On Certain Families of Analytic Functions in the Hornich Space

  • Md Firoz Ali
  • A. Vasudevarao


Let \((H,\oplus ,\odot )\) denote the Hornich space consisting of all locally univalent and analytic functions f on the unit disk \({\mathbb {D}}:=\{z\in {\mathbb {C}}:\,|z| <1\}\) with \(f(0)=0= f'(0)-1\) for which \(\arg f'\) is bounded in \({\mathbb {D}}\). For \(f,g\in H\) and \(r,s\in \mathbb {R}\), we consider the integral operator \(I_{r,s}(z):= \int _{0}^{z} (f'(\xi ))^r (g'(\xi ))^s\,\mathrm{d}\xi \) and determine all values of r and s for which the operator \((f,g)\mapsto I_{r,s}\) maps a specified subclass of H into another specified subclass of H. We also determine the set of extreme points for different subclasses of H with respect to the Hornich space structure. Using the extreme points, we develop a new approach to obtain the pre-Schwarzian norm estimate for different subclasses of H. We also consider a larger space \({\widetilde{H}}\), whose linear structure is same as that of H and study the same problems as stated above for some subclasses of \({\widetilde{H}}\).


Univalent Starlike Convex Close-to-convex Spiral-like functions Extreme point Pre-Schwarzian norm Banach space Hornich space 

Mathematics Subject Classification

Primary 30C45 30C55 



The first author thanks University Grants Commission for financial support through a UGC-SRF Fellowship. The second author thanks SERB, Govt of India for support.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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