Toeplitz Determinant for a Subclass of Analytic Functions Involving Touchard Polynomials
Toeplitz Determinant for a Subclass of Analytic Functions
Resumen
In this paper, we introduce a new subclass of univalent functions that generalizes existing subclasses of univalent functions. By employing subordination principles, we derive initial Taylor–Maclaurin coefficient estimates for functions in this subclass. Additionally, we establish bounds for the Fekete-Szeg\"{o} functional and Toeplitz determinants. To further strengthen the applicability of our findings, we incorporate Touchard polynomials, demonstrating their role in Geometric Function Theory (GFT). Our results unify and generalize several known subclasses, offering potential applications of Touchard polynomials in the field of GFT.
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\bibitem{Ali-Allu-Thomas-2018} M. F. Ali, V. Allu, and D. Thomas, Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, \textit{Bull. Aust. Math. Soc.}, \textbf{97} (2), (2018), \url{https://doi.org/10.1017/S0004972717001174}.
\bibitem{Ali-Thomas-Vasudevarao-2018} M. F. Ali, D. K. Thomas, and A. Vasudevarao, Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, \textit{Bull. Aust. Math. Soc.}, \textbf{97} (2), (2018), 253--264.
\bibitem{Allu-2022} V. Allu, A. Lecko, and D. K. Thomas, Hankel, Toeplitz and Hermitian-Toeplitz determinants for certain close-to-convex functions,
\textit{Mediterr. J. Math.}, \textbf{19}, (2022), 1--17, \url{https://doi.org/10.1007/s00009-022-01988-y}.
\bibitem{Altinkaya-Magesh-Yalcin-2019-Toeplitz} S. Alt\i nkaya, N. Magesh, and S. Yal\c{c}\i n, Construction of Toeplitz matrices whose elements are the coefficients of univalent functions associated with $Q$-derivative operator, \textit{Casp. J. Math. Sci.}, \textbf{8} (1), (2019), 51--57, \url{https://doi.org/10.22080/cjms.2018.14151.1342}.
\bibitem{Al-Oboudi-Haidan-2000} F. M. Al-Oboudi and M. M. Haidan, Spirallike functions of complex order, \textit{J. Natur. Geom.}, \textbf{19}, (2000), 53--72.
\bibitem{Altintas-1991} O. Alt\i nta\c{s}, On a subclass of certain starlike functions with negative coefficients, \textit{Math. Japon.}, \textbf{36} (3), (1991), 489--495.
\bibitem{Altintas-Irmak-Srivastava-1995} O. Alt\i nta\c{s}, H. Irmak, and H. M. Srivastava, Fractional calculus and certain starlike functions with negative coefficients, \textit{Comput. Math. Appl.}, \textbf{30} (2), (1995), 9--15.
\bibitem{EEA-WYK-RMEA-AMA-FEM-RAT-2023} E. E. Ali, W. Y. Kota, R. M. El-Ashwah, A. M. Albalahi, F. E. Mansour, and R. A. Tahira, An application of Touchard polynomials on subclasses of analytic functions, \textit{Symmetry}, \textbf{15} (12), (2023), 1--14, \url{https://doi.org/10.3390/sym15122125}.
\bibitem{KNB-2009} K. N. Boyadzhiev, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, \textit{Abstr. Appl. Anal.}, \textbf{2009} (1), (2009), \url{https://doi.org/10.1155/2009/168672}.
\bibitem{Caratheodory-1907} C. Carath\'eodory, \"Uber den Variabilit\"atsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, \textit{Math. Ann.}, \textbf{64} (1), (1907), 95--115.
\bibitem{Cotirla-Wanas-2022} L.-I. Cot\^irl\u{a} and A. K. Wanas, Symmetric Toeplitz matrices for a new family of prestarlike functions, \textit{Symmetry}, \textbf{14}, (2022), \url{https://doi.org/10.3390/sym14071413}.
\bibitem{Efraimidis-2016} I. Efraimidis, A generalization of Livingston’s coefficient inequalities for functions with positive real part, \textit{J. Math. Anal. Appl.}, \textbf{435} (1), (2016), 369--379, \url{https://doi.org/10.1016/j.jmaa.2015.10.050}.
