Startseite Mathematik Close-to-convex functions associated with a rational function
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Close-to-convex functions associated with a rational function

  • Swati Anand , Pratima Rai und Sushil Kumar EMAIL logo
Veröffentlicht/Copyright: 24. Mai 2024
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

In this paper, we consider the class 𝓚𝓢(ψ0) of close-to-convex functions associated to a rational function ψ0(z) = (k2 + z2)/(k2kz), where k = 2 + 1 and the rational function ψ0 is related to cardioid shaped bounded domain. We compute the radius of convexity, growth, distortion and certain coefficient inequalities of such functions. We determine bounds on second and third order Toeplitz determinants. Moreover, we also compute bounds on second order Hankel determinants and second and third order Vandermonde determinants.

MSC 2010: 30C45

This work was supported by the Institute of Eminence, University of Delhi, Delhi, India–110007, Grant No. /IoE/2021/12/FRP


Acknowledgement

The authors are thankful to the referee for his valuable comments.

  1. Communicated by Stanisława Kanas

References

[1] Ahuja, O.—Bohra, N.—Çetinkaya, A.—Kumar, S.: Univalent functions associated with the symmetric points and cardioid-shaped domain involving (p, q)-calculus, Kyungpook Math. J. 61(1) (2021), 75–98.Suche in Google Scholar

[2] Ahuja, O.—Khatter, K.—Ravichandran V.: Toeplitz determinants associated with Ma-Minda classes of starlike and convex functions, Iran. J. Sci. Technol. Trans. A Sci. 45(6) (2021), 2021–2027.10.1007/s40995-021-01173-6Suche in Google Scholar

[3] Ali, M. F.—Thomas, D. K.—Vasudevarao, A.: Toeplitz determinants whose elements are the coefficients of analytic and univalent functions, Bull. Aust. Math. Soc. 97(2) (2018), 253–264.10.1017/S0004972717001174Suche in Google Scholar

[4] Anand, S.—Jain, N. K.—Kumar, S.: Certain estimates of normalized analytic functions, Math. Slovaca 72(1) (2022), 85–102.10.1515/ms-2022-0006Suche in Google Scholar

[5] Bulut, S.: Coefficient estimates for Libera type bi-close-to-convex functions, Math. Slovaca 71(6) (2021), 1401–1410.10.1515/ms-2021-0060Suche in Google Scholar

[6] Duren, P. L.: Univalent Functions. Grundlehren Math. Wiss., Vol. 259, Springer-Verlag, New York, 1983.Suche in Google Scholar

[7] Gao, C.—Zhou, S.: On a class of analytic functions related to the starlike functions, Kyungpook Math. J. 45(1) (2005), 123–130.Suche in Google Scholar

[8] Grunsky, H.: Neue Abschätzungen zur konformen Abbildung ein-und mehrfach zusammenhängender bereiche, Schr. Deutsche Math.-Ver. 43 (1934), 140-143.Suche in Google Scholar

[9] Güney, H.—Murugusundaramoorthy, G.—Srivastava, H. M.: The second Hankel determinant for a certain class of bi-close-to-convex functions, Results Math. 74(3) (2019), Art. No. 93.10.1007/s00025-019-1020-0Suche in Google Scholar

[10] Janteng, A.—Halim, S. A.—Darus, M.: Coefficient inequality for a function whose derivative has a positive real part, JIPAM J. Inequal. Pure Appl. Math. 7(2) (2006), Art. No. 50.Suche in Google Scholar

[11] Kaplan, W.: Close-to-convex schlicht functions, Michigan Math. J. 1(2) (1952), 169–185.10.1307/mmj/1028988895Suche in Google Scholar

[12] Kanas, S. R.—Masih, V. S.—Ebadian, A.: Coefficients problems for families of holomorphic functions related to hyperbola, Math. Slovaca 70(3) (2020), 605–616.10.1515/ms-2017-0375Suche in Google Scholar

[13] Kowalczyk, B.—Lecko, A.: Radius problem in classes of polynomial close-to-convex functions I, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 63(1) (2013), 65–77.Suche in Google Scholar

[14] Kowalczyk, B.—Lecko, A.: Radius problem in classes of polynomial close-to-convex functions II. Partial solutions, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 63(2) (2013), 23–34.Suche in Google Scholar

[15] Kowalczyk, J.—Leś-Bomba, E.: On a subclass of close-to-convex functions, Appl. Math. Lett. 23(10) (2010), 1147–1151.10.1016/j.aml.2010.03.004Suche in Google Scholar

[16] Kowalczyk, J.— Leś, E.—Sokół, J.: Radius problems in a certain subclass of close-to-convex functions, Houston J. Math. 40(4) (2014), 1061–1072.Suche in Google Scholar

