(z) \ast
\biggl(
(1 +B\zeta ) z
(1 - z)2
- (1 +A\zeta ) z
1 - z
\biggr)
-
- ( - 1)\lambda +1 J\lambda
\scrH g (z) \ast
\biggl(
(1 +B\zeta ) z
(1 - z)2
+
(1 +A\zeta ) z
1 - z
\biggr)
=
= J\lambda
\scrH f(z) \ast \phi (z; \zeta ) \not = 0.
Let f \in \scrS \scrH (k) be of the form (1). Then f \in \scrS \lambda ,\tau
\scrH (k;A,B) if and only if it satisfies (5) or
equivalently
J\lambda +1, - \tau
\scrH f(z)
J\lambda ,\tau
\scrH f(z)
\not = 1 +A\xi
1 +B\xi
, z \in \BbbU 0, | \xi | = 1. (9)
Since
J\lambda +1, - \tau
\scrH h(z) = J\lambda ,\tau
\scrH h(z) \ast z
(1 - z)2
, J\lambda ,\tau
\scrH h(z) = J\lambda ,\tau
\scrH h(z) \ast z
1 - z
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
36 J. DZIOK
the above inequality yields
0 \not = (1 +B\xi ) J\lambda +1, - \tau
\scrH f (z) - (1 +A\xi ) J\lambda ,\tau
\scrH f (z) =
= (1 +B\xi ) J\lambda +1, - \tau
\scrH h(z) - (1 +A\xi ) J\lambda ,\tau
\scrH h(z) -
- \tau
\Bigl[
(1 +B\xi ) J\lambda +1, - \tau
\scrH g(z) + (1 +A\xi ) J\lambda ,\tau
\scrH h(z)
\Bigr]
=
= J\lambda ,\tau
\scrH h(z) \ast
\biggl(
(1 +B\xi ) z
(1 - z)2
- (1 +A\xi ) z
1 - z
\biggr)
-
- \tau g(z) \ast
\biggl(
(1 +B\zeta ) z
(1 - z)2
+
(1 +A\zeta ) z
1 - z
\biggr)
=
= f(z) \ast J\lambda ,\tau
\scrH \psi \xi (z) , z \in \BbbU 0, | \xi | = 1.
Thus, f \in \scrS \lambda ,\tau
\scrH (k;A,B) if and only if f(z) \ast J\lambda ,\tau
\scrH \psi \xi (z) \not = 0 for z \in \BbbU 0, | \xi | = 1, i.e.,
\scrS \lambda ,\tau
\scrH (k;A,B) =
\bigl\{
J\lambda ,\tau
\scrH (\psi \xi ) : | \xi | = 1
\bigr\} \ast
.
Similarly as Theorem 1 we prove the following theorem.
Theorem 2.
\scrR \lambda ,\tau
\scrH (k;A,B) =
\Bigl\{
J\lambda ,\tau
\scrH (\delta \xi ) : | \xi | = 1
\Bigr\} \ast
,
where
\delta \xi (z) := z
1 +B\xi - (1 +A\xi ) (1 - z)
1 - z
+ \tau z
1 +B\xi
1 - z
, z \in \BbbU . (10)
If we put \lambda = 0 or \lambda = 1, \tau = ( - 1)n in Theorems 1 and 2 we obtain the following results
(see [5]).
Theorem 3.
\scrS \scrH (k;A,B) = \{ \psi \xi : | \xi | = 1\} \ast ,
where \psi \xi is defined by (7).
Theorem 4.
\scrK \scrH (k;A,B) = \{ \psi \xi : | \xi | = 1\} \ast ,
where
\psi \xi (z) := z
(B - A) \xi + (2 +A\xi +B\xi ) z
(1 - z)3
- z
2 + (A+B) \xi + (B - A) \xi z
(1 - z)3
, z \in \BbbU .
Theorem 5.
