1. Introduction
Let
=
be the unit disc. The set of all analytic functions of the form
in
with normalization
is denoted by
. The subfamily of
which are univalent in
is symbolized by
. For
, we denote by
and
, the family of
-pseudo-starlike functions and the family of
-pseudo-convex functions, respectively, where
and
The family
was investigated by Babalola [
1] and he proved that all
-pseudo-starlike are univalent in
. The family
was defined by Guney and Murugusundaramoorthy [
2]. In 1969, Mocanu [
3] examined the family
of
-convex functions
satisfying
Clearly
and
. It was shown in [
4] that for all
,
.
In [
5], the Koebe one-quarter theorem ensures that
contains a disc of radius 1/4 for every function
. Hence, every function
admits an inverse
defined by
, where
and a compuatation shows that
g has the expansion of the form
A member
s of
is called bi-univalent in
If both
s and
are univalent in
. Let
be the family of bi-univalent functions in
given by (
1). The systematic study of the family
has its origin in a paper authored by Lewin [
6], where coefficient-related investigations for elements of the family
are examined. Lewin was the first to investigate the family
and it was proved that
for members of the family
. Few years later, the estimation for
was further investigated by Brannan and Clunie [
7] and they proved that
, if
. In 1984, Tan [
8] found initial coefficient estimates of functions in the family
. Brannan and Taha in [
9], investigated bi-starlike and bi-convex functions, which are analogous to the concepts of starlike and convex functions. An investigation by Srivastava et al. [
10] resurfaced the interest in the study of family
and it opened the space for many thinkings in the topics of discussion of the paper. The trend in the last decade was to investigate coefficient-related non-sharp bounds for members of certain subfamilies of
as it can be seen in papers [
11,
12,
13,
14,
15].
An holomorphic function
s in
is said to be
-fold symmetric if
. For each
, the function
s given by
,
, is univalent and maps
into a region with
-fold symmetry. The class of
-fold symmetric univalent functions in
is symbolized by
. A function
has the form given by
Clearly .
Following the concept of
,
Srivastava et al. [
16] examined the family
of
-fold symmetric bi-univalent functions. They found some interesting results, such as the series for
, which is as follows:
We obtain (
2) from (
4) on taking
. Note that
. Some examples in the class
are
Inverse functions of them are as follows:
The momentum on investigations of functions in certain subfamilies of
was gained in recent years due to the paper [
16] and it has led to a large number of papers on subfamilies of
[
17,
18,
19,
20]. Inspired by these works, many researchers have investigated several interesting subfamilies of
and found non-sharp estimates on initial coefficients and the Fekete–Szegö functional problem
,
[
21] of functions belonging to these subfamilies (see, for example [
22,
23,
24,
25]) and this continued to appear in [
26,
27], showing the developments in the subject area of this paper.
Motivated by the works of [
3,
4,
20,
28], we define a
-pseudo-
-convex
-fold symmetric bi-univalent function family
, in
Section 2. We derive estimations for
,
and
, for functions
. In
Section 3, we investigate bounds on
,
and
, for functions ∈
. In
Section 4, we initiate bounds on
,
and
, for functions ∈
. The results obtained are not sharp. We discuss several related families and indicate connections to earlier defined classes.
2. The Function Family
Throughout our investigations in the present paper, it is assumed that
is as stated in (
4) and
is an analytic function with
such that
is symmetric with respect to the real axis,
and
. The function
has an expansion given by
We symbolize by
the family of analytic functions having the series form
satisfying
In view of Pommerenke [
29], the
-fold symmetric function
p is of the form
Let
and
be analytic in
with
and
< 1. We suppose that
and
. Additionally, we know that
After simple calculations using (
5), we get
and
Definition 1. A function of the form (3) is said to be in the class , , if it fulfills the following subordination conditions:andwhere . For , a function in the class is called -pseudo-bi--convex -fold summetric function of Ma-Minda type. For , a function in the class is called -pseudo-bi-Mocanu-convex function of Ma-Minda type of complex order .
In [
30], Mishra and Soren have illustrated that
is univalent starlike function of orde
,
if
. They have further shown that
is a univalent starlike function of order
. Therefore
is bi-starlike function of order
. On similar lines of [
30], one can show that
is bi-convex function of order
. Hence, the function
.
Remark 1. The family , was eximined by Aldawish et al. [24]. We observe that certain choices of and in lead to the subfamilies and , as given below:
(i). If we set
in the class
, then we get the subclass
of functions
s satisfying
and
where
and
.
(ii). If we allow
in the class
, then we get the subclass
of functions
s satisfying
where
and
.
