Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3775–3806 | Cite as

Slice Starlike Functions Over Quaternions

  • Zhenghua XuEmail author
  • Guangbin Ren
Article

Abstract

In this paper, we initiate the study of the geometric function theory for slice starlike functions over quaternions and its subclasses. This allows us to answer negatively some questions about the Bieberbach conjecture, the growth, distortion, and covering theorems for slice regular functions. Precisely, we find that the Bieberbach conjecture holds true for slice starlike functions in contrast to the fact that the Bieberbach conjecture fails for biholomorphic starlike mappings in higher dimensions. We also establish some sharp versions of the growth, distortion, and covering theorems for quaternions.

Keywords

Starlike function Quaternion Bieberbach conjecture Fekete–Szegö inequality Growth and distortion theorems Bloch–Landau theorem 

Mathematics Subject Classification

Primary 30G35 Secondary 30C50 

Notes

Funding

This work was supported by the NNSF of China (11371337).

References

  1. 1.
    Abu-Muhanna, Y., Ali, R.M., Ponnusamy, S.: On the Bohr inequality. In: Govil, N.K., Mohapatra, R., Qazi, M.A., Schmeisser, G. (eds.) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol. 117, pp. 269–300. Springer, Cham (2017)CrossRefGoogle Scholar
  2. 2.
    Aizenberg, L., Elin, M., Shoikhet, D.: On the Rogosinski radius for holomorphic mappings and some of its applications. Stud. Math. 168(2), 147–158 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arcozzi, N., Sarfatti, G.: Invariant metrics for the quaternionic Hardy space. J. Geom. Anal. 25(3), 2028–2059 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bayart, F., Pellegrino, D., Seoane-Sepúlveda, J.B.: The Bohr radius of the n-dimensional polydisk is equivalent to \(\sqrt{(logn)/n}\). Adv. Math. 264, 726–746 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bisi, C., Stoppato, C.: The Schwarz-Pick lemma for slice regular functions. Indiana Univ. Math. J. 61, 297–317 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cartan, H.: Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalents. In: Montel, P. (ed.) Leçons sur les Fonctions Univalents ou Mutivalents, pp. 129–155. Gauthier-Villars, Paris (1933)Google Scholar
  7. 7.
    Colombo, F., Sabadini, I., Struppa, D.C.: Slice monogenic functions. Israel J. Math. 171, 385–403 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus. Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, vol. 289. Birkhäuser/Springer, Basel (2011)Google Scholar
  9. 9.
    de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Defant, A., Maestre, M., Schwarting, U.: Bohr radii of vector valued holomorphic functions. Adv. Math. 231(5), 2837–2857 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Della Rocchetta, C., Gentili, G., Sarfatti, G.: The Bohr theorem for slice regular functions. Math. Nachr. 285(17–18), 2093–2105 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Della Rocchetta, C., Gentili, G., Sarfatti, G.: A Bloch-Landau theorem for slice regular functions. In: Advances in Hypercomplex Analysis. Springer INdAM Series, vol. 1, pp. 55–74. Springer, Milan (2013)Google Scholar
  13. 13.
    Duren, P.L.: Univalent Functions, vol. 259. Springer, New York (1983)zbMATHGoogle Scholar
  14. 14.
    Fekete, M., Szegö, G.: Eine Bemerkung über Ungerade Schlichte Funktionen. J. Lond. Math. Soc. 8, 85–89 (1933)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gal, S.G., González-Cervantes, J.O., Sabadini, I.: Univalence results for slice regular functions of a quaternion variable. Complex Var. Elliptic Equ. 60(10), 1346–1365 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gal, S.G., González-Cervantes, J.O., Sabadini, I.: On some geometric properties of slice regular functions of a quaternion variable. Complex Var. Elliptic Equ. 60(10), 1431–1455 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gentili, G., Stoppato, C.: The zero sets of slice regular functions and the open mapping theorem. In: Hypercomplex Analysis and Applications. Trends in Mathematics, pp. 95–107. Birkhäuser/Springer, Basel (2011)Google Scholar
  18. 18.
