Generalized uniformly close-to-convex functions of order ? and type ?

Generalized uniformly close-to-convex functions of order ? and type ?

In this paper, a class of analytic functions f defined on the open unit disc satisfying Re z(Dn,??f(z))0Dn,??g(z)> ?z(Dn,??f(z))0Dn,??g(z)- 1+ ?, is studied, where ? >= 0, -1 = 0. and g is a certain analytic function associated with conic domains. Among other results, inclusion relations and the coefficients bound are studied. Various known special cases of these results are pointed out. A subclass of uniformly quasi-convex functions is also studied.

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  • M. Acu, On a subclass of n-uniformly close to convex functions, General Mathematics, 14(1) (2006), 55-64.
  • M. Acu and D. Blezu, Bounds of the coefficients for uniformly close-to-convex functions, Libertas Matematica XXII (2002), 81-86.
  • R. Aghalary and GH. Azadi, The Dziok-Srivastava operator and k-uniformly starlike func- tions, J. Inequal. Pure Appl. Math. 6(2) (2005), 1-7, Article 52 (electronic).
  • F.M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci. 27 (2004), 1429-1436.
  • F.M. Al-Oboudi and K.A. Al-Amoudi On classes of analytic functions related to conic domains, J. Math. Anal. Appl. 339(1) (2008), 655-667.
  • F.M. Al-Oboudi and K.A. Al-Amoudi, Subordination results for classes of analytic functions related to conic domains defined by a fractional operator, J. Math. Anal. Appl. 354(2) (2009), 412-420.
  • R. Bharti, R. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28(1) (1997), 17-32.
  • D. Blezu, On the n-uniformly close to convex functions with respect to a convex domain, General Mathematics 9(3-4) (2001), 3-10.
  • A.W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92.
  • S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, II, Folia Sci. Univ. Tehn. Resov. 170 (1998), 65-78.
  • S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, Comput. Appl. Math. 105 (1999), 327-336.
  • W. Kaplan, Close-to-convex Schlicht functions, Mich. Math. J. 15 (1968), 277-282.
  • S. Kumar and C. Ramesha, Subordination properties of uniformly convex and uniformly close to convex functions, J. Ramanujan Math. Soc. 9(2) (1994), 203-214.
  • S.S. Milller and PT. Mocanu, General second order inequalities in the complex plane. "Babes-Bolya" Univ. Fac. of Math. Research Seminars, Seminar on Geometric Function Theory, 4 (1982), 96-114.
  • K.I. Noor and D.K. Thomas, Quasi-convex univalent functions, Int. J. Math. & Math. Sci. 3 (1980), 255-266.
  • K.I. Noor, M. Arif, and W. Ul-Haq, On k-uniformly close-to-convex functions of complex order, Applied Mathematics and Computation, 215 (2009), 629-635.
  • S. Owa, On the distortion theorems, I, Kyungpook Math. J. 18(1) (1978), 53-59.
  • S. Owa and H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39(5) (1987), 1057-1077.
  • W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943), 48-82.
  • F. Rİnning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 45(14) (1991), 117-122.
  • F. Rİnning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118(1) (1993), 189-196.
  • St. Ruscheweyh, Convolutions in Geometric Function Theory, Sem. Math. Sup., vol. 83, Presses Univ. de Montreal, 1982.
  • G.S. Salagean, Subclasses of univalent functions, in: Complex Analysis - Fifth Romanian- Finish Seminar, Part 1, Bucharest, 1981, in: Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983, pp. 362-372.
  • H. Silverman and D.N. Telage, Extreme points of subclasses of close-to-convex functions, Proc. Amer. Math. Soc. 74 (1979), 59-65.
  • H.M. Srivastava and A.K. Mishra, Applications of fractional calculus to parabolic starlike and uniformly convex functions, J. Comput. Math. Appl. 39(3/4) (2000), 57-69.
  • H.M. Srivastava, Shu-Hai Li and Huo Tang, Certain classes of K-uniformly close-to-convex functions and other related functions defined by using the Dziok-Srivastava operator, Bull. Math. Anal. 1(3) (2009), 49-63.
  • K.G. Subramanian, T.V. Sudharsan and H. Silverman, On uniformly close-to-convex func- tions and uniformly quasi-convex functions, IJMMS. 48 (2003), 3053-3058.