Startseite Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables

  • Zhenhan Tu und Liangpeng Xiong EMAIL logo
Veröffentlicht/Copyright: 19. Juli 2019
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

Let Sψ be a subclass of starlike functions in the unit disk 𝕌, where ψ is a convex function such that ψ(0) = 1, ψ′(0) > 0, ℜ(ψ(ξ)) > 0 and ψ(𝕌) is symmetric with respect to the real axis. We obtain the sharp solution of Fekete-Szegö problem for the family Sψ, and then extend the result to the case of corresponding subclass defined on the bounded starlike circular domain Ω in several complex variables, which give an unified answer of Fekete-Szegö problem for the kinds of subclasses of starlike mappings defined on Ω. At last, we propose two conjectures related the same problems on the unit ball in a complex Banach space and on the unit polydisk in ℂn.

  1. (Communicated by Stanisława Kanas)

Acknowledgement

The project is supported by the National Natural Science Foundation of China (No. 11671306).

References

[1] Ali, R. M.—Jain, N.K.—Ravichandran, V.: On the radius constants for classes of analytic functions, Bull. Malays. Math. Sci. Soc. 36(1) (2013), 23–38.Suche in Google Scholar

[2] Bieberbach, L.: Über die Koeffizienten der einigen Potenzreihen welche eine schlichte Abbildung des Einheitskreises vermitten, S. B. Preuss. Akad. Wiss, 1916.Suche in Google Scholar

[3] Bhowmik, B.—Ponnusamy, S.—Wirths, K. J.: On the Fekete-Szegö problem for concave univalent functions, J. Math. Anal. Appl. 373 (2011), 432–438.10.1016/j.jmaa.2010.07.054Suche in Google Scholar

[4] Cartan, H.: Sur la possibilité détendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes. In: Lecons sur les Fonctions Univalentes ou Multivalentes (P. Montel, ed.), Gauthier-Villars, Paris, 1933.Suche in Google Scholar

[5] Fekete, M.—Szegö, G.: Eine Bemerkunguber ungerade schlichte Funktionen, J. Lond. Math. Soc. 8 (1933), 85–89.10.1112/jlms/s1-8.2.85Suche in Google Scholar

[6] Graham, I.—Hamada, H.—Kohr, G.: Parametric representation of univalent mappings in several complex variables, Canad. J. Math. 54 (2002), 324–351.10.4153/CJM-2002-011-2Suche in Google Scholar

[7] Graham, I.—Hamada, H.—Honda, T.—Shon, K. H.: Growth, distortion and coefficient bounds for Carathéodory families in ℂn and complex Banach spaces, J. Math. Anal. Appl. 416 (2014), 449–469.10.1016/j.jmaa.2014.02.033Suche in Google Scholar

[8] Gong, S.: The Bieberbach Conjecture, Amer. Math. Soc., International Press, Providence, RI, 1999.10.1090/amsip/012Suche in Google Scholar

[9] Hamada, H.—Kohr, G.—Liczberski, P.: Starlike mappings of order α on the unit ball in complex Banach spaces, Glas. Mat. Ser. 36(3) (2001), 39–48.Suche in Google Scholar

[10] Hamada, H.—Honda, T.—Kohr, G.: Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representationt, J. Math. Anal. Appl. 317 (2006), 302–319.10.1016/j.jmaa.2005.08.002Suche in Google Scholar

[11] Kanas, S.—Darwish, H. E.: Fekete-Szegö problem for starlike and convex functions of complex order, Appl. Math. Lett. 23 (2010), 777–782.10.1016/j.aml.2010.03.008Suche in Google Scholar

[12] Kanas, S.: An unified approach to the Fekete-Szegö problem, Appl. Math. Comput. 218 (2012), 8453–8461.10.1016/j.amc.2012.01.070Suche in Google Scholar

[13] Kohr, G.: On some best bounds for coefficients of subclasses of univalent holomorphic mappings in ℂn. Complex Var. 36 (1998), 261–284.10.1080/17476939808815113Suche in Google Scholar

[14] Koepf, W.: On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc. 101 (1987), 89–95.10.2307/2046556Suche in Google Scholar

[15] Kowalczyk, B.—Lecko, A.: Fekete-Szegö problem for a certain subclass of close-to-convex functions, Bull. Malays. Math. Sci. Soc. 38 (2015), 1393–1410.10.1007/s40840-014-0091-zSuche in Google Scholar