    Gentili, G., Struppa, D.C.: A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris 342(10), 741–744 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gentili, G., Struppa, D.C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gentili, G., Struppa, D.C.: Regular functions on the space of Cayley numbers. Rocky Mt. J. Math. 40, 225–241 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gentili, G., Sarfatti, G.: Landau-Toeplitz theorems for slice regular functions over quaternions. Pac. J. Math. 265(2), 381–404 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gentili, G., Stoppato, C., Struppa, D.C.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer, Berlin (2013)Google Scholar
  23. 23.
    Ghiloni, R., Perotti, A.: Slice regular functions on real alternative algebras. Adv. Math. 226(2), 1662–1691 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gong, S.: The Bieberbach Conjecture. AMS/IP Studies in Advanced Mathematics, vol. 12. American Mathematical Society, Providence, RI (1999)Google Scholar
  25. 25.
    Gong, S.: Convex and Starlike Mappings in Several Complex Variables. Mathematics and its Applications, vol. 435. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  26. 26.
    Goodman, A.W.: Univalent functions and nonanalytic curves. Proc. Am. Math. Soc. 8, 598–601 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gori, A., Vlacci, F.: Starlikeness for functions of a hypercomplex variable. Proc. Am. Math. Soc. 145(2), 791–804 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 255. Marcel Dekker Inc, New York (2003)Google Scholar
  29. 29.
    Graham, I., Varolin, D.: Bloch constants in one and several variables. Pacific J. Math. 174(2), 347–357 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hamada, H., Honda, T., Kohr, G.: Bohr’s theorem for holomorphic mappings with values in homogeneous balls. Israel J. Math. 173, 177–187 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hayman, W.K.: The asymptotic behaviour of p-valent functions. Proc. Lond. Math. Soc. (3) 5, 257–284 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kaplan, W.: Close-to-convex schlicht functions. Michigan Math. J. 1, 169–185 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Koepf, W.: On the Fekete-Szegö problem for close-to-convex functions. Proc. Am. Math. Soc. 101(1), 89–95 (1987)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kythe, P.K.: Complex analysis. Conformal Inequalities and the Bieberbach Conjecture. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL (2016)Google Scholar
  35. 35.
    Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liu, X., Liu, T., Xu, Q.: A proof of a weak version of the Bieberbach conjecture in several complex variables. Sci. China Math. 58(12), 2531–2540 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liu, T., Ren, G.: Growth theorem of convex mappings on bounded convex circular domains. Sci. China Ser. A 41, 123–130 (1998)Google Scholar
  38. 38.
    Reade, M.O.: On close-to-convex univalent functions. Michigan Math. J. 3, 59–62 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Ren, G., Wang, X.: Carathéodory theorems for slice regular functions. Complex Anal. Oper. Theory 9(5), 1229–1243 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ren, G., Wang, X.: Julia theory for slice regular functions. Trans. Am. Math. Soc. 369(2), 861–885 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ren, G., Wang, X.: The growth and distortion theorems for slice monogenic functions. Pacific J. Math. 290(1), 169–198 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ren, G., Wang, X., Xu, Z.: Slice regular functions on regular quadratic cones of real alternative algebras. Trends in Mathematics, pp. 227–245. Birkhäuser, Basel (2016)Google Scholar
  43. 43.
    Robertson, M.S.: On the theory of univalent functions. Ann. Math. (2) 37(2), 374–408 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rogosinski, W.: Über Bildschranken bei Potenzreihen und ihren Abschnitten. Math. Z. 17, 260–276 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Roper, K.A., Suffridge, T.J.: Convexity properties of holomorphic mappings in \({{\bf C}}^n\). Trans. Am. Math. Soc. 351(5), 1803–1833 (1999)CrossRefzbMATHGoogle Scholar
  46. 46.
    Stoppato, C.: Regular Möbius transformations of the space of quaternions. Ann. Glob. Anal. Geom. 39(4), 387–401 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Stoppato, C.: A new series expansion for slice regular functions. Adv. Math. 231(3–4), 1401–1416 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Xu, Z., Wang, X.: On two Bloch type theorems for quaternionic slice regular functions. arXiv:1601.02338 (2016)

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.School of MathematicsHeFei University of TechnologyHefeiChina
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations