Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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SOME EXTENSION RESULTS CONCERNING ANALYTIC AND MEROMORPHIC MULTIVALENT FUNCTIONS

  • Received : 2018.07.19
  • Accepted : 2018.10.29
  • Published : 2019.07.31
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Abstract

Let $\mathscr{B}^{{\eta},{\mu}}_{p,n}\;({\alpha});\;({\eta},{\mu}{\in}{\mathbb{R}},\;n,\;p{\in}{\mathbb{N}})$ denote all functions f class in the unit disk ${\mathbb{U}}$ as $f(z)=z^p+\sum_{k=n+p}^{\infty}a_kz^k$ which satisfy: $$\|\[{\frac{f^{\prime}(z)}{pz^{p-1}}}\]^{\eta}\;\[\frac{z^p}{f(z)}\]^{\mu}-1\| <1-{\frac{\alpha}{p}};\;(z{\in}{\mathbb{U}},\;0{\leq}{\alpha}<p)$$. And $\mathscr{M}^{{\eta},{\mu}}_{p,n}\;({\alpha})$ indicates all meromorphic functions h in the punctured unit disk $\mathbb{U}^*$ as $h(z)=z^{-p}+\sum_{k=n-p}^{\infty}b_kz^k$ which satisfy: $$\|\[{\frac{h^{\prime}(z)}{-pz^{-p-1}}}\]^{\eta}\;\[\frac{1}{z^ph(z)}\]^{\mu}-1\|<1-{\frac{\alpha}{p}};\;(z{\in}{\mathbb{U}},\;0{\leq}{\alpha}<p)$$. In this paper several sufficient conditions for some classes of functions are investigated. The authors apply Jack's Lemma, to obtain this conditions. Furthermore, sufficient conditions for strongly starlike and convex p-valent functions of order ${\gamma}$ and type ${\beta}$, are also considered.

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References

  1. R. M. Ali and V. Ravichandran, Integral operators on Ma-Minda type starlike and convex functions, Math. Comput. Modelling 53 (2011), no. 5-6, 581-586. https://doi.org/10.1016/j.mcm.2010.09.007
  2. D. Breaz and H. Guney, The integral operator on the classes $S^{*}_{a}(b)\;and\;C_{\alpha}(b)$, J. Math. Inequal. 2 (2008), no. 1, 97-100. https://doi.org/10.7153/jmi-02-09
  3. N. E. Cho and S. Owa, Suffcient conditions for meromorphic starlikeness and close-to-convexity of order ${\alpha}$, IJMMS 26 (2001), no. 5, 317-319.
  4. B. A. Frasin, Family of analytic functions of complex order, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 22 (2006), no. 2, 179-191.
  5. B. A. Frasin, New general integral operator, Comput. Math. Appl. 62 (2011), no. 11, 4272-4276. https://doi.org/10.1016/j.camwa.2011.10.016
  6. B. A. Frasin and M. Darus, On certain analytic univalent functions, Int. J. Math. Math. Sci. 25 (2001), no. 5, 305-310. https://doi.org/10.1155/S0161171201004781
  7. B. A. Frasin and J. M. Jahangiri, A new and comprehensive class of analytic functions, An. Univ. Oradea Fasc. Mat. 15 (2008), 59-62.
  8. M. D. Ganigi and B. A. Uralegaddi, Subclasses of meromorphic close-to-convex functions, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 33(81) (1989), no. 2, 105-109.
  9. A. W. Goodman, On the Schwarz-Christoffel transformation and p-valent functions, Trans. Amer. Math. Soc. 68 (1950), 204-223. https://doi.org/10.2307/1990442
  10. P. Goswami, T. Bulboaca, and S. K. Bansal, Some relations between certain classes of analytic functions, J. Class. Anal. 1 (2012), no. 2, 157-173. https://doi.org/10.7153/jca-01-16
  11. S. P. Goyal and J. K. Prajapat, A new class of meromorphic multivalent functions involving certain linear operator, Tamsui Oxf. J. Math. Sci. 25 (2009), no. 2, 167-176.
  12. H. Irmak, Certain inequalities and their applications to multivalently analytic functions, Math. Inequal. Appl. 8 (2005), no. 3, 451-458. https://doi.org/10.7153/mia-08-41
  13. H. Irmak and F. Cetin, Some theorems involving inequalities on p-valent functions, Turkish J. Math. 23 (1999), no. 3, 453-459.
  14. I. S. Jack, Functions starlike and convex of order ${\alpha}$, J. London Math. Soc. (2) 3 (1971), 469-474. https://doi.org/10.1112/jlms/s2-3.3.469
  15. V. Kumar and S. L. Shukla, Certain integrals for classes of p-valent meromorphic functions, Bull. Austral. Math. Soc. 25 (1982), no. 1, 85-97. https://doi.org/10.1017/S0004972700005062
  16. J. Liu, The Noor integral and strongly starlike functions, J. Math. Anal. Appl. 261 (2001), no. 2, 441-447. https://doi.org/10.1006/jmaa.2001.7489
  17. S. S. Miller and P. T. Mocanu, Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker, Inc., New York, 2000.
  18. P. T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 1-2, 117-124.
  19. S. Owa, H. E. Darwish, and M. K. Aouf, Meromorphic multivalent functions with positive and fixed second coeffcients, Math. Japon. 46 (1997), no. 2, 231-236.
  20. S. Owa, M. Nunokawa, and H. M. Srivastava, A certain class of multivalent functions, Appl. Math. Lett. 10 (1997), no. 2, 7-10. https://doi.org/10.1016/S0893-9659(97)00002-5
  21. S. Owa, M. Nunokawa, H. Saitoh, and H. M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett. 15 (2002), no. 1, 63-69. https://doi.org/10.1016/S0893-9659(01)00094-5
  22. S. Ponnusamy, Polya-Schoenberg conjecture for Caratheodory functions, J. London Math. Soc. (2) 51 (1995), no. 1, 93-104. https://doi.org/10.1112/jlms/51.1.93
  23. S. Ponnusamy and S. Rajasekaran, New suffcient conditions for starlike and univalent functions, Soochow J. Math. 21 (1995), no. 2, 193-201.
  24. S. Ponnusamy and V. Singh, Criteria for strongly starlike functions, Complex Variables Theory Appl. 34 (1997), no. 3, 267-291. https://doi.org/10.1080/17476939708815053
  25. M. San and H. Irmak, Ordinary differential operator and some of its applications to certain meromorphically p-valent functions, Appl. Math. Comput. 218 (2011), no. 3, 817-821. https://doi.org/10.1016/j.amc.2011.02.037
  26. O. Singh, P. Goswami, and B. Frasin, Suffcient conditions for certain subclasses of meromorphic p-valent functions, Bol. Soc. Parana. Mat. (3) 33 (2015), no. 2, 9-16. https://doi.org/10.5269/bspm.v33i2.21919
  27. R. Singh and S. Singh, Some suffcient conditions for univalence and starlikeness, Colloq. Math. 47 (1982), no. 2, 309-314 (1983). https://doi.org/10.4064/cm-47-2-309-314
  28. H. M. Srivastava, H. M. Hossen, and M. K. Aouf, A unified presentation of some classes of meromorphically multivalent functions, Comput. Math. Appl. 38 (1999), no. 11-12, 63-70. https://doi.org/10.1016/S0898-1221(99)00285-0