\bibitem{Frasin-2006} B. A. Frasin, Family of analytic functions of complex order, \textit{Acta Math. Acad. Paedagog. Nyh\'{a}zi.}, \textbf{22} (2), (2006), 179--191, \url{http://eudml.org/doc/53409}.
\bibitem{Jahangiri-1986} M. Jahangiri, On the coefficients of powers of a class of Bazilevi\v{c} functions, \textit{Indian J. Pure Appl. Math.}, \textbf{17}, (1986), 1140--1144.
\bibitem{Jahangiri-Magesh-Yamini-2015} J. M. Jahangiri, N. Magesh, and J. Yamini, Fekete–Szeg\"{o} inequalities for classes of bi-starlike and bi-convex functions, \textit{Electron. J. Math. Anal. Appl.}, \textbf{3} (1), (2015), 133--140.
\bibitem{SJH-ARJ-HHE-2024} S. J. Hamaad, A. R. S. Juma, and H. H. Ebrahim, Subclass of bi-starlike function associated with Touchard polynomials, \textit{J. Interdiscip. Math.}, \textbf{27} (4), (2024), 793--797, \url{https://doi.org/10.47974/JIM-1873}.
\bibitem{Hadi-Saleem-Lupas-2025} S. H. Hadi, Y. H. Saleem, A. A. Lupaş, K. M. K. Alshammari, and A. Alatawi, Toeplitz determinants for inverse of analytic functions, \textit{Mathematics}, \textbf{13} (4), (2025), \url{https://doi.org/10.3390/math13040676}.
\bibitem{Kamali-Orhan-2004} M. Kamali and H. Orhan, On a subclass of certain starlike functions with negative coefficients, \textit{Bull. Korean Math. Soc.}, \textbf{41} (1), (2004), 53--71, \url{https://doi.org/10.4134/BKMS.2004.41.1.053}.
\bibitem{Kamali-2022} M. Kamali and A. Riskulova, On bounds of Toeplitz determinants for a subclass of analytic functions, \textit{Bull. Math. Anal. Appl.}, \textbf{14}, (2022), 36--48, \url{https://doi.org/10.54671/BMAA-2022-3-3}.
\bibitem{Li_Gou_2024} Z. Li, D. Gou, Toeplitz Determinant for the inverse of a function whose derivative has a positive real part, Appl. Math.
Nonlinear Sci., (9) (2024), 1-–7.
\bibitem{Maminda-1992-INC} W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, In Proceeding of the International Conference on Complex Analysis, Tianjin, China, 1992.
\bibitem{GMS-SP-2020} G. Murugusundaramoorthy and S. Porwal, Univalent functions with positive coefficients involving Touchard polynomials, \textit{Al-Qadisiyah J. Pure Sci.}, \textbf{25} (4), (2020), 1--8, \url{https://doi.org/10.29350/qjps.2020.25.4.1176}.
\bibitem{Nasr-Aouf-1982-convex} M. A. Nasr and M. K. Aouf, On convex functions of complex order, \textit{Mansoura Bull. Sci.}, \textbf{8}, (1982), 565--582.
\bibitem{Nasr-Aouf-1985} M. A. Nasr and M. K. Aouf, Starlike functions of complex order, \textit{J. Natur. Sci. Math.}, \textbf{25}, (1985), 1--12.
\bibitem{Nandeesh-Salestina-Archana-Murugusundaramoorthy-2025} M. Nandeesh, M. Ruby Salestina, Archana, and G. Murugusundaramoorthy, Toeplitz matrices whose elements are coefficients of new subclasses of analytical functions, \textit{Commun. Appl. Nonlinear Anal.}, \textbf{32} (2), (2025), \url{https://doi.org/10.52783/cana.v32.1750}.
\bibitem{Orhan-Powal-Magesh-2022} H. Orhan, S. Porwal, and N. Magesh, The Fekete-Szeg\H{o} problem for a generalized class of analytic functions of complex order associated with $q$-calculus, Palest. J. Math., \textbf{11}, (2022), 39--47.
\bibitem{Pei-Long-Liu-Gangadharan-2024} K. Pei, P. Long, J. Liu, and G. Murugusundaramoorthy, Fekete–Szeg\"{o} inequalities and the symmetric Toeplitz determinants for certain analytic function class involving $q$-differintegral operator, \textit{Chin. Quart. J. Math.}, \textbf{39} (4), (2024), 366--378.