\scrR \scrH (k;A,B) = \{ \delta \xi : | \xi | = 1\} \ast ,
where
\delta \xi (z) := z
1 +B\xi - (1 +A\xi ) (1 - z)2
(1 - z)2
- z
1 +B\xi
(1 - z)2
, z \in \BbbU .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
CLASSES OF HARMONIC FUNCTIONS DEFINED BY EXTENDED SĂLĂGEAN OPERATOR 37
3. Correlated coefficients. Let us consider the function \varphi \in \scrH of the form
\varphi = u+ v, u(z) =
\infty \sum
n=0
unz
n, v(z) =
\infty \sum
n=1
vnz
nm, z \in \BbbU . (11)
We say that a function f \in \scrH of the form (4) has coefficients correlated with the function \varphi if
unan = - | un| | an| , vnbn = | vn| | bn| , n \in \BbbN k. (12)
In particular, if there exists a real number \eta such that
\varphi (z) =
z
1 - ei\eta z
+
e2i\eta z
1 - ei\eta z
=
\infty \sum
n=1
\Bigl(
ei(n - 1)\eta zn + ei(n+1)\eta zn
\Bigr)
, z \in \BbbU ,
then we obtain functions with varying coefficients defined by Jahangiri and Silverman [12] (see also
[8]). Moreover, if we take
\varphi (z) = 2\mathrm{R}\mathrm{e}
z
1 - z
=
\infty \sum
n=1
(zn + zn) , z \in \BbbU ,
then we obtain functions with negative coefficients introduced by Silverman [21]. These functions
were intensively investigated by many authors (see, for example, [4, 5, 8 – 10, 12, 14, 24]).
Let \scrT \lambda ,\tau (k, \eta ) denote the class of functions f \in \scrH with coefficients correlated with respect to
the function
\varphi (z) = J\lambda ,\tau
\scrH
\biggl(
z
1 - ei\eta z
+
e2i\eta z
1 - ei\eta z
\biggr)
=
\infty \sum
n=1
ei\eta (n - 1)n\lambda zn + \tau
\infty \sum
n=2
ei\eta (n+1)n\lambda zn, z \in \BbbU . (13)
Moreover, let us define
\scrS \lambda ,\tau
\scrT (k, \eta ;A,B) := \scrT \lambda ,\tau (k, \eta ) \cap \scrS \lambda ,\tau
\scrH (k;A,B), \scrR \lambda ,\tau
\scrT (k, \eta ;A,B) := \scrT \lambda ,\tau (k, \eta ) \cap \scrR \lambda ,\tau
\scrH (k;A,B),
where \eta , A, and B are real parameters with B > \mathrm{m}\mathrm{a}\mathrm{x}\{ 0, A\} . Finally, let us assume\bigm| \bigm| \bigm| n\lambda \bigm| \bigm| \bigm| \geq \bigm| \bigm| \bigm| k\lambda \bigm| \bigm| \bigm| \geq 1, n \in \BbbN k. (14)
Theorem 6. If a function f \in \scrH of the form (4) satisfies the condition
\infty \sum
n=k
(| \alpha n| | an| + | \beta n| | bn| ) \leq B - A, (15)
where
\alpha n = n\lambda \{ n (1 +B) - (1 +A)\} , \beta n = n\lambda \{ n (1 +B) + (1 +A)\} , (16)
then f \in \scrS \lambda ,\tau
\scrH (k;A,B).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
38 J. DZIOK
Proof. It is clear that the theorem is true for the function f(z) \equiv z. Let f \in \scrH be a function of
the form (4) and let there exist n \in \BbbN k such that an \not = 0 or bn \not = 0. By (14) we have
| \alpha n|
B - A
\geq n,
| \beta n|
B - A
\geq n, n \in \BbbN k, (17)
Thus, by (15) we get
\infty \sum
n=k
(n| an| + n| bn| ) \leq 1 (18)
and
\bigm| \bigm| h\prime (z)\bigm| \bigm| - \bigm| \bigm| g\prime (z)\bigm| \bigm| \geq 1 -
\infty \sum
n=k
n | an| | z| n -
\infty \sum
n=k
n| bn| | z| n \geq 1 - | z|
\infty \sum
n=k
(n| an| + n| bn| ) \geq
\geq 1 - | z|
B - A
\infty \sum
n=k
(| \alpha n| | an| + | \beta n| | bn| ) \geq 1 - | z| > 0, z \in \BbbU .
