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Averaging operators and the classes of starlike functions related to parabola

Averaging operators and the classes of starlike functions related to parabola An operator I is said to be an averaging (or mean-value) operator on a set $${\mathcal {K}}$$ K of analytic functions in $$\Delta =\{z: |z|<1\}$$ Δ = { z : | z | < 1 } , if $$I[f](0)=f(0)$$ I [ f ] ( 0 ) = f ( 0 ) and $$I[f](\Delta )$$ I [ f ] ( Δ ) is contained in the convex hull of $$f(\Delta )$$ f ( Δ ) for all $$f\in {\mathcal {K}}$$ f ∈ K . In this work we consider the class $$\mathcal {SP}(\alpha )$$ SP ( α ) of functions defined by us (Folia Sci Univ Technol Resov 28:35–42, 1993), which is connected with the class of uniformly convex functions introduced by Goodman (Ann Polon Math 56:87–92, 1991). We describe an interesting new construction of averaging operators which might attract a considerable attention of mathematicians working in the field. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Averaging operators and the classes of starlike functions related to parabola

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00351-5
Publisher site
See Article on Publisher Site

Abstract

An operator I is said to be an averaging (or mean-value) operator on a set $${\mathcal {K}}$$ K of analytic functions in $$\Delta =\{z: |z|<1\}$$ Δ = { z : | z | < 1 } , if $$I[f](0)=f(0)$$ I [ f ] ( 0 ) = f ( 0 ) and $$I[f](\Delta )$$ I [ f ] ( Δ ) is contained in the convex hull of $$f(\Delta )$$ f ( Δ ) for all $$f\in {\mathcal {K}}$$ f ∈ K . In this work we consider the class $$\mathcal {SP}(\alpha )$$ SP ( α ) of functions defined by us (Folia Sci Univ Technol Resov 28:35–42, 1993), which is connected with the class of uniformly convex functions introduced by Goodman (Ann Polon Math 56:87–92, 1991). We describe an interesting new construction of averaging operators which might attract a considerable attention of mathematicians working in the field.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 23, 2019

References