\bibitem{Powal-2014-Poisson} S. Porwal, An application of a Poisson distribution series on certain analytic functions, \textit{J. Complex Anal.}, (2014), 1--3, \url{https://doi.org/10.1155/2014/984135}.
\bibitem{SP-GM-2022} S. Porwal and G. Murugusundaramoorthy, Unified classes of starlike and convex functions associated with Touchard polynomials, \textit{Sci. Technol. Asia}, \textbf{27} (4), (2022), 207--214, \url{https://doi.org/10.14456/scitechasia.2022.80}.
\bibitem{Radhika-SS-GMS-2018} V. Radhika, S. Sivasubramanian, G. Murugusundaramoorthy, and J. M. Jahangiri, Toeplitz matrices whose elements are coefficients of Bazilevi\v{c} functions, \textit{Open Math.}, \textbf{16}, (2018), 1161--1169, \url{https://doi.org/10.1515/math-2018-0093}.
\bibitem{Ramachandran-Kavitha-2017} C. Ramachandran and D. Kavitha, Toeplitz determinant for some subclasses of analytic functions, \textit{Glob. J. Pure Appl. Math.}, \textbf{13} (2), (2017), 785--793.
\bibitem{Ravichandran-2005-complex-order}V. Ravichandran, Y. Polatoglu, M. Bolcal and A. Sen: Certain subclasses of starlike and convex functions of complex order. Hacettepe J. Math. Stat. 34 (2005), 9-15.
\bibitem{Robertson-1936} M. S. Robertson, On the theory of univalent functions, \textit{Ann. Math.}, \textbf{37}, (1936), 374--408.
\bibitem{HMS-Toeplitz-2019} H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan, and B. Khan, Hankel and Toeplitz determinants for a subclass of $q$-starlike functions associated with a general conic domain, \textit{Mathematics}, \textbf{7}, (2019), 181, \url{https://doi.org/10.3390/math7090181}.
\bibitem{Sun-Wang-2023} Y. Sun and Z. G. Wang, Sharp bounds on Hermitian Toeplitz determinants for Sakaguchi classes, \textit{Bull. Malays. Math. Sci. Soc.}, \textbf{46} (2), (2023), 59.
\bibitem{Sun-Wang-Tang-2023} Y. Sun, Z. G. Wang, and H. Tang, Sharp bounds on the fourth-order Hermitian Toeplitz determinant for starlike functions of order $1/2$, \textit{J. Math. Inequal.}, \textbf{17}, (2023), 985--996.
\bibitem{TS-CR-KAS-2021} T. Soupramanien, C. Ramachandran, and K. Al-Shaqsi, Certain subclasses of univalent functions with positive coefficients involving Touchard polynomials, \textit{Adv. Math. Sci. J.}, \textbf{10} (2), (2021), 981--990, \url{https://doi.org/10.37418/amsj.10.2.27}.
\bibitem{Thomas-Halim-2017-Toeplitz} D. K. Thomas and S. A. Halim, Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions, \textit{Bull. Malays. Math. Sci. Soc.}, \textbf{40}, (2017), 1781--1790, \url{https://doi.org/10.1007/s40840-016-0385-4}.
\bibitem{Touchard-1939} J. Touchard, Sur les cycles des substitutions, \textit{Acta Math.}, \textbf{70}, (1939), 243--297, \url{https://doi.org/10.1007/BF02547349}.
\bibitem{Wahid-2022} N. H. A. A. Wahid, D. Mohamad, N. M. Kamarozzaman, and A. A. Shahminan, Toeplitz determinants for the class of functions with bounded turning, \textit{Eur. J. Pure Appl. Math.}, \textbf{15}, (2022), 1937--1947, \url{https://doi.org/10.29020/nybg.ejpam.v15i4.455}.
\bibitem{Ye-Lim-2016} K. Ye and L. H. Lim, Every matrix is a product of Toeplitz matrices, \textit{Found. Comput. Math.}, \textbf{16}, (2016), 577--598, \url{https://doi.org/10.1007/s10208-015-9254-z}.
\bibitem{Zhang-Tang-2021} H. Y. Zhang and H. Tang, Fourth Toeplitz determinants for starlike functions defined by using the sine function, \textit{J. Funct. Spaces}, \textbf{2021}, (2021), \url{https://doi.org/10.1155/2021/4103772}.
