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Notes on Convex Functions of Order $$\alpha $$ α

Notes on Convex Functions of Order $$\alpha $$ α Marx and Strohhäcker showed in 1933 that f(z) / z is subordinate to $$1/(1-z)$$ 1 / ( 1 - z ) for a normalized convex function f on the unit disk $$|z|<1.$$ | z | < 1 . In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to $$k_\alpha (z)/z$$ k α ( z ) / z if f is convex of order $$\alpha $$ α for $$1/2\le \alpha <1$$ 1 / 2 ≤ α < 1 and conjectured