Article
Kyungpook Mathematical Journal 2024; 64(4): 591-606
Published online December 31, 2024 https://doi.org/10.5666/KMJ.2024.64.4.591
Copyright © Kyungpook Mathematical Journal.
Investigation of Toeplitz Determinants in Ma-Minda Classes of Starlike and Convex Functions using q-Calculus
Pradeep Kumar and Amit Soni∗
Department of Mathematics, Government Engineering College, Bikaner-334001, Rajasthan, India
e-mail : prdeepbhadu007@gmail.com and aamitt1981@gmail.com
Received: July 22, 2023; Accepted: November 16, 2023
Abstract
This research paper focuses on obtaining sharp bounds for Toeplitz determinants whose entries are the coefficients of starlike functions satisfying specific conditions. The conditions are related to the quantity
Keywords: Univalent functions, Subordination, Toeplitz determinants, Fekete-Szegȍ inequality, q-Calculus
1. Introduction
The paragraph discusses classes of analytic functions in the open unit disk
Denote subclass
The field of quantum calculus, also known as q-calculus, does not rely on limits in its computations. Its widespread use in mathematics and physics stems from its application in various areas, including but not limited to, ordinary fractional calculus, basic hypergeometric functions, orthogonal polynomials, and combinatorics. Its importance is due to its versatility and ability to address complex problems in a variety of disciplines.
Let
We note that
where
For definitions and properties of q-calculus, one may refer to [9], [16], [18].
Definition 1.1. The class
In the limiting case
The class
In the limiting case
Toeplitz matrices and their determinants are of significant importance in various areas of mathematics, and they find many applications in diverse fields. (See [32]). A survey article by Ye and Lim [34] provides information on the application of Toeplitz matrices in several areas of pure and applied mathematics. It is worth noting that Toeplitz symmetric matrices are those matrices that have constant entries along the diagonal. For a given function
The bounds of
This article presents sharp estimates for Toeplitz determinants
and
respectively, belong to the classes
The primary outcomes of this research are established based on the aforementioned estimate by associating coefficients of the functions in the classes
The condition can be waived by using the definition of
2. Main Results
The sharp bound for
Theorem 2.1. If f
Proof. Since
Corresponding to the function w, define the function
so that
Clearly, the function
Since
Using (2.4), (2.7), (2.8) the coefficients
and
.
The equations (2.9) and (2.10) (see Ali et al. [4] for a general result for p-valent functions) readily shows that
Since
and, when
Using these estimates for the second and third coefficients given in (2.12) and (2.13) , we have
The result is sharp for the function
proving the sharpness.
Theorem 2.2. If
Proof. Let
The Taylor series expansion of the function f given by
Then using (2.14), (2.15) and (2.7), the coefficients
Using the well-known estimate
For a function
Since
The result is sharp for the function
proving the sharpness of the result.
Theorems
Theorem 2.3. If f
Proof. Since
it follows that
Since
Since
Using these estimates for the second and third coefficients given in (2.12) and (2.21), and the bound for
Theorem 2.4. If f
Proof. The given conditions on
and
Since
Using the bound for
The result is sharp for the function
3. Exploring Some Special Cases
3.1. The Ma and Minda classes of starlike and convex functions encompass various notable subclasses that have been extensively examined by different researchers (see, for instance, Kargar et al. [20] and Mahzoon [25]). Theorems
For
which implies
and for
If
If
The classes
and
For
and
In particular, as
3.2. Mendiratta et al. [26] introduced and studied the class
shows that
and
for
and
For
3.3. Sharma et al. [30] defined and studied the class of functions defined by
and
Whereas
and
3.4. The class
3.5. The class
Therefore,
3.6. Ronning [28], inspired by Goodman [13], introduced and analyzed the parabolic starlike class
This yields
and
for
3.7. Yunus et al. [35] investigated the class
3.8. Wani and Swaminathan [33] investigated the class of functions defined by
3.9. S.S. Kumar and K. Gangania [22] examined the class of functions defined by
3.10. J. Sokol and J. Stankiewicz [31] examined the class of functions defined by
3.11.
P. Goel and S. Sivaprasad Kumar [11] investigated the class of functions defined by
4. Conclusion
This research paper established precise bounds
References
- O. P. Ahuja. The Bieberbach conjecture and its impact on the developments in geometric function theory, Math. Chronicle, 15(1986), 1-28.
- 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, D. K. Thomas and A. Vasudevarao. Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, Bull. Aust. Math. Soc., 97(2)(2018), 253-264.
- R. M. Ali, V. Ravichandran and N. Seenivasagan. Coefficient bounds for p-valent functions, Appl. Math. Comput., 187(1)(2007), 35-46.
- V. Allu and A. Pandey. The Zalcman conjecture for certain analytic and univalent functions, J. Math. z Appl., 492(2)(2020), 124466.