[16] Luo, H.—Xu, Q. H.: On the Fekete and Szegö inequality for the subclass of strongly starlike mappings of order α, Results Math. 72 (2017), 343–357.10.1007/s00025-017-0650-3Suche in Google Scholar

[17] Liu, T. S.—Xu, Q. H.: Fekete and Szegö inequality for a subclass of starlike mappings of order α on the bounded starlike circular domain in ℂn, Acta Math. Sci. 37B (2017), 722–731.10.1016/S0252-9602(17)30033-4Suche in Google Scholar

[18] Liu, T. S.—Ren, G. B.: The growth theorem for starlike mappings on the bounded starlike circular domain, Chin. Ann. Math. 19B (1998), 401–408.Suche in Google Scholar

[19] Liu, X. S.—Liu, T. S.: The estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation in several complex variables, Acta Math. Sin. (Engl. Ser.) 33 (2017), 287–300.10.1007/s10114-016-5226-8Suche in Google Scholar

[20] Liu, H.—Lu, K. P.: Two subclasses of starlike mappings in several complex variables, Chin. Ann. Math. 21B (2000), 533–546.Suche in Google Scholar

[21] Orhan, H.—Deniz, E.—Răducanu, D.: The Fekete-Szegö problem for subclasses of analytic functions defined by a differential operator related to conic domains, Comput. Math. Appl. 59 (2010), 283–295.10.1016/j.camwa.2009.07.049Suche in Google Scholar

[22] Pommerenke, C.: Univalent Functions. In: Studia Mathematica/Mathematische Lehrbucher, Vandenhoeck and Ruprecht, 1975.Suche in Google Scholar

[23] Pfluger, A.: The Fekete-Szegö inequality for complex parameters, Complex Variables Theory Appl. 7 (1986), 149–160.10.1080/17476938608814195Suche in Google Scholar

[24] Raina, R. K.—Sokól, J.: Fekete-szegö problem for some starlike functions related to shell-like curves, Math. Slovaca 66 (2016), 135–140.10.1515/ms-2015-0123Suche in Google Scholar

[25] Srivastava, H. M.—Gaboury, S.—Ghanim, F.: Initial coefficient estimates for some subclasses of m-Fold symmetric bi-univalent functions, Acta Math. Sci. 36B (2016), 863–871.10.1016/S0252-9602(16)30045-5Suche in Google Scholar

[26] Xu, Q. H.—Liu, T. S.: Biholomorphic mappings on bounded starlike circular domains, J. Math. Anal. Appl. 366 (2010), 153–163.10.1016/j.jmaa.2009.12.025Suche in Google Scholar

[27] Xu, Q. H.—Fang, F.—Liu, T. S.: On the Fekete and Szegö problem for starlike mappings of order α, Acta Math. Sin. (Engl. Ser.) 33 (2017), 554–564.10.1007/s10114-016-5762-2Suche in Google Scholar

[28] Xu, Q. H.—Liu, T. S.: On the Fekete and Szegö problem for the class of starlike mappings in several complex variables, Abstr. Appl. Anal. (2014), Art. ID 807026.10.1155/2014/807026Suche in Google Scholar

Received: 2018-05-27
Accepted: 2019-02-22
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Regular papers
  2. On the finite embeddability property for quantum B-algebras
  3. A dual Ramsey theorem for finite ordered oriented graphs
  4. On EMV-Semirings
  5. A nonsymmetrical matrix and its factorizations
  6. On weakly 𝓗-permutable subgroups of finite groups
  7. The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions
  8. A geometrical version of Hardy-Rellich type inequalities
  9. On a Choquet-Stieltjes type integral on intervals
  10. Uniqueness of meromorphic functions sharing four small functions on annuli
  11. A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator
  12. Functions of bounded variation related to domains bounded by conic sections
  13. Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables
  14. Oscillatory criteria for the second order linear ordinary differential equations
  15. When deviation happens between rough statistical convergence and rough weighted statistical convergence
  16. The retraction of certain banach right modules associated to a character
  17. On sequence spaces generated by binomial difference operator of fractional order
  18. Some refinements of young type inequality for positive linear map
  19. On the Gromov-Hausdorff limit of metric spaces
  20. The beta exponentiated Nadarajah-Haghighi distribution: theory, regression model and application
  21. Hilbert algebras with supremum generated by finite chains
Heruntergeladen am 15.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0273/html
Button zum nach oben scrollen