Therefore, by (2) the function f is locally univalent and sense-preserving in \BbbU . Moreover, if z1, z2 \in
\in \BbbU , z1 \not = z2, then\bigm| \bigm| \bigm| \bigm| zn1 - zn2
z1 - z2
\bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
l=1
zl - 1
1 zn - l
2
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
n\sum
l=1
| z1| l - 1 | z2| n - l < n, n \in \BbbN k.
Hence, by (18) 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
n=k
an (z
n
1 - zn2 )
\bigm| \bigm| \bigm| \bigm| \bigm| -
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
n=k
bn (zn1 - zn2 )
\bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq | z1 - z2| -
\infty \sum
n=k
| an| | zn1 - zn2 | -
\infty \sum
n=k
| bn| | zn1 - zn2 | =
= | z1 - z2|
\Biggl(
1 -
\infty \sum
n=k
| an|
\bigm| \bigm| \bigm| \bigm| zn1 - zn2
z1 - z2
\bigm| \bigm| \bigm| \bigm| - \infty \sum
n=k
| bn|
\bigm| \bigm| \bigm| \bigm| zn1 - zn2
z1 - z2
\bigm| \bigm| \bigm| \bigm|
\Biggr)
>
> | z1 - z2|
\Biggl(
1 -
\infty \sum
n=k
n| an| -
\infty \sum
n=k
n| bn|
\Biggr)
\geq 0.
This leads to the univalence of f, i.e., f \in \scrS \scrH . Therefore, f \in \scrS \lambda ,\tau
\scrT (k;A,B) if and only if there
exists a complex-valued function \omega , \omega (0) = 0, | \omega (z)| < 1, z \in \BbbU , such that
J\lambda +1, - \tau
\scrH (z)
J\lambda ,\tau
\scrH f(z)
=
1 +A\omega (z)
1 +B\omega (z)
, z \in \BbbU ,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
CLASSES OF HARMONIC FUNCTIONS DEFINED BY EXTENDED SĂLĂGEAN OPERATOR 39
or equivalently \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
D\scrH
\Bigl(
J\lambda ,\tau
\scrH f
\Bigr)
(z) - J\lambda ,\tau
\scrH f(z)
BD\scrH
\bigl(
J\lambda \tau
\scrH f
\bigr)
(z) - A
\bigl(
J\lambda \tau
\scrH f(z)
\bigr)
(z)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < 1, z \in \BbbU . (19)
Thus, it is suffice to prove that\bigm| \bigm| \bigm| D\scrH
\Bigl(
J\lambda \tau
\scrH f
\Bigr)
(z) - J\lambda \tau
\scrH f(z)
\bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| BD\scrH
\Bigl(
J\lambda \tau
\scrH f
\Bigr)
(z) - AJ\lambda \tau
\scrH f (z)
\bigm| \bigm| \bigm| < 0, z \in \BbbU \setminus \{ 0\} .
Indeed, letting | z| = r, 0 < r < 1, we have\bigm| \bigm| \bigm| D\scrH
\Bigl(
J\lambda \tau
\scrH f
\Bigr)
(z) - J\lambda \tau
\scrH f(z)
\bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| BD\scrH
\Bigl(
J\lambda \tau
\scrH f
\Bigr)
(z) - AJ\lambda \tau
\scrH f (z)
\bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
n=k
(n - 1)n\lambda anz
n -
\infty \sum
n=k
(n+ 1)\tau n\lambda bnz
n
\bigm| \bigm| \bigm| \bigm| \bigm| -
-
\bigm| \bigm| \bigm| \bigm| \bigm| (B - A) z +
\infty \sum
n=k
(Bn - A)n\lambda anz
n +
\infty \sum
n=k
(Bn+A) \tau n\lambda bnz
n
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\infty \sum
n=k
(n - 1)
\bigm| \bigm| \bigm| n\lambda an\bigm| \bigm| \bigm| rn +
\infty \sum
n=k
(n+ 1)
\bigm| \bigm| \bigm| n\lambda bn\bigm| \bigm| \bigm| rn - (B - A) r+
+
\infty \sum
n=k
(Bn - A)
\bigm| \bigm| \bigm| n\lambda an\bigm| \bigm| \bigm| rn +
\infty \sum
n=k
(Bn+A)
\bigm| \bigm| \bigm| n\lambda bn\bigm| \bigm| \bigm| rn \leq
\leq r
\Biggl\{ \infty \sum
n=k
(| \alpha n| | an| + | \beta n| | bn| ) rn - 1 - (B - A)
\Biggr\}
< 0,
whence f \in \scrS \lambda ,\tau
\scrH (k;A,B).
