Subclasse of starlike and convex functions associated to domain bounded by a nephroid

Document Type : Promotional Paper

Author

Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

One of the most important and interesting topics in the geometric function theory is the classes of starlike and convex functions of Ma-Minda type on the unit disk $ \mathbb{D} = \left \{z \in \mathbb {C} \colon | z | <1 \right \}$ which are defined by subordination. Let $ \mathcal {A} $ be the class of functions $f$, analytic in the unit disc $\mathbb{D}=\{z:|z|<1\}$ on the complex plane $ \mathbb {C}$, with the normalization $f(0)=f'(0)-1=0$, and $ \mathcal {ST} _N (s)$ and $ \mathcal {CV} _N (s) $ be a family of starlike and convex functions $ f \in \mathcal {A} $ so that for each $ z \in \mathbb {D} $, such that the quantity $zf'(z)/f(z)$ or $1+zf''(z)/f'(z)$ respectively are lying in the region bounded by the Nephroid \[\left [(u-1)^ 2 + v ^ 2-4s ^ 2 \right] ^ 3 = 108 s ^4v ^ 2, \quad 0 \] In this paper, some properties and characteristics of classes $ \mathcal {ST} _N (s) $ and $ \mathcal {CV} _N (s) $ defined by the Ma-Minda type, like the structure of functions in these classes, we study extreme functions, growth theorem, distoration, and rotation theorem.

Keywords

References

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Mathematics and Society
Volume 6, Issue 4 - Serial Number 4
December 2021
Pages 71-87

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History

  • Receive Date: 18 June 2022
  • Revise Date: 05 September 2022
  • Accept Date: 06 September 2022
  • Publish Date: 20 February 2022

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