Starlike cases of the generalized goodman conjecture

详细信息    查看全文

• 作者：F. G. Avkhadiev (1)
  K. -J. Wirths (2)
  
• 关键词：Starlike meromorphic function ; subordination
• 刊名：Lobachevskii Journal of Mathematics
• 出版年：2013
• 出版时间：April 2013
• 年：2013
• 卷：34
• 期：2
• 页码：142-147
• 全文大小：514KB
• 参考文献：1. F. G. Avkhadiev and K.-J. Wirths, Monatsh. Math. g class="a-plus-plus">147g>, 103 (2006). f="http://dx.doi.org/10.1007/s00605-005-0334-z">CrossRef
  2. F. G. Avkhadiev and K.-J. Wirths, Complex Var. Elliptic Equ. g class="a-plus-plus">52g>, 299 (2007). f="http://dx.doi.org/10.1080/17476930600863325">CrossRef
  3. F. G. Avkhadiev and K.-J. Wirths, / Schwarz-Pick Type Inequalities (Frontiers in Mathematics, Birkh?user Verlag, Basel, Boston, Berlin, 2009). f="http://dx.doi.org/10.1007/978-3-0346-0000-2">CrossRef
  4. B. Bhowmik, S. Ponnusamy, and K.-J. Wirths, Cubo g class="a-plus-plus">12g>, 15 (2010). f="http://dx.doi.org/10.4067/S0719-06462010000100003">CrossRef
  5. C. Carathéodory, / Theory of functions of a complex variable (Chelsea Publ. Comp., New York, 1960).
  6. L. de Branges, Acta Math. g class="a-plus-plus">154g>, 137 (1985). f="http://dx.doi.org/10.1007/BF02392821">CrossRef
  7. G. M. Goluzin, Mat. Sb. g class="a-plus-plus">3g>(45), 321 (1938) (in Russian)
  8. A.W. Goodman, Trans. Amer. Math. Soc. g class="a-plus-plus">81g>, 92 (1956). f="http://dx.doi.org/10.1090/S0002-9947-1956-0075299-2">CrossRef
  9. J. A. Jenkins, Michigan Math. J. g class="a-plus-plus">9g>, 25 (1962). f="http://dx.doi.org/10.1307/mmj/1028998616">CrossRef
  10. S. S. Miller and P. T. Mocanu, / Differential Subordinations, Theory and Applications (Pure and Applied Mathematics, Marcel Dekker, New York, Basel, 2000).
  11. J. Miller, Proc. Amer. Math. Soc. g class="a-plus-plus">31g>, 446 (1972). f="http://dx.doi.org/10.1090/S0002-9939-1972-0288267-7">CrossRef
  12. Ch. Pommerenke, / Univalent Functions (Vandenhoeck and Ruprecht, G?ttingen, 1975).
  13. M. S. Robertson, Bull. Amer. Math. Soc. g class="a-plus-plus">76g>, 1 (1970). f="http://dx.doi.org/10.1090/S0002-9904-1970-12356-4">CrossRef
  14. G. Schober, / Univalent Functions -Selected Topics, Lecture Notes in Mathematics 478 (Springer-Verlag, Berlin, Heidelberg, New York, 1975). f="http://dx.doi.org/10.1007/BFb0077279">CrossRef
  15. K.-J. Wirths, Ann. Polon. Math. g class="a-plus-plus">83g>(1), 87 (2004). f="http://dx.doi.org/10.4064/ap83-1-10">CrossRef
  16. C. Yuh Lin, Proc. Amer. Math. Soc. g class="a-plus-plus">103g>, 517 (1988). f="http://dx.doi.org/10.2307/2047172">CrossRef
  
• 作者单位：F. G. Avkhadiev (1)
  K. -J. Wirths (2)
  
  1. Depart. of Mech. and Math., Kazan Federal University, 420008, Kazan, Russia
  2. Institut für Analysis and Algebra, TU Braunschweig, 38106, Braunschweig, Germany
  
• ISSN：1818-9962

文摘

We consider functions f that are meromorphic and univalent in the unit disc $\mathbb{D}$ with a simple pole at the point p ?(0, 1) and normalized by f(0) = f-0) ?1 = 0. A function g is called subordinated under such a function f, if there exists a function ω holomorphic in $\mathbb{D}$ , ω( $\mathbb{D}$ ) ⊿ , such that g(z) = f(zω(z)), z ? $\mathbb{D}$ , and we use the abbreviation g ?f to indicate this relationship between two functions. We conjectured that for g ?f, the inequalities $|a_n (g)| \leqslant \frac{1} {{p^{n - 1} }}\sum\limits_{k = 0}^{n - 1} {p^{2k} } ,n \in \mathbb{N}, $ are valid. Here f is as above and the expansion $g(z)\sum\limits_{n = 1}^\infty {a_n (g)z^n } $ is valid in some neighbourhod of the origin. In the present article, we prove that this is true for two classes of functions f for which \f( $\mathbb{D}$ ) is starlike.