The next theorem, shows that the condition (15) is also the sufficient condition for a function
f \in \scrH of correlated coefficients to be in the class \scrS \lambda ,\tau
\scrT (k, \eta ;A,B).
Theorem 7. Let f \in \scrT \lambda ,\tau (k, \eta ) be a function of the form (4). Then f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) if and
only if the condition (15) holds true.
Proof. In view of Theorem 6 we need only show that each function f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) satisfies
the coefficient inequality (15). If f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), then it is of the form (4) with (12) and it
satisfies (19) or equivalently\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\sum \infty
n=k
(n - 1)n\lambda anz
n - (n+ 1)\tau n\lambda bnz
n
(B - A) z +
\sum \infty
n=k
\Bigl\{
(Bn - A)n\lambda anz
n - (Bn+A)\tau n\lambda bnz
n
\Bigr\}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < 1, z \in \BbbU .
Therefore, putting z = rei\eta , 0 \leq r < 1, by (13) and (12) we obtain\sum \infty
n=k
(n - 1)
\bigm| \bigm| \bigm| n\lambda \bigm| \bigm| \bigm| | an| + (n+ 1)
\bigm| \bigm| \bigm| n\lambda \bigm| \bigm| \bigm| | bn| rn - 1
(B - A) -
\sum \infty
n=k
\Bigl\{
(Bn - A)
\bigm| \bigm| \bigm| n\lambda \bigm| \bigm| \bigm| | an| + (Bn+A)
\bigm| \bigm| \bigm| n\lambda \bigm| \bigm| \bigm| | bn| rn - 1
\Bigr\} < 1. (20)
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40 J. DZIOK
It is clear that the denominator of the left-hand side cannot vanish for r \in \langle 0, 1) . Moreover, it is
positive for r = 0, and in consequence for r \in \langle 0, 1) . Thus, by (20) we have
\infty \sum
n=k
(| \alpha n| | an| + | \beta n| | bn| ) rn - 1 < B - A, 0 \leq r < 1. (21)
The sequence of partial sums \{ Sn\} associated with the series
\sum \infty
n=k
(| \alpha n| | an| + | \beta n| | bn| ) is non-
decreasing sequence. Moreover, by (21) it is bounded by B - A. Hence, the sequence \{ Sn\} is
convergent and
\infty \sum
n=k
(| \alpha n| | an| + | \beta n| | bn| ) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Sn \leq B - A,
which yields the assertion (15).
The following result may be proved in much the same way as Theorem 7.
Theorem 8. Let f \in \scrT \lambda ,\tau (k, \eta ) be a function of the form (4). Then f \in \scrR \lambda ,\tau
\scrT (k, \eta ;A,B) if and
only if
\infty \sum
n=k
\bigm| \bigm| \bigm| n\lambda \bigm| \bigm| \bigm| (| an| + | bn| ) \leq
B - A
1 +B
.
By Theorems 7 and 8 we have the following corollary.
Corollary 1. Let a =
1 +A
1 +B
and
i\phi (z) = z +
\infty \sum
n=k
\biggl(
1
n - a
zn +
1
n+ a
zn
\biggr)
, z \in \BbbU , i
i\omega (z) = z +
\infty \sum
n=k
((n - a) zn + (n+ a) zn) , z \in \BbbU .i
Then
f \in \scrR \lambda ,\tau
\scrT (k, \eta ;A,B) \leftrightarrow f \ast \phi \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B),
f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) \leftrightarrow f \ast \omega \in \scrR \lambda ,\tau
\scrT (k, \eta ;A,B).
In particular,
\scrR \lambda +1,\tau
\scrT (k, \eta ; - 1, B) = \scrS \lambda ,\tau
\scrT (k, \eta ; - 1, B).
4. Topological properties. We consider the usual topology on \scrH defined by a metric in which
a sequence \{ fn\} in \scrH converges to f if and only if it converges to f uniformly on each compact
subset of \BbbU . It follows from the theorems of Weierstrass and Montel that this topological space is
complete.
