Article
Kyungpook Mathematical Journal 2021; 61(1): 75-98
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
Univalent Functions Associated with the Symmetric Points and Cardioid-shaped Domain Involving (p,q)-calculus
Om Ahuja, Nisha Bohra, Asena Çetinkaya, Sushil Kumar*
Department of Mathematical Sciences, Kent State University, Ohio, 44021, U.S.A
e-mail : oahuja@kent.edu
Department of Mathematics, Sri Venkateswara College, University of Delhi, Delhi-110 021, India
email : nishib89@gmail.com
Department of Mathematics and Computer Science, İstanbul Kültür University, İstanbul, Turkey
e-mail : asnfigen@hotmail.com
Bharati Vidyapeeth's college of Engineering, Delhi-110063, India
e-mail : sushilkumar16n@gmail.com
Received: March 19, 2020; Revised: July 28, 2020; Accepted: July 28, 2020
Abstract
In this paper, we introduce new classes of post-quantum or (p,q)-starlike and convex functions with respect to symmetric points associated with a cardiod-shaped domain. We obtain (p,q)-Fekete-Szegö inequalities for functions in these classes. We also obtain estimates of initial (p,q)-logarithmic coefficients. In addition, we get q-Bieberbach-de-Branges type inequalities for the special case of our classes when p=1. Moreover, we also discuss some special cases of the obtained results.
Keywords: (p, q)-Fekete-Szegö, inequalities, (p,q)-starlike functions, (p,q)-convex functions, (p,q)-logarithmic coefficient bounds, cardioid-shaped domain, q-Bie
1. Introduction
We first recall the basic definitions in univalent function theory. Let
Let
Note that the image of the open unit disk under the rational function
-
Figure 1. Image of unit disk
under the function
For
In particular,
Quantum Calculus or
If
then the
If
We now introduce two new classes of
Definition 1.1.
A function
where
The functions in the class
which is equivalent to
or
In view of convolution, we obtain
and
On substituting (1.5) and (1.6) into (1.4), we conclude that the function
We remark that for technique used in proving (1.7), one may refer to [1, 30].
Remark 1.2.
If we replace the rational function
and it gives rise to a new class of
was introduced and studied by Sakaguchi [26].
Definition 1.3.
A function
where
The functions in the class
Remark 1.4.
If we replace the rational function
This condition defines a new class of
was introduced and studied by Das and Singh [10].
Remark 1.5.
For special values of parameters
-
(1)
, -
(2)
, -
(3)
-
(4)
.
The sharp bounds of the initial coefficients yield the information regarding the geometric properties like the growth, distortion and covering estimates of the functions. Sharp estimates of the coefficient functional
The
Motivated by the stated research papers, we determine the sharp bounds on
2. (p,q) -Fekete-Szegö Functional Inequalities
In this section, we investigate the behaviour of the Fekete-Szegö functionals defined on
Theorem 2.1.
Let
In order to prove Theorem 2.1, we need next two results.
Lemma 2.2.
([20]) If
Lemma 2.3.
([20]) If
When
or one of its rotations. If
Also the upper bound in inequality (2.1) is sharp, and it can be improved as follows:
and
where
where
or equivalently
Then
Using (2.3) in (2.2), and on comparing both sides we get
Since
and
Using (2.5) and (2.6), we obtain
where
In view of Lemma 2.2, we get
where ν is given by (2.8). After substituting the value of
Putting
Corollary 2.4.
If a function
Theorem 2.5.
Let
then
Further, if
If
It is a routine to verify that these functions belong to
Again replacing
Corollary 2.6.
Let
Further, if
If
Theorem 2.7.
Let
where
or equivalently
Then ψ is analytic in
Using (2.9) and (2.10), we get
Since
Using the method of proof used in Theorem 2.1, we obtain
where
Using Lemma 2.2, we get
Substituting the value of
Replacing
Corollary 2.8.
If the function
Theorem 2.9.
Let
Then
Moreover, if
and if
If
If
If
Since
Similarly, if
Similar to the results in previous corollaries, putting
3. (p,q) -Logarithmic Coefficients
In this section, we obtain the estimates of the initial logarithmic coefficients of the functions
Theorem 3.1.
Let
and
where
and
In order to prove Theorem 3.1, we need following result due to Prokhorov and Szynal.
Lemma 3.2.
([23]) Let
where
and for
By substituting the value of
where
Using Lemma 2.3, we get the required bound for
where
On simplifying,
where
The proof of next theorem is similar to the proof of previous theorem and hence it is omitted.
Theorem 3.3.
Let
and
where
and
4. q -Bieberbach-de-Branges Type Coefficient Inequalities
In view of the work done in [6, 27], we investigate
Theorem 4.1.
If a function
where
We need the following result for the proof of Theorem 4.1
Lemma 4.2.
Let
for all
where
Using Parseval identity, and
If
This gives desired result for all
where
we have
where
Equating both sides of the equality (4.6), we get
and for
In view of Lemma 4.2 and using (4.7), (4.8), (4.9) and (4.10) we get the following coefficient inequalities.
These bounds show that (4.1) and (4.2) are valid for
We now prove (4.1) and (4.2) by using induction. Taking into account Lemma 4.2, (4.11) and (4.12) we get
and
Now, suppose that (4.1) and (4.2) hold for
In order to complete the proof, it is sufficient to show that
for
Therefore, (4.15) is valid for
Theorem 4.3.
If a function
where
where
Equating both sides, we get
and for
Using Lemma 4.2 and (4.18), (4.19), (4.20) and (4.21) we get the following coefficient inequalities.
