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FEKETE-SZEGÖ PROBLEM FOR CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS

  • VASUDEVARAO, ALLU (Department of Mathematics Indian Institute of Technology Khargpur)
  • Received : 2014.08.23
  • Published : 2015.11.30

Abstract

For $1{\leq}{\alpha}<2$, let $\mathcal{F}({\alpha})$ denote the class of locally univalent normalized analytic functions $f(z)=z+{\Sigma}_{n=2}^{\infty}{a_nz^n}$ in the unit disk ${\mathbb{D}}=\{z{\in}{\mathbb{C}}:{\left|z\right|}<1\}$ satisfying the condition $Re\(1+{\frac{zf^{{\prime}{\prime}}(z)}{f^{\prime}(z)}}\)>{\frac{{\alpha}}{2}}-1$. In the present paper, we shall obtain the sharp upper bound for Fekete-$Szeg{\ddot{o}}$ functional $|a_3-{\lambda}a_2^2|$ for the complex parameter ${\lambda}$.

Keywords

univalent functions;starlike;convex;close-to-convex and Fekete-$Szeg{\ddot{o}}$ problem

References

  1. H. R. Abdel-Gawad and D. K. Thomas, The Fekete-Szego problem for strongly closeto- convex functions, Proc. Amer. Math. Soc. 114 (1992), no. 2, 345-349.
  2. B. Bhowmik, S. Ponnusamy, and K.-J Wirths, On the Fekete-Szego problem for concave univalent functions, J. Math. Anal. Appl. 373 (2011), no. 2, 432-438. https://doi.org/10.1016/j.jmaa.2010.07.054
  3. J. H. Choi, Y. C. Kim, and T. Sugawa, A general approach to the Fekete-Szego problem, J. Math. Soc. Japan. 59 (2007), no. 3, 707-727. https://doi.org/10.2969/jmsj/05930707
  4. P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften 259, New York, Berlin, Heidelberg, Tokyo, Springer-Verlag, 1983.
  5. M. Fekete and G. Szego, Eine Bemerkung uber ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85-89.
  6. J. A. Jenkins, A general coefficient theorem, Trans. Amer. Math. Soc. 77 (1954), 262- 280. https://doi.org/10.1090/S0002-9947-1954-0064146-9
  7. J. A. Jenkins, On Certain Coefficients of Univalent Funcions, Analytic Functions, Princeton University press, Princeton, N.J., 1960.
  8. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), no. 1, 8-12. https://doi.org/10.1090/S0002-9939-1969-0232926-9
  9. Y. C. Kim and T. Sugawa, Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions, Proc. Edinb. Math. Soc. 49 (2006), no. 1, 131-143. https://doi.org/10.1017/S0013091504000306
  10. W. Koepf, On the Fekete-Szego problem for close-to-convex functions, Proc. Amer. Math. Soc. 101 (1987), no. 1, 89-95.
  11. W. Koepf, On the Fekete-Szego problem for close-to-convex functions II, Arch. Math. (Basel) 49 (1987), no. 5, 420-433. https://doi.org/10.1007/BF01194100
  12. R. R. London, Fekete-Szego inequalities for close-to-convex functions, Proc. Amer. Math. Soc. 117 (1993), no. 4, 947-950.
  13. W. Ma and D. Minda, An internal geometric characterization of strongly starlike functions, Ann. Univ. Mariae Curie-Sk lodowska, Sect. A 45 (1991), 89-97.
  14. W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis, pp. 157-169, (eds. Z. Li, F. Ren, L. Yang and S. Zhang), International Press Inc., 1992.
  15. W. Ma and D. Minda, Coefficient inequalities for strongly close-to-convex functions, J. Math. Anal. Appl. 205 (1997), no. 2, 537-553. https://doi.org/10.1006/jmaa.1997.5234
  16. Y. A. Muhanna, L. Li, and S. Ponnusamy, Extremal problems on the class of convex functions of order -$\frac{1}{2}$, Arch. Math. (Basel) 103 (2014), no. 6, 461-471.
  17. Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. https://doi.org/10.1090/S0002-9904-1949-09241-8
  18. A. Pfluger, The Fekete-Szego inequality by a variational method, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 447-454. https://doi.org/10.5186/aasfm.1985.1049
  19. A. Pfluger, The Fekete-Szego inequality for complex parameters, Complex Variables Theory Appl. 7 (1986), no. 1-3, 149-160. https://doi.org/10.1080/17476938608814195
  20. S. Ponnusamy and S. Rajasekaran, New sufficient conditions for starlike and univalent functions, Soochow J. Math. 21 (1995), no. 2, 193-201.
  21. S. Ponnusamy and V. Singh, Univalence of certain integral transforms, Glas. Mat. Ser. III (51) 31 (1996), no. 2, 253-261.
  22. S. Ponnusamy and A. Vasudevarao, Region of variability of two subclasses of univalent functions, J. Math. Anal. Appl. 332 (2007), no. 2, 1323-1334. https://doi.org/10.1016/j.jmaa.2006.11.019
  23. A. Vasudevarao, An arclength problem for some subclasses of univalent functions, J. Analysis (2014) to appear.

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