Let \scrF be a subclass of the class \scrH . A functions f \in \scrF is called an extreme point of \scrF if the
condition
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CLASSES OF HARMONIC FUNCTIONS DEFINED BY EXTENDED SĂLĂGEAN OPERATOR 41
f = \gamma f1 + (1 - \gamma )f2, f1, f2 \in \scrF , 0 < \gamma < 1,
implies f1 = f2 = f. 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 .
We say that \scrF is locally uniformly bounded if for each r, 0 < r < 1, there is a real constant
M =M(r) so that
| f(z)| \leq M, f \in \scrF , | z| \leq r.
We say that a class \scrF is convex if
\gamma f + (1 - \gamma )g \in \scrF , f, g \in \scrF , 0 \leq \gamma \leq 1.
Moreover, we define the closed convex hull of \scrF as the intersection of all closed convex subsets of
\scrH that contain \scrF . We denote the closed convex hull of \scrF by co\scrF .
A real-valued functional \scrJ : \scrH \rightarrow \BbbR is called convex on a convex class \scrF \subset \scrH if
\scrJ (\gamma f + (1 - \gamma )g) \leq \gamma \scrJ (f) + (1 - \gamma )\scrJ (g), f, g \in \scrF , 0 \leq \gamma \leq 1.
The Krein – Milman theorem (see [15]) is fundamental in the theory of extreme points. In parti-
cular, it implies the following lemma.
Lemma 1 [5, p. 45]. Let \scrF be a nonempty 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, Montel’s theorem implies the following lemma.
Lemma 2. A class \scrF \subset \scrH is compact if and only if \scrF is closed and locally uniformly bounded.
Theorem 9. The class \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) is convex and compact subset of \scrH .
Proof. Let f1, f2 \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) be functions of the form (6), 0 \leq \gamma \leq 1. Since
\gamma f1(z) + (1 - \gamma )f2(z) = z +
\infty \sum
n=k
\Bigl\{
(\gamma a1,n + (1 - \gamma )a2,n) z
n + (\gamma b1,n + (1 - \gamma )b2,n) zn
\Bigr\}
,
and by Theorem 7 we have
\infty \sum
n=k
\{ | \alpha n| | \gamma a1,n + (1 - \gamma )a2,n| + | \beta n| | \gamma b1,n + (1 - \gamma ) b2,nz
n| \} \leq
\leq \gamma
\infty \sum
n=k
\{ | \alpha na1,n| + | \beta nb1,n| \} + (1 - \gamma )
\infty \sum
n=k
\{ | \alpha na2,n| + | \beta nb2,n| \} \leq
\leq \gamma (B - A) + (1 - \gamma ) (B - A) = B - A,
the function \phi = \gamma f1 + (1 - \gamma )f2 belongs to the class \scrS \lambda ,\tau
\scrT (k, \eta ;A,B). Hence, the class is convex.
Furthermore, for f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), | z| \leq r, 0 < r < 1, we obtain
| f(z)| \leq r +
\infty \sum
n=k
(| an| + | bn| ) rn \leq r +
\infty \sum
n=k
(| \alpha n| | an| + | \beta n| | bn| ) \leq r + (B - A) .
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42 J. DZIOK
Thus, we conclude that the class \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) is locally uniformly bounded. By Lemma 2,
we only need to show that it is closed, i.e., if fl \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), l \in \BbbN , and fl \rightarrow f, then
f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B). Let fl and f are given by (6) and (4), respectively. Using Theorem 7 we get
\infty \sum
n=k
(| \alpha nal,n| + | \beta nbl,n| ) \leq B - A, l \in \BbbN . (22)
Since fl \rightarrow f, we conclude that | al,n| \rightarrow | an| and | bl,n| \rightarrow | bn| as l \rightarrow \infty , n \in \BbbN . The sequence of
partial sums \{ Sn\} associated with the series
\sum \infty
n=k
(| \alpha nan| + | \beta nbn| ) is nondecreasing sequence.
Moreover, by (22) it is bounded by B - A. Therefore, the sequence \{ Sn\} is convergent and
\infty \sum
n=k
(| \alpha nan| + | \beta nbn| ) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Sn \leq B - A.
This gives the condition (15), and, in consequence, f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), which completes the proof.