These bounds show that (4.16) and (4.17) are valid for
We now prove (4.16) and (4.17) by using induction. Using Lemma 4.2 and equations (4.22) and (4.23) we get
and
Now, suppose that (4.16) and (4.17) hold for
In order to complete the proof, it is sufficient to show that
for
Concluding Remarks
In Remark 1.2 and Remark 1.4, we defined two new classes
Acknowledgements.
All the authors are thankful to the referees for giving helpful comments/suggestions.
References
- O. P. Ahuja. Hadamard products and neighbourhoods of spirallike functions, Yokohama Math. J. 40(2)(1993), 95-103.
- O. P. Ahuja and A. Çetinkaya. Use of quantum calculus approach in mathematical sciences and its role in geometric function theory, AIP Conference Proceedings 2095, 020001 (2019); https://doi.org/10.1063/1.5097511.
- O. P. Ahuja, A. Çetinkaya, and Y. Polatoglu. Bieberbach-de Branges and Fekete-Szegö inequalities for certain families of q-convex and q-close to convex functions, J. Comput. Anal. Appl. 26(4)(2019), 639-649.
- M. F. Ali and A. Vasudevarao. On logarithmic coefficients of some close-to-convex functions, Proc. Amer. Math. Soc. 146(3)(2018), 1131-1142.
- A. Aral, V. Gupta, and R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, 2013.
- R. Bucur and D. Breaz. On a new class of analytic functions with respect to symmetric points involving the q-derivative operator, J. Phys.: Conf. Series 1212 (2019), doi:10.1088/1742-6596/1212/1/012011.
- A. Çetinkaya, Y. Kahramaner, and Y. Polatoglu. Fekete-Szegö inequalities for q-starlike and q-convex functions, Acta Univ. Apulensis Math. Inform. 53(2018), 55-64.
- A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A 24(13)(1991), L711-L718.
- N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim. On the third logarith-mic coefficient in some subclasses of close-to-convex functions, Yokohama Math. J.Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(2)(2020). Paper No. 52, 14 pp.
- R. N. Das and P. Singh. On subclasses of schlicht mapping, Indian J. Pure Appl. Math. 8(8)(1977), 864-872.
- P. L. Duren and Y. J. Leung. Logarithmic coefficients of univalent functions, J. Analyse Math. 36(1979), 36-43.
- M. Fekete and G. Szegö. Eine Bemerkung Uber Ungerade Schlichte Funktionen, J. London Math. Soc. 8(2)(1933), 85-89.
- M. E. H. Ismail, E. Merkes, and D. Styer. A generalization of starlike functions, Complex Variables Theory Appl. 14(1-4)(1990), 77-84.
- F. H. Jackson. On q-functions and certain difference operators, Transactions of the Royal Society of Edinburgh 46(1908), 253-281.
- F. H. Jackson. On q-definite integrals, Quar. J. Pure Appl. Math. 41(1910), 193-203.
- B. Kowalczyk, A. Lecko, and H. M. Srivastava. A note on the Fekete-Szegö problem for close-to-convex functions with respect to convex functions, Publ. Inst. Math.(Beograd) (N.S.) 101(115)(2017), 143-149.
- S. Kumar and V. Ravichandran. A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40(2)(2016), 199-212.
- S. Kumar and V. Ravichandran. Functions defined by coefficient inequalities, Malays. J. Math. Sci. 11(3)(2017), 365-375.
- S. Kumar, V. Ravichandran, and S. Verma. Initial coefficients of starlike functions with real coefficients, Bull. Iranian Math. Soc. 43(6)(2017), 1837-1854.
- W. C. Ma and D. Minda. A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994.
- H. E. özkan Uçar. Coefficient inequality for q-starlike functions, Appl. Math. Comput. 276(2016), 122-126.
- U. Pranav Kumar and A. Vasudevarao. Logarithmic coefficients for certain subclasses of close-to-convex functions, Monatsh. Math. 187(3)(2018), 543-563.
- D. V. Prokhorov and J. Szynal. Inverse coefficients for (α, β)-convex functions, Ann. Univ. Mariae Curie-Sk lodowska Sect 35(1981), 125-143.
- C. Ramachandran, D. Kavitha, and T. Soupramanien. Certain bound for q-starlike and q-convex functions with respect to symmetric points, Int. J. Math. Math. Sci. (2015). Art. ID 205682, 7 pp.
- C. Ramachandran, D. Kavitha, and W. Ul-Haq. Fekete Szegö theorem for a close-to-convex error function, Math. Slovaca 69(2)(2019), 391-398.
- K. Sakaguchi. On a certain univalent mapping, J. Math. Soc. Japan 11(1959), 72-75.
- C. Selvaraj and N. Vasanthi. Subclasses of analytic functions with respect to symmetric and conjugate points, Tamkang J. Math. 42(1)(2011), 87-94.
- S. Sivasubramanian and P. Gurusamy. The Fekete-Szegö coefficient functional problems for q-starlike and q-convex functions related with lemniscate of Bernoulli, Asian-Eur. J. Math. 12(2), 1950019 (2019). 14 pp.
- J. Sokółand and D. K. Thomas. Further results on a class of starlike functions related to the Bernoulli lemniscate, Houston J. Math. 44(1)(2018), 83-95.
- E. Yaşar and S. Yalçın. Coefficient inequalities for certain classes of Sakaguchi-type harmonic functions, Southeast Asian Bull. Math. 38(6)(2014), 925-931.