\bibitem{Zulfiqar-2022} F. Zulfiqar, S. N. Malik, M. Raza, and M. Ali, Fourth-order Hankel determinants and Toeplitz determinants for convex functions connected with sine functions, \textit{J. Math.}, \textbf{2022}, (2022), \url{https://doi.org/10.1155/2022/2871511}.
\bibitem{Ali-Thomas-Vasudevarao-2018} M. F. Ali, D. K. Thomas, and A. Vasudevarao, Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, \textit{Bull. Aust. Math. Soc.}, \textbf{97} (2), (2018), 253--264.
\bibitem{Allu-2022} V. Allu, A. Lecko, and D. K. Thomas, Hankel, Toeplitz and Hermitian-Toeplitz determinants for certain close-to-convex functions,
\textit{Mediterr. J. Math.}, \textbf{19}, (2022), 1--17, \url{https://doi.org/10.1007/s00009-022-01988-y}.
\bibitem{Altinkaya-Magesh-Yalcin-2019-Toeplitz} S. Alt\i nkaya, N. Magesh, and S. Yal\c{c}\i n, Construction of Toeplitz matrices whose elements are the coefficients of univalent functions associated with $Q$-derivative operator, \textit{Casp. J. Math. Sci.}, \textbf{8} (1), (2019), 51--57, \url{https://doi.org/10.22080/cjms.2018.14151.1342}.
\bibitem{Al-Oboudi-Haidan-2000} F. M. Al-Oboudi and M. M. Haidan, Spirallike functions of complex order, \textit{J. Natur. Geom.}, \textbf{19}, (2000), 53--72.
\bibitem{Altintas-1991} O. Alt\i nta\c{s}, On a subclass of certain starlike functions with negative coefficients, \textit{Math. Japon.}, \textbf{36} (3), (1991), 489--495.
\bibitem{Altintas-Irmak-Srivastava-1995} O. Alt\i nta\c{s}, H. Irmak, and H. M. Srivastava, Fractional calculus and certain starlike functions with negative coefficients, \textit{Comput. Math. Appl.}, \textbf{30} (2), (1995), 9--15.
\bibitem{EEA-WYK-RMEA-AMA-FEM-RAT-2023} E. E. Ali, W. Y. Kota, R. M. El-Ashwah, A. M. Albalahi, F. E. Mansour, and R. A. Tahira, An application of Touchard polynomials on subclasses of analytic functions, \textit{Symmetry}, \textbf{15} (12), (2023), 1--14, \url{https://doi.org/10.3390/sym15122125}.
\bibitem{KNB-2009} K. N. Boyadzhiev, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, \textit{Abstr. Appl. Anal.}, \textbf{2009} (1), (2009), \url{https://doi.org/10.1155/2009/168672}.
\bibitem{Caratheodory-1907} C. Carath\'eodory, \"Uber den Variabilit\"atsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, \textit{Math. Ann.}, \textbf{64} (1), (1907), 95--115.
\bibitem{Cotirla-Wanas-2022} L.-I. Cot\^irl\u{a} and A. K. Wanas, Symmetric Toeplitz matrices for a new family of prestarlike functions, \textit{Symmetry}, \textbf{14}, (2022), \url{https://doi.org/10.3390/sym14071413}.
\bibitem{Efraimidis-2016} I. Efraimidis, A generalization of Livingston’s coefficient inequalities for functions with positive real part, \textit{J. Math. Anal. Appl.}, \textbf{435} (1), (2016), 369--379, \url{https://doi.org/10.1016/j.jmaa.2015.10.050}.
\bibitem{Frasin-2006} B. A. Frasin, Family of analytic functions of complex order, \textit{Acta Math. Acad. Paedagog. Nyh\'{a}zi.}, \textbf{22} (2), (2006), 179--191, \url{http://eudml.org/doc/53409}.
\bibitem{Jahangiri-1986} M. Jahangiri, On the coefficients of powers of a class of Bazilevi\v{c} functions, \textit{Indian J. Pure Appl. Math.}, \textbf{17}, (1986), 1140--1144.