Theorem 10.
E\scrS \lambda ,\tau
\scrT (k, \eta ;A,B) = \{ hn : n \in \BbbN k - 1\} \cup \{ gn : n \in \BbbN k\} ,
where
hk - 1(z) = z, hn(z) = z - B - A
\alpha nei(n - 1)\eta
zn, gn(z) = z +
B - A
\tau \beta ne - i(n+1)\eta
zn, z \in \BbbU . (23)
Proof. Suppose that 0 < \gamma < 1 and
gn = \gamma f1 + (1 - \gamma )f2,
where f1, f2 \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) are functions of the form (6). Then, by (15) we have | b1,n| = | b2,n| =
=
B - A
| \beta n|
, and, in consequence, a1,l = a2,l = 0 for l \in \BbbN k and b1,l = b2,l = 0 for l \in \BbbN k\diagdown \{ n\} .
It follows that gn = f1 = f2, and consequently gn \in E\scrS \ast
\scrT (k, \eta ;A,B). Similarly, we verify that the
functions hn of the form (23) are the extreme points of the class \scrS \lambda ,\tau
\scrT (k, \eta ;A,B). Now, suppose that
a function f belongs to the set E\scrS \lambda ,\tau
\scrT (k, \eta ;A,B) and f is not of the form (23). Then there exists
m \in \BbbN k 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 =
| \alpha mam|
B - A
, \varphi =
1
1 - \gamma
(f - \gamma hm) ,
we have that 0 < \gamma < 1, hm \not = \varphi and
f = \gamma hm + (1 - \gamma )\varphi .
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CLASSES OF HARMONIC FUNCTIONS DEFINED BY EXTENDED SĂLĂGEAN OPERATOR 43
Thus, f /\in E\scrS \lambda ,\tau
\scrT (k, \eta ;A,B). Similarly, if 0 < | bm| < B - A
| \beta n|
, then putting
\gamma =
| \beta mbm|
B - A
, \phi =
1
1 - \gamma
(f - \gamma gm),
we have that 0 < \gamma < 1, gm \not = \phi and
f = \gamma gm + (1 - \gamma )\phi .
It follows that f /\in E\scrS \lambda ,\tau
\scrT (k, \eta ;A,B), and the proof is completed.
5. Applications. It is clear that if the class
\scrF = \{ fn \in \scrH : n \in \BbbN \} ,
is locally uniformly bounded, then
co\scrF =
\Biggl\{ \infty \sum
n=1
\gamma nfn :
\infty \sum
n=1
\gamma n = 1, \gamma n \geq 0, n \in \BbbN
\Biggr\}
.
Thus, by Theorem 7 we have the following corollary.
Corollary 2.
\scrS \lambda ,\tau
\scrT (k, \eta ;A,B) =
\Biggl\{ \infty \sum
n=k - 1
(\gamma nhn + \delta ngn) :
\infty \sum
n=k - 1
(\gamma n + \delta n) = 1 (\delta k - 1 = 0, \gamma n, \delta n \geq 0)
\Biggr\}
,
where hn, gn are defined by (23).
For each fixed value of m,n \in \BbbN k, z \in \BbbU , the following real-valued functionals are continuous
and convex on \scrH :
\scrJ (f) = | an| , \scrJ (f) = | bn| , \scrJ (f) = | f(z)| \scrJ (f) = | D\scrH f(z)| , f \in \scrH .
Moreover, for \gamma \geq 1, 0 < r < 1, 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 ,
is also continuous and convex on \scrH .
Therefore, by Lemma 1 and Theorem 7 we have the following corollaries.
Corollary 3. Let f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B) be a function of the form (4). Then
| an| \leq
B - A
| \alpha n|
, | bn| \leq
B - A
| \beta n|
, n \in \BbbN k,
where \alpha n, \beta n are defined by (16). The result is sharp. The functions hn, gn of the form (23) are the
extremal functions.
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44 J. DZIOK
Corollary 4. Let f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), | z| = r < 1. Then
r - B - A
| k\lambda | (k - 1 + kB - A)
rk \leq | f(z)| \leq r +
B - A
| k\lambda | (k - 1 + kB - A)
rk,
r - k (B - A)
| k\lambda | (k - 1 + kB - A)
rk \leq | D\scrH f(z)| \leq r +
k (B - A)
| k\lambda | (k - 1 + kB - A)
rk.