Remark 2. The family was investigated by Tang et al. [20]. A function in the class is called bi-Mocanu-convex function of Ma-Minda type studied by Ali et al. [31]. Theorem 1. Let and . If a function s in belongs to , thenandwhereand Proof. Let the function
s in
belongs to
. Then there are holomorphic functions
with
satisfying
and
where
.
Using (
3) in (
16) and (
17), we obtain
and
where L, M and N are given by (
13), (
14) and (
15), respectively.
Comparing (
7) and (
18), we get
Comparing (
8) and (
19), we obtain
and
From (
20) and (
22), we get
and
For finding the bound on
, we add (
21) and (
23) and then we use (
25) to obtain
By using (
6), (
20) and (
24) in (
26) for the coefficients
and
, we obtain
Inequality (
27) implies the assertion (
9).
To obtain bound on
, we subtract (
23) from (
21):
In view of (
20), (
24), (
28) and applying (
6), it follows that
Inequality (
29) gets the desired estimate (
10).
From (
26) and (
28), for
, we get
where
In view of (
6), we conclude that
from which we obtain the desired assertion (
11) with
J as in (
12). So the proof is completed.
Remark 3. (i). If we take in Theorem 1, then we obtain Corollary 2 of Aldawish et al. [24]. (ii). If we set in Theorem 1, we get Corollaries 2.2 and 2.11 of [32]. Further, we get Corollaries 2.6 and 2.13 of [32], when . We obtain the results stated below, if we set in Theorem 1.
Corollary 1. Let . If a function s in belongs to , thenand Remark 4. (i). We obtain the bound on stated in Theorems 5 and 6 of Tang et al. [20] from Corollary 1, if we allow . (ii). We also obtain Corollary 19 of [20] from Corollary 1, when and . Further for , we get Corollary 20 of [20]. We obtain the results stated below, if we set in Theorem 1.
Corollary 2. Let and . If a function s in , thenandwhereand If in the above corollary, then we have
Corollary 3. Let and . If a function s in , thenand Taking
and
in Definition 1, we get the subfamily
of functions
satisfying
where
and
.
Corollary 4. Let and . If a function s in , thenand 3. The Function Family
Let
=
. Then we have from Definition 1 the subclass
of functions
satisfying
and
where
,
.
Remark 5. The family, was investigated by Aldawish et al. [24]. We observe that the choice and in lead to the subfamilies and , respectively, as given below:
(i).
), is the family of
s satisfying
and
where
,
.
(ii).
is the family of
s satisfying
where
,
.
Remark 6. We note that . The function class was considered by Kumar et al. [33]. The function family coincides with the family of strongly bi-starlike functions of order ϱ, which was studied by Branan and Taha [9]. If , then Theorem 1 reduce to the corollary stated below:
Corollary 5. Let and . If a function s in belongs to , thenandwhere L, M, and N are as in (13), (14) and (15), respectively. Remark 7. (a). If we take in Corollary 5, then we obtain Corollary 5 of Aldawish et al. [24]. (b). (i) The bound on stated in Corollary 6 of [34] is got, if we set in Corollary 5. (ii) Our result on is better than the bound stated in Corollary 6 of [34], in terms of ϱ ranges as well as the bounds, if in Corollary 5. Setting , Corollary 5 reduce to the next corollary.
Corollary 6. Let . If a function s in belongs to , thenand We obtain the results stated below, if we take in Corollary 5.
Corollary 7. Let . If a function s in belongs to , thenandwhere and are as in (30), (31) and (32), respectively. Results analogous to Corollary 3 and Corollary 4 can be obtained, by taking in Corollary 7 and in Corollary 5, respectively.
4. The Function Family
Let
. then from Definition 1, we obtain the subclass
of functions
satisfying
and
where
.
Remark 8. The family ),, was investigated by Aldawish et al. [24]. We remark that certain values of and in lead to the subfamilies as mentioned below:
(i).
) is the family of
s satisfying
and
where
and
.
(ii).
, is the family of
satisfying
and
where
and
.
Remark 9. We note that . The function class was considered by Kumar et al. [33]. The function family coincides with the family of bi-starlike functions of order ξ, which was studied by Branan and Taha [9]. If we take in Theorem 1, we get
Corollary 8. Let . If a function s in belongs to , thenandwhere L, M, and N are as in (13), (14) and (15), respectively Remark 10. (a) If we allow in Corollary 8, then we have Corollary 8 of Aldawish et al. [24]. (b) (i) The bound on stated in Corollary 7 of [34] is obtained, if in Corollary 8. (ii) Our result on in Corollary 8 is better than the bound stated in Corollary 7 of [34] in terms of ξ ranges as well as the bounds, when . We obtain the results stated below, if we set in Corollary 8.