\bibitem{Jahangiri-Magesh-Yamini-2015} J. M. Jahangiri, N. Magesh, and J. Yamini, Fekete–Szeg\"{o} inequalities for classes of bi-starlike and bi-convex functions, \textit{Electron. J. Math. Anal. Appl.}, \textbf{3} (1), (2015), 133--140.
\bibitem{SJH-ARJ-HHE-2024} S. J. Hamaad, A. R. S. Juma, and H. H. Ebrahim, Subclass of bi-starlike function associated with Touchard polynomials, \textit{J. Interdiscip. Math.}, \textbf{27} (4), (2024), 793--797, \url{https://doi.org/10.47974/JIM-1873}.
\bibitem{Hadi-Saleem-Lupas-2025} S. H. Hadi, Y. H. Saleem, A. A. Lupaş, K. M. K. Alshammari, and A. Alatawi, Toeplitz determinants for inverse of analytic functions, \textit{Mathematics}, \textbf{13} (4), (2025), \url{https://doi.org/10.3390/math13040676}.
\bibitem{Kamali-Orhan-2004} M. Kamali and H. Orhan, On a subclass of certain starlike functions with negative coefficients, \textit{Bull. Korean Math. Soc.}, \textbf{41} (1), (2004), 53--71, \url{https://doi.org/10.4134/BKMS.2004.41.1.053}.
\bibitem{Kamali-2022} M. Kamali and A. Riskulova, On bounds of Toeplitz determinants for a subclass of analytic functions, \textit{Bull. Math. Anal. Appl.}, \textbf{14}, (2022), 36--48, \url{https://doi.org/10.54671/BMAA-2022-3-3}.
\bibitem{Li_Gou_2024} Z. Li, D. Gou, Toeplitz Determinant for the inverse of a function whose derivative has a positive real part, Appl. Math.
Nonlinear Sci., (9) (2024), 1-–7.
\bibitem{Maminda-1992-INC} W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, In Proceeding of the International Conference on Complex Analysis, Tianjin, China, 1992.
\bibitem{GMS-SP-2020} G. Murugusundaramoorthy and S. Porwal, Univalent functions with positive coefficients involving Touchard polynomials, \textit{Al-Qadisiyah J. Pure Sci.}, \textbf{25} (4), (2020), 1--8, \url{https://doi.org/10.29350/qjps.2020.25.4.1176}.
\bibitem{Nasr-Aouf-1982-convex} M. A. Nasr and M. K. Aouf, On convex functions of complex order, \textit{Mansoura Bull. Sci.}, \textbf{8}, (1982), 565--582.
\bibitem{Nasr-Aouf-1985} M. A. Nasr and M. K. Aouf, Starlike functions of complex order, \textit{J. Natur. Sci. Math.}, \textbf{25}, (1985), 1--12.
\bibitem{Nandeesh-Salestina-Archana-Murugusundaramoorthy-2025} M. Nandeesh, M. Ruby Salestina, Archana, and G. Murugusundaramoorthy, Toeplitz matrices whose elements are coefficients of new subclasses of analytical functions, \textit{Commun. Appl. Nonlinear Anal.}, \textbf{32} (2), (2025), \url{https://doi.org/10.52783/cana.v32.1750}.
\bibitem{Orhan-Powal-Magesh-2022} H. Orhan, S. Porwal, and N. Magesh, The Fekete-Szeg\H{o} problem for a generalized class of analytic functions of complex order associated with $q$-calculus, Palest. J. Math., \textbf{11}, (2022), 39--47.
\bibitem{Pei-Long-Liu-Gangadharan-2024} K. Pei, P. Long, J. Liu, and G. Murugusundaramoorthy, Fekete–Szeg\"{o} inequalities and the symmetric Toeplitz determinants for certain analytic function class involving $q$-differintegral operator, \textit{Chin. Quart. J. Math.}, \textbf{39} (4), (2024), 366--378.
\bibitem{Powal-2014-Poisson} S. Porwal, An application of a Poisson distribution series on certain analytic functions, \textit{J. Complex Anal.}, (2014), 1--3, \url{https://doi.org/10.1155/2014/984135}.
\bibitem{SP-GM-2022} S. Porwal and G. Murugusundaramoorthy, Unified classes of starlike and convex functions associated with Touchard polynomials, \textit{Sci. Technol. Asia}, \textbf{27} (4), (2022), 207--214, \url{https://doi.org/10.14456/scitechasia.2022.80}.