The result is sharp. The function hk of the form (23) is the extremal function.
Corollary 5. Let 0 < r < 1, \gamma \geq 1. If f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), then
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| f(rei\theta )\bigm| \bigm| \bigm| \gamma d\theta \leq 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| hk(rei\theta )\bigm| \bigm| \bigm| \lambda d\theta ,
1
2\pi
2\pi \int
0
| D\scrH f(z)| \gamma d\theta \leq
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| D\scrH hk(re
i\theta )
\bigm| \bigm| \bigm| \gamma d\theta ,
where hk is the function defined by (23).
The following covering result follows from Corollary 4.
Corollary 6. If f \in \scrS \lambda ,\tau
\scrT (k, \eta ;A,B), then \BbbU (r) \subset f (\BbbU ) , where
r = 1 - B - A
| k\lambda | (k - 1 + kB - A)
.
By using Corollary 1 and the results above we obtain corollaries listed below.
Corollary 7. The class \scrR \lambda ,\tau
\scrT (k, \eta ;A,B) is convex and compact subset of \scrH . Moreover,
E\scrR \lambda ,\tau
\scrT (k, \eta ;A,B) = \{ hn : n \in \BbbN k - 1\} \cup \{ gn : n \in \BbbN k\}
and
\scrR \lambda ,\tau
\scrT (k, \eta ;A,B) =
\Biggl\{ \infty \sum
n=k - 1
(\gamma nhn + \delta ngn) :
\infty \sum
n=k - 1
(\gamma n + \delta n) = 1 (\delta k - 1 = 0, \gamma n, \delta n \geq 0)
\Biggr\}
,
where
hk - 1(z) = z, hn(z) = z - (B - A) ei(n - 1)\eta
(1 +B)n\lambda
zn, gn(z) = z +
(B - A)ei(n+1)\eta
\tau (1 +B)n\lambda
zn, z \in \BbbU .
(24)
Corollary 8. Let f \in \scrR \lambda ,\tau
\scrT (k, \eta ;A,B) be a function of the form (4). Then
| an| \leq
B - A
(1 +B) | n\lambda |
, | bn| \leq
B - A
(1 +B) | n\lambda |
, n \in \BbbN k,
r - B - A
(1 +B) | k\lambda |
rk \leq | f(z)| \leq r +
B - A
(1 +B) | k\lambda |
rk, | z| = r < 1,
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CLASSES OF HARMONIC FUNCTIONS DEFINED BY EXTENDED SĂLĂGEAN OPERATOR 45
r - k (B - A)
(1 +B) | k\lambda |
rk \leq | D\scrH f(z)| \leq r +
k (B - A)
(1 +B) | k\lambda |
rk, | z| = r < 1,
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| f(rei\theta )\bigm| \bigm| \bigm| \gamma d\theta \leq 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| hk(rei\theta )\bigm| \bigm| \bigm| \lambda d\theta ,
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| D\scrH f(re
i\theta )
\bigm| \bigm| \bigm| \gamma d\theta \leq 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| D\scrH hk(re
i\theta )
\bigm| \bigm| \bigm| \gamma d\theta .
The results are sharp. The functions hn, gn of the form (24) are the extremal functions.
Corollary 9. Let us assume (14). If f \in \scrR \lambda ,\tau
\scrT (k, \eta ;A,B), then \BbbU (r) \subset f (\BbbU ) , where
r = 1 - B - A
k (1 +B) | k\lambda |
.
The class \scrS \lambda
\scrH (\tau , n;A,B) generalize classes of starlike functions of complex order. The class
\scrC \scrS \scrH (\gamma ) := \scrS \scrH (\varphi ; 1 - 2\gamma , 1) (\gamma \in \BbbC \smallsetminus \{ 0\} ) was defined by Yalçin and Öztürk [23]. In particular,
if we put \gamma :=
1 - \alpha
1 + ei\eta
, then we obtain the class \scrR \scrS \scrH (\alpha , \eta ) := \scrS \scrH
\biggl(
2\alpha - 1 + ei\eta
1 + ei\eta
, 1
\biggr)
studied by
Yalçin et al. [24]. It is the class of functions f \in \scrH 0 such that
\mathrm{R}\mathrm{e}
\biggl\{ \bigl(
1 + ei\eta
\bigr) D\scrH f (z)
f(z)
- ei\eta
\biggr\}
> \alpha , z \in \BbbU , \eta \in \BbbR .