Corollary 9. Let . If a function s in belongs to , thenand We obtain the results stated below, if we set in Corollary 8.
Corollary 10. Let . If a function s in belongs to , thenandwhere and are as in (30), (31) and (32), respectively. Results analogous to Corollary 3 and Corollary 4 can be obtained, by taking in Corollary 10 and in Corollary 8, respectively.
5. Conclusions
In the current study, a
-fold bi-univalent function family
is introduced and the original results about the upper bounds of
and
are estimated for functions belonging to this family. Furthermore, the estimate of Fekete–Szegö problem
,
, for functions in
is also examined. Various subfamilies of
are also discussed. The problem to determine bound on
,
for the classes that have been examined in this paper remain open. Since the only investigation on the defined family was related to coefficient bounds, it could inspire many researchers for further investigations related to different other aspects associated with (i) q-derivative operator [
35], (ii) integrodifferential operator [
36], (iii) Hohlov operator linked with legendary polynomials [
37] and so on.
Author Contributions
Formal analysis, conceptualization, investigation, methodology and writing—original draft preparation, S.R.S.; software, resources, visualization, data curation, writing—review, editing, validation and funding acqquisition, L.-I.C. All authors have agreed to the final version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Acknowledgments
The authors would like to thank all the referres for their helpful comments and suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Babalola, K.O. On λ-pseudo-starlke functions. J. Class. Anal. 2012, 3, 137–147. [Google Scholar]
- Guney, H.O.; Murugusundaramoorthy, G. New classes of pseudo-type bi-univalent fumctions. RACSAM 2020, 114, 65. [Google Scholar] [CrossRef]
- Moacnu, P.T. Une propriete de convexite generalisee dans la theorie de la representation conforme. Mathematica 1969, 11, 127–133. [Google Scholar]
- Miller, S.S.; Moacnu, P.T.; Reade, M.O. All α-convex functions are univalent and starlike. Proc. Am. Math. Soc. 1973, 37, 553–554. [Google Scholar] [CrossRef]
- Duren, P.L. Univalent Functions; Springer: New York, NY, USA, 1983. [Google Scholar]
- Lewin, M. On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 1967, 18, 63–68. [Google Scholar] [CrossRef]
- Brannan, D.A.; Clunie, J.; Kirwan, W.E. Coefficient estimates for a class of starlike functions. Can. J. Math. 1970, 22, 476–485. [Google Scholar] [CrossRef]
- Tan, D.L. Coefficient estimates for bi-univalent functions. Chin. Ann. Math. Ser. A 1984, 5, 559–568. [Google Scholar]
- Brannan, D.A.; Taha, T.S. On some classes of bi-univalent functions. Math. Anal. Its Appl. 1985, 3, 18–21. [Google Scholar]
- Srivastava, H.M.; Mishra, A.K.; Gochhayat, P. Certain subclasses analytic and bi-univalent functions. Appl. Math. Lett. 2010, 23, 1188–1192. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Gaboury, S.; Ghanim, F. Coefficients estimate for some general subclasses of analytic and bi-univalent functions. Afr. Mat. 2017, 28, 693–706. [Google Scholar] [CrossRef]
- Frasin, B.A.; Aouf, M.K. New subclass of bi-univalent functons. Appl. Math. Lett. 2011, 24, 1569–1573. [Google Scholar] [CrossRef]
- Deniz, E. Certain subclasses of bi-univalent functions satisfying subordinate conditions. J. Class. Anal. 2013, 2, 49–60. [Google Scholar] [CrossRef]
- Tang, H.; Deng, G.; Li, S. Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions. J. Ineq. Appl. 2013, 2013, 317. [Google Scholar] [CrossRef]
- Frasin, B.A. Coefficient bounds for certain classes of bi-univalent functions. Hacet. J. Math. Stat. 2014, 43, 383–389. [Google Scholar] [CrossRef] [Green Version]