- N. E. Cho, S. Kumar and V. Kumar. Hermitian-Toeplitz and Hankel determinants for certain starlike functions, Asian-Eur. J. Math., 15(3)(2022), 2250042:1-16. https://doi.org/10.1142/S1793557122500425.
- N. E. Cho, V. Kumar, S. S. Kumar and V. Ravichandran. Radius problems for starlike functions associated with the sine function, Bull. Iran. Math. Soc., 45(1)(2019), 213-232.
- K. Cudna, O. S. Kwon, A. Lecko, Y. J. Sim and B. S´miarowska. The second and third-order Hermitian Toeplitz determinants for starlike and convex functions of order α, Bol. Soc. Mat. Mex., 26(2)(2020), 361-375.
- G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, (2004).
- S. Giri and S. S. Kumar. Hermitian-Toeplitz determinants for certain univalent functions, Anal. Math. Phys., 13(2)(2023), 1-19.
- P. Goel and S. S. Kumar. Certain Class of Starlike Functions Associated with Modified Sigmoid Function, Bull. Malays. Math. Sci. Soc., 43(1)(2019), 957-991. https://doi.org/10.1007/s40840-019-00784-y.
- A. W. Goodman, Univalent functions, volume I and II, Mariner Pub. Co. Inc., Tampa Florida, (1984).
- A. W. Goodman. On uniformly convex functions, Ann. Polon. Math, 56(1)(1991), 87-92.
- U. Grenander and G. Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, (1958).
- M. E. H. Ismail, E. Markes and D. Styer. A generalization of starlike functions, Complex Var. Theor. Appl., 14(1990), 77-84.
- F. H. Jackson. On q-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46(2)(1909), 253-281.
- W. Janowski. Some extremal problems for certain families of analytic functions, Ann. Polon. Math., 28(1973), 297-326.
- V. Kac and P. Cheung, Quantum calculus, Springer, 2002.
- K. Khatter, V. Ravichandran and S. S. Kumar. Starlike functions associated with exponential function and the lemniscate of Bernoulli, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 113(1)(2019), 233-253.
- R. Kargar, A. Ebadian and L. Trojnar-Spelina. Further results for starlike functions related with booth lemniscate, Iran. J. Sci. Technol. Trans. A Sci., 4(3)(2019), 1235-1238.
- B. Kowalczyk, A. Lecko and B. Smiarowska. On some coefficient inequality in the subclass of close-to-convex functions, Bull. Soc. Sci. Lettres Łódź Sér. Rech. Déform., 67(1)(2017), 79-90.
- S. S. Kumar and G. Kamaljeet. A cardioid domain and starlike functions, Anal. Math. Phys., 11(2)(2021), 1-34. https://doi.org/10.1007/s13324-021-00483-7.
- A. Lecko, Y. J. Sim and B. Śmiarowska. The fourth-order Hermitian Toeplitz determinant for convex functions, Anal. Math. Phys., 10(3)(2020), 39:1-11. https://doi.org/10.1007/s13324-020-00382-3.
- W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis, Tianjin (1992), 157-169, Conf. Proc. Lecture Notes Anal. 1. Int. Press, Cambridge, 1994.
- H. Mahzoon. Further Results for α-Spirallike Functions of Order β, Iran. J. Sci. Technol. Trans. A Sci., 44(2020), 1085-1089. https://doi.org/10.1007/s40995-020-00898-0.
- R. Mendiratta, S. Nagpal and V. Ravichandran. On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38(1)(2015), 365-386.
- R. K. Raina and J. Sokół. Some properties related to a certain class of starlike functions., C. R. Math., 353(11)(2015), 973-978.
- F. Ronning. A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie-Skłodowska, A Math., 47(1993), 123-134.
- M. S. Robertson. Certain classes of starlike functions, Michigan Math. J., 32(2)(1985), 135-140.
- K. Sharma, N. K. Jain and V. Ravichandran. Starlike functions associated with a cardioid, Afr. Mat., 27(5-6)(2016), 923-939.
- J. Sokol and S. Stankiewicz. Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat., 19(1996), 101-105.
- O. Toeplitz. Zur Transformation der Scharen bilinearer Formen von unendlichvielen Veränderlichen, Mathematischphysikalische, Klasse, Nachr. der Kgl. Gessellschaft der Wissenschaften zu Göttingen, 1907, 110-115.
- L. A. Wani and A. Swaminathan. Starlike and convex functions associated with a nephroid domain, Bull. Malays. Math. Sci. Soc., 44(1)(2021), 79-104.
- K. Ye and L.-H. Lim. Every matrix is a product of Toeplitz matrices, Found Comput Math, 16(3)(2016), 577-598.
- Y. Yunus, S. A. Halim and A. B. Akbarally. Subclass of starlike functions associated with a limacon, AIP Conf. Proc., 1974(1)(2018), 030023.
- H.-Y. Zhang, R. Srivastava and H. Tang. Third-order Hankel and Toeplitz determinants for starlike functions connected with the sine function, Mathematics, 7(5)(2019), 404.