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A Note on Successive Coefficients of Convex Functions

A Note on Successive Coefficients of Convex Functions In this note, we investigate the supremum and the infimum of the functional $$|a_{n+1}|-|a_{n}|$$ | a n + 1 | - | a n | for functions, convex and analytic on the unit disk, of the form $$f(z)=z+a_2z^2+a_3z^3+\cdots .$$ f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ . We also consider the related problem of maximizing the functional $$|a_{n+1}-a_{n}|$$ | a n + 1 - a n | for convex functions f with $$f''(0)=p$$ f ′ ′ ( 0 ) = p for a prescribed $$p\in [0,2].$$ p ∈ [ 0 , 2 ] . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Note on Successive Coefficients of Convex Functions

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-016-0177-8
Publisher site
See Article on Publisher Site

Abstract

In this note, we investigate the supremum and the infimum of the functional $$|a_{n+1}|-|a_{n}|$$ | a n + 1 | - | a n | for functions, convex and analytic on the unit disk, of the form $$f(z)=z+a_2z^2+a_3z^3+\cdots .$$ f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ . We also consider the related problem of maximizing the functional $$|a_{n+1}-a_{n}|$$ | a n + 1 - a n | for convex functions f with $$f''(0)=p$$ f ′ ′ ( 0 ) = p for a prescribed $$p\in [0,2].$$ p ∈ [ 0 , 2 ] .

Journal

Computational Methods and Function TheorySpringer Journals

Published: Aug 2, 2016

References