Advertisement

Complex Analysis and Operator Theory

, Volume 11, Issue 7, pp 1639–1649 | Cite as

Radius Problems for Some Subclasses of Analytic Functions

  • R. Kargar
  • A. Ebadian
  • J. SokółEmail author
Article

Abstract

In this work, we define the class \({\mathcal {M}}(\alpha )\) of normalized analytic functions which satisfy the following two-sided inequality:
$$\begin{aligned} 1+\frac{\alpha -\pi }{2\sin \alpha }< {{\mathfrak {R}}}{{\mathfrak {e}}}\left\{ \frac{zf'(z)}{f(z)}\right\}<1+ \frac{\alpha }{2\sin \alpha } \quad |z|<1, \end{aligned}$$
where \(\pi /2\le \alpha <\pi \). We obtain a sufficient condition for functions to be in the class \({\mathcal {M}}(\alpha )\) and solve several radius problems related to other well-known function classes.

Keywords

Analytic Univalent Starlike function Subordination Radius problems 

Mathematics Subject Classification

30C45 

References

  1. 1.
    Dorff, M.: Convolutions of planar harmonic convex mappings. Complex Var. Theory Appl. 45(3), 263–271 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Duren, P.L.: Univalent Functions, Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)Google Scholar
  3. 3.
    Dziok, J., Raina, R.K., Sokół, J.: On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers. Math. Comp. Model 57, 1203–1211 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eenigenburg, P.J., Miller, S.S., Mocanu, P.T., Reade, O.M.: Second order differential inequalities in the complex plane. J. Math. Anal. Appl. 65, 289–305 (1978)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rønning, F.: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 118, 189–196 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sokół, J.: On some subclass of strongly starlike functions. Demonstr. Math. XXXI(1), 81–86 (1998)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Sokół, J., Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike function. Folia Sci. Univ. Tech. Resoviensis 147, 101–105 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran
  2. 2.Young Researchers and Elite Club, Urmia BranchIslamic Azad UniversityUrmiaIran
  3. 3.University of Rzeszów, Faculty of Mathematics and Natural SciencesRzeszówPoland

Personalised recommendations