- Srivastava, H.M.; Sivasubramanian, S.; Sivakumar, R. Initial coefficients estimate for some subclasses of m-fold symmetric bi-univalent functions. Tbilisi Math. J. 2014, 7, 1–10. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Gaboury, S.; Ghanim, F. Initial coefficients estimate for some subclasses of m-fold symmetric bi-univalent functions. Acta Math. Sci. Ser. B 2016, 36, 863–971. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Zireh, A.; Hajiparvaneh, S. Coefficients estimate for some subclasses of m-fold symmetric bi-univalent functions. Filomat 2018, 32, 3143–3153. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Wanas, A.K. Initial Maclaurin coefficients bounds for new subclasses of m-fold symmetric bi-univalent functions defined by a linear combination. Kyungpook Math. J. 2019, 59, 493–503. [Google Scholar]
- Tang, H.; Srivastava, H.M.; Sivasubramanian, S.; Gurusamy, P. Fekete–Szegö functional problems of m-fold symmetric bi-univalent functions. J. Math. Ineq. 2016, 10, 1063–1092. [Google Scholar] [CrossRef]
- Fekete, M.; Szegö, G. Eine Bemerkung Über Ungerade Schlichte Funktionen. J. Lond. Math. Soc. 1933, 89, 85–89. [Google Scholar] [CrossRef]
- Wanas, A.K.; Páll-Szabó, A.O. Coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions. Stud. Univ. Babes-Bolai Math. 2021, 66, 659–666. [Google Scholar] [CrossRef]
- Swamy, S.R.; Frasin, B.A.; Aldawish, I. Fekete–Szegö functional problem for a special family of m-fold symmetric bi-univalent functions. Mathematics 2022, 10, 1165. [Google Scholar] [CrossRef]
- Aldawish, I.; Swamy, S.R.; Frasin, B.A. A special family of m-fold symmetric bi-univalent functions satisfying subordination condition. Fractal Fract. 2022, 6, 271. [Google Scholar] [CrossRef]
- Breaz, D.; Cotîrlă, L.I. The study of the new classes of m-fold symmetric bi-univalent Functions. Mathematics 2022, 10, 75. [Google Scholar] [CrossRef]
- Oros, G.I.; Cotîrlă, L.I. Coefficient Estimates and the Fekete–Szegö problem for new classes of m-fold symmetric bi-univalent functions. Mathematics 2022, 10, 129. [Google Scholar] [CrossRef]
- Shehab, N.H.; Juma, A.R.S. Coefficient bounds of m-fold symmetric bi-univalent functions for certain subclasses. Int. J. Nonlinear Anal. Appl. 2021, 12, 71–82. [Google Scholar] [CrossRef]
- Minda, W.C.; Minda, D. A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis; Conference Proceedings and Lecture Notes in Analysis; Li, J., Ren, F., Yang, L., Zhang, S., Eds.; International Press: Cambridge, MA, USA, 1994; Volume I, pp. 157–169. [Google Scholar]
- Pommerenke, C. Univalent Functions; Vandenhoeck and Ruprecht: Gottingen, Germany, 1975. [Google Scholar]
- Mishra, A.K.; Soren, M.M. Coefficient bounds for bi-starlike analytic functions. Bull. Belg. Math. Soc. Simon Stevin 2014, 21, 157–167. [Google Scholar] [CrossRef]
- Ali, R.M.; Lee, S.K.; Ravichandran, V.; Subramanian, S. Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Appl. Math. Lett. 2012, 25, 344–351. [Google Scholar] [CrossRef]
- Akgul, A. Fekete–Szegö coefficient inequality for a new class of m-fold symmetric bi-univalent functions satisfying subordination conditions. Honam Math. J. 2018, 40, 733–748. [Google Scholar]
- Kumar, T.R.K.; Karthikeyan, S.; Vijayakumar, S.; Ganapathy, G. Initial coefficient estimates for certain subclasses of m-fold symmetric bi-univalent functions. Adv. Dyn. Syst. Appl. 2021, 16, 789–800. [Google Scholar]
- Altınkaya, Ş.; Yalçın, S. Coefficients bounds for certain subclasses of m-fold symmetric bi-univalent functions. J. Math. 2015, 2015, 241683. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Mostafa, A.O.; Aouf, M.K.; Zayed, H.M. Basic and fractional q-calculus and associated Fekete–Szegö problem for p-valently q-starlike functions and p-valently q-convex functions of complex order. Miskolc Math. Notes 2019, 20, 489–509. [Google Scholar] [CrossRef]
- Páll-Szabó, Á.O.; Oros, G.I. Coefficient related studies for new classes of bi-univalent functions. Mathematics 2020, 8, 1110. [Google Scholar] [CrossRef]
- Murugusundaramoorthy, G.; Cotîrlă, L.I. Bi-univalent functions of complex order defined by Hohlov operator associated with legendrae polynomial. AIMS Math. 2022, 7, 8733–8750. [Google Scholar] [CrossRef]
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