[17] Kumar, S.—Ravichandran, V.: A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40(2) (2016), 199–212.Suche in Google Scholar

[18] Kumar, S.—Rai, P.—Çetinkaya, A.: Radius estimates of certain analytic functions, Honam Math. J. 43(4) (2021), 627–639.Suche in Google Scholar

[19] Kumar, S.—Anand, S.—Jain, N. K.: Hankel and Symmetric Toeplitz determinants for Sakaguchi starlike functions, Stud. Univ. Babeş-Bolyai Math., to appear.Suche in Google Scholar

[20] Kumar, S.—Ravichandran, V.—Verma, S.: Initial coefficients of starlike functions with real coefficients, Bull. Iranian Math. Soc. 43(6) (2017), 1837–1854.Suche in Google Scholar

[21] Kumar, S.—Ravichandran, V.: Functions defined by coefficient inequalities, Malays. J. Math. Sci. 11(3) (2017), 365–375.Suche in Google Scholar

[22] Kumar, V.—Kumar, S.: Bounds on Hermitian-Toeplitz and Hankel determinants for strongly starlike functions, Bol. Soc. Mat. Mex. (3) 27(2) (2021), Art. No. 55.10.1007/s40590-021-00362-ySuche in Google Scholar

[23] Nevanlinna, R.: Über die schlichten Abbildungen des Einheits-Kreises, Översikt Finska Vetenskaps-Soc. Förh 62A (1920), 1-14.Suche in Google Scholar

[24] Prajapat, J. K.—Maharana, S.—Bansal, D.: Radius of starlikeness and Hardy space of Mittag-Leffler functions, Filomat 32(18) (2018), 6475–6486.10.2298/FIL1818475PSuche in Google Scholar

[25] Pommerenke, Ch.: On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc. 41 (1966), 111–122.10.1112/jlms/s1-41.1.111Suche in Google Scholar

[26] Pommerenke, Ch.: On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108–112.10.1112/S002557930000807XSuche in Google Scholar

[27] Rai, P.—Kumar, S.: Coefficient Inequalities for a subfamily of Sakaguchi starlike functions, Asian-Eur. J. Math. 16(5) (2023), Art. ID 2350084.10.1142/S1793557123500845Suche in Google Scholar

[28] Ravichandran, V.—Verma, S.: Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353(6) (2015), 505–510.10.1016/j.crma.2015.03.003Suche in Google Scholar

[29] Robertson, M. I. S.: On the theory of univalent functions, Ann. of Math. (2) 37(2) (1936), 374–408.10.2307/1968451Suche in Google Scholar

[30] Răducanu, D.—Zaprawa, P.: Second Hankel determinant for close-to-convex functions, C. R. Math. Acad. Sci. Paris 355(10) (2017), 1063–1071.10.1016/j.crma.2017.09.006Suche in Google Scholar

[31] Sharma, P.—Raina, R. K.—Sokół, J.: On a set of close-to-convex functions, Studia Sci. Math. Hungar. 55(2) (2018), 190–202.10.1556/012.2018.55.2.1381Suche in Google Scholar

[32] Şeker, B.: On certain new subclass of close-to-convex functions, Appl. Math. Comput. 218(3) (2011), 1041–1045.10.1016/j.amc.2011.03.018Suche in Google Scholar

[33] Vasudevarao, A.—Lecko, A.—Thomas, D. K.: Hankel, Toeplitz, and Hermitian-Toeplitz determinants for certain close-to-convex functions, Mediterr. J. Math. 19(1) (2022), Art. No. 22.10.1007/s00009-021-01934-ySuche in Google Scholar

[34] Vijayalakshmi, S. P.—Bulut, S.—Sudharsan, T. V.: Vandermonde determinant for a certain Sakaguchi type function in Limaçon domain, Asian-Eur. J. Math. 15(12) (2022), Art. ID 22502126.10.1142/S1793557122502126Suche in Google Scholar

[35] Wang, Z.—Gao, C.—Yuan, S.: On certain subclass of close-to-convex functions, Acta Math. Acad. Paedagog. Nyházi (N.S.) 22(2) (2006), 171–177.Suche in Google Scholar

[36] Xu, Q.-H.—Srivastava, H. M.—Li, Z.: A certain subclass of analytic and close-to-convex functions, Appl. Math. Lett. 24(3) (2011), 396–401.10.1016/j.aml.2010.10.037Suche in Google Scholar

Received: 2023-03-21
Accepted: 2023-08-24
Published Online: 2024-05-24
Published in Print: 2024-04-25

© 2024 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 16.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0026/pdf
Button zum nach oben scrollen