Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 1887–1903 | Cite as

Certain Ma–Minda type classes of analytic functions associated with the crescent-shaped region

  • Poonam SharmaEmail author
  • Ravinder Krishna Raina
  • Janusz Sokół


In this paper, we study certain classes of analytic functions which satisfy a Ma–Minda type subordination condition and are associated with the crescent-shaped region. We first give the extremal functions of these function classes and related to them, we present various characteristic properties. It is shown that for certain range of coefficients, a bilinear transformation belongs to a certain class (defined below). Also, for this and related classes, we present certain results and give some specific examples. Further, we obtain results on coefficient inequalities and coefficient estimates. Finally, using the subordination theory, we derive various results and corollaries for functions belonging to the classes studied in this paper.


Analytic functions Convex functions Starlike functions Subordination 

Mathematics Subject Classification

Primary 30C45 30C50 30C80 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no competing interests.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Poonam Sharma
    • 1
    Email author
  • Ravinder Krishna Raina
    • 2
    • 3
  • Janusz Sokół
    • 4
  1. 1.Department of Mathematics and AstronomyUniversity of LucknowLucknowIndia
  2. 2.M.P. University of Agriculture and TechnologyUdaipurIndia
  3. 3.UdaipurIndia
  4. 4.Faculty of Mathematics and Natural SciencesUniversity of RzeszówRzeszówPoland

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