Generalized Close-to-Convexity Related with Bounded Boundary Rotation

Main Article Content

Khalida Inayat Noor
Muhammad Aslam Noor
Muhammad Uzair Awan


The class Pα,m[A, B] consists of functions p, analytic in the open unit disc E with p(0) = 1 and satisfy

p(z) = (m/4 + ½) p1(z) – (m/4 – 1/2) p2(z), m ≥ 2,

and p1, p2 are subordinate to strongly Janowski function (1+Az/1+Bz)α, α ∈ (0, 1] and −1 ≤ B < A ≤ 1. The class Pα,m[A, B] is used to define Vα,m[A, B] and Tα,m[A, B; 0; B1], B1 ∈ [−1, 0). These classes generalize the concept of bounded boundary rotation and strongly close-to-convexity, respectively. In this paper, we study coefficient bounds, radius problem and several other interesting properties of these functions. Special cases and consequences of main results are also deduced.

Article Details


  1. D. A. Brannan, J. G. Clunie and W. E. Kirwan, On the coefficient problem for the functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Series AI Math. 523(1973), 1-18.
  2. D. A. Brannan, On functions of bounded boundary rotation, Proc. Edinburgh Math. Soc. 16(1969), 339-347.
  3. A. W. Goodman, Univalent Functions, Vol 1, 11, Polygonal Publishing House, Washington, New Jersey, 1983.
  4. A. W. Goodman, on Close-to-Convex Functions of Higher order, Ann. Univ. Sci. Budapest, Evotous Sect. Math. 25(1972), 17-30.
  5. W. K. Hayman, on functions with positive real part, J. Lond. Math. Soc. 36(1961), 34-48.
  6. W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28(1973), 297-326.
  7. S. S. Miller and P. T. Mocanu, Differential Subordination: Theory and Applications, Dekker, New York, 2000.
  8. K. I. Noor, On a generalization of Close-to-Convexity, Int. J. Math. Math. Sci. 6(1983), 327-334.
  9. K. I. Noor, Some properties of certain classes of functions with bounded radius rotation, Honam Math. J. 19(1997), 97-105.
  10. K. I. Noor, B. Malik and S. Mustafa, A survey on functions of bounded boundary and bounded radius rotation, Appl. Math. E-Notes, 12(2012), 136-152.
  11. K. I. Noor, On some univalent integral operations, J. Math. Anal. Appl. 128(1987), 586-592.
  12. K. I. Noor, Higher order close-to-convex functions, Math. Japonica, 37(1992), 1-8.
  13. K. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon Math. 31(1975), 311-323.
  14. R. Parvatham and S. Radha, On certain classes of analytic functions, Ann. Polon Math. 49(1988), 31-34.
  15. B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math. 10(1971), 7-16.
  16. Ch. Pommerenke, Linear-invarient families analytischer funktionen 1, Math. Ann. 155(1964), 108-154.
  17. S. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Polya-Shoenberg conjecture, Comment. Math. Helv. 48(1973), 119-135.