\bibitem{Radhika-SS-GMS-2018} V. Radhika, S. Sivasubramanian, G. Murugusundaramoorthy, and J. M. Jahangiri, Toeplitz matrices whose elements are coefficients of Bazilevi\v{c} functions, \textit{Open Math.}, \textbf{16}, (2018), 1161--1169, \url{https://doi.org/10.1515/math-2018-0093}.
\bibitem{Ramachandran-Kavitha-2017} C. Ramachandran and D. Kavitha, Toeplitz determinant for some subclasses of analytic functions, \textit{Glob. J. Pure Appl. Math.}, \textbf{13} (2), (2017), 785--793.
\bibitem{Ravichandran-2005-complex-order}V. Ravichandran, Y. Polatoglu, M. Bolcal and A. Sen: Certain subclasses of starlike and convex functions of complex order. Hacettepe J. Math. Stat. 34 (2005), 9-15.
\bibitem{Robertson-1936} M. S. Robertson, On the theory of univalent functions, \textit{Ann. Math.}, \textbf{37}, (1936), 374--408.
\bibitem{HMS-Toeplitz-2019} H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan, and B. Khan, Hankel and Toeplitz determinants for a subclass of $q$-starlike functions associated with a general conic domain, \textit{Mathematics}, \textbf{7}, (2019), 181, \url{https://doi.org/10.3390/math7090181}.
\bibitem{Sun-Wang-2023} Y. Sun and Z. G. Wang, Sharp bounds on Hermitian Toeplitz determinants for Sakaguchi classes, \textit{Bull. Malays. Math. Sci. Soc.}, \textbf{46} (2), (2023), 59.
\bibitem{Sun-Wang-Tang-2023} Y. Sun, Z. G. Wang, and H. Tang, Sharp bounds on the fourth-order Hermitian Toeplitz determinant for starlike functions of order $1/2$, \textit{J. Math. Inequal.}, \textbf{17}, (2023), 985--996.
\bibitem{TS-CR-KAS-2021} T. Soupramanien, C. Ramachandran, and K. Al-Shaqsi, Certain subclasses of univalent functions with positive coefficients involving Touchard polynomials, \textit{Adv. Math. Sci. J.}, \textbf{10} (2), (2021), 981--990, \url{https://doi.org/10.37418/amsj.10.2.27}.
\bibitem{Thomas-Halim-2017-Toeplitz} D. K. Thomas and S. A. Halim, Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions, \textit{Bull. Malays. Math. Sci. Soc.}, \textbf{40}, (2017), 1781--1790, \url{https://doi.org/10.1007/s40840-016-0385-4}.
\bibitem{Touchard-1939} J. Touchard, Sur les cycles des substitutions, \textit{Acta Math.}, \textbf{70}, (1939), 243--297, \url{https://doi.org/10.1007/BF02547349}.
\bibitem{Wahid-2022} N. H. A. A. Wahid, D. Mohamad, N. M. Kamarozzaman, and A. A. Shahminan, Toeplitz determinants for the class of functions with bounded turning, \textit{Eur. J. Pure Appl. Math.}, \textbf{15}, (2022), 1937--1947, \url{https://doi.org/10.29020/nybg.ejpam.v15i4.455}.
\bibitem{Ye-Lim-2016} K. Ye and L. H. Lim, Every matrix is a product of Toeplitz matrices, \textit{Found. Comput. Math.}, \textbf{16}, (2016), 577--598, \url{https://doi.org/10.1007/s10208-015-9254-z}.
\bibitem{Zhang-Tang-2021} H. Y. Zhang and H. Tang, Fourth Toeplitz determinants for starlike functions defined by using the sine function, \textit{J. Funct. Spaces}, \textbf{2021}, (2021), \url{https://doi.org/10.1155/2021/4103772}.
\bibitem{Zulfiqar-2022} F. Zulfiqar, S. N. Malik, M. Raza, and M. Ali, Fourth-order Hankel determinants and Toeplitz determinants for convex functions connected with sine functions, \textit{J. Math.}, \textbf{2022}, (2022), \url{https://doi.org/10.1155/2022/2871511}.
Publicado
2025-09-18
Número
Sección
Research Articles
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
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