The classes \scrS n
\scrH (k;A,B) and \scrR n
\scrH (k;A,B) are related to harmonic starlike functions, harmonic
convex functions and harmonic Janowski functions.
The classes \scrS \scrH (\alpha ) := \scrS 0
\scrH (2; 2\alpha - 1, 1) and \scrK \scrH (\alpha ) := \scrS 1
\scrH (2; 2\alpha - 1, 1) were investigated by
Jahangiri [10] (see also [2, 18]). They are the classes of starlike and convex functions of order
\alpha , respectively. The classes N\scrH (\alpha ) := \scrR 1
\scrH (2; 2\alpha - 1, 1) and R\scrH (\alpha ) := \scrR 2
\scrH (2; 2\alpha - 1, 1) were
studied in [1] (see also [14]). Finally, the classes \scrS \scrH := \scrS \scrH (0) and \scrK \scrH := \scrK \scrH (0) are the classes of
functions which are starlike and convex in \BbbU (r), respectively, for all r \in (0, 1\rangle . We should notice
that the classes \scrS (A,B) := \scrS \scrH (A,B) \cap \scrA and \scrR (A,B) := \scrR \scrH (A,B) \cap \scrA were introduced by
Janowski [13].
Using obtained results to the classes defined above we can obtain new and also well-known
results (see, for example, [1 – 9, 10 – 14, 18, 21 – 24]).
Remark . The results obtained in classes of harmonic functions can be transfered to correspon-
ding classes of analytic functions.
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Received 17.04.18
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| id | umjimathkievua-article-78 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:03Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/36/46d6ef6f6e8ecdd73d8643f045091a36.pdf |
| spelling | umjimathkievua-article-782025-03-31T08:49:21Z Classes of harmonic functions defined by extended Sălăgean operator Classes of harmonic functions defined by extended Sălăgean operator Dziok, J. Dziok, J. harmonic function Sălăgean operator subordinatio extreme point starlike function harmonic function Sălăgean operator subordinatio extreme point starlike function UDC 517.57 The object of the present paper is to investigate classes of harmonic functions defined by the extended Sălăgea operator. By using the extreme points theory we obtain coefficients estimates and distortion theorems for these classes of functions. Some integral mean inequalities are also pointed out. &nbsp; УДК 517.57 Класи гармонiчних функцiй, якi визначенi розширеним оператором Саладжана Досліджуються класи гармонічних функцій, які визначені розширеним оператором Саладжана. За допомогою теорії екстремальних точок отримано оцінки для коефіцієнтів та теореми деформації для класів функцій. Також наведено деякі нерівності для інтегральних середніх. Institute of Mathematics, NAS of Ukraine 2021-01-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/78 10.37863/umzh.v73i1.78 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 1 (2021); 33 - 46 Український математичний журнал; Том 73 № 1 (2021); 33 - 46 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/78/8895 Copyright (c) 2021 J. Dziok |
| spellingShingle | Dziok, J. Dziok, J. Classes of harmonic functions defined by extended Sălăgean operator |
| title | Classes of harmonic functions defined by extended Sălăgean operator |
| title_alt | Classes of harmonic functions defined by extended Sălăgean operator |
| title_full | Classes of harmonic functions defined by extended Sălăgean operator |
| title_fullStr | Classes of harmonic functions defined by extended Sălăgean operator |
| title_full_unstemmed | Classes of harmonic functions defined by extended Sălăgean operator |
| title_short | Classes of harmonic functions defined by extended Sălăgean operator |
| title_sort | classes of harmonic functions defined by extended sălăgean operator |
| topic_facet | harmonic function Sălăgean operator subordinatio extreme point starlike function harmonic function Sălăgean operator subordinatio extreme point starlike function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/78 |
| work_keys_str_mv | AT dziokj classesofharmonicfunctionsdefinedbyextendedsalageanoperator AT dziokj classesofharmonicfunctionsdefinedbyextendedsalageanoperator |