Alpha Convex Functions Associated with Conic Domains

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Khalida Inayat Noor, Nasir Khan, Krzysztof Piejko


In this paper we define a new class k -Umα [A,B] of Janowski type k-uniformly alpha convex functions. We use the method of differential subordinations theory to obtain some new results like sufficient condition, inclusion relations, coefficient estimate and covering properties. The results presented here include a number of well-known results as their special cases.

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  1. D. A. Brannan., W. E. Kirwan., On some classes of bounded univalent functions, J. London. Math. Soc. 1(2)(1969), 431-443.
  2. P. L. Duren., Univalent functions Grundlehren der Math. Wissenchanften, Springer-Verlag, New York-Berlin (1983).
  3. W. Janowski., Some external problem for certain families of analytic functions, I. Ann. Polon. Math. 28(1973), 298-326.
  4. S. Kanas, A. Wi ´sniowska., Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105(1999), 327-336.
  5. S. Kanas, A. Wi ´sniowska., Conic domains and starlike functions, Rev. Roumaine Math. Pure. Appl. 45(2000), 647-657.
  6. S. Kanas., Coefficient estimates in subclasses of the Caratheodory class related to conical domains, Acta. Math. Univ. Comenian. 74(2)(2005), 149-161.
  7. S. Kanas., Alternative characterization of the class k - UCV and related classes of univalent functions, Serdica Math. J. 25(1999), 341-350.
  8. S. Kanas., Techniques of the differential subordination for domains bounded by conic sections, Int. J. Math. Math. Sci. 38(2003), 2389-2400.
  9. E. Lindelöf., Mémorie sur certaines inequalitiés dans la théorie des fonctions monogènes et sur quelques propiétés nouvelles de ces fonctions dans le voisinage d'un point singulier essentiel, Acta Soc. Sci. Fenn. 35(1908) 1-35.
  10. J. E. Littelwood., On inequalities in the theory of functions, Proc. London. Math. Soc. 23(1925), 481-519.
  11. M. S. Liu., On certain subclass of analytic functions; J. South. China Normal Univ (in chinese). 4(2002), 15-20.
  12. S. S. Miller., P. T. Mocanu., Differential subordination and univalent functions, Michigan. Math. J. 28(2)(1981), 157-172.
  13. P. T. Mocanu., Une propriete de convexite generlise dans la theorie de la representation conforme, Mathematica (Cluj). 11(1969), 127-133.
  14. S. Nawaz., Certain subclasses of analytic functions associated with conic domains. Ph.D Thesis(2012), Comsats Institute of Information Technology Islamabad, Pakistan.
  15. K. I. Noor, S. N. Malik., On coefficient inequalities of functions associated with conic domains, Comput. Math. Appl. 62(2011), 2209-2217.
  16. K. I. Noor, S. N. Malik., M. Arif, M. Raza., On bounded boundary and bounded radius rotation related with Janowski function, World. Appl. Sci. J. 12(6) (2011), 895-902.
  17. K. I. Noor, M. Arif, W. Ul-Haq., On k-uniformly close-to-convex functions of complex order, Appl. Math. Comput. 215(2)(2009), 629-635.
  18. M. Nunokawa, J. Sokol., On order of strongly starlikeness in the class of uniformly convex functions, Math. Nachr. 288(2015), 1003C1008.
  19. W. Rogosinski., On the coefficients of subordinate functions, Proc. Lond. Math. Soc. 48(2)(1943), 48-82.
  20. W. Rogosinski., On subordinate functions, Proc. Camb. Phil. Soc. 35(1939), 1-26.
  21. M. A. Rosihan., V. Ravichandran., Convolution and Differential subordination for mulitivalent functions, Bull. Malays. Math. Sci. 32(3) (2009), 351-360.
  22. S. Shams, S. R. Kulkarni, J. M. Jahangiri., Classes of uniformly starlike and convex functions, Int. J. Math. Math. Sci. 55(2004), 2959-2961.
  23. H. Silverman., Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51(1975), 109-116.
  24. J. Stankiewicz., Quelques probl”˜emes extr”˜emaux dans les classes des fonctions α-angulairement ”˜etoil”˜ees, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 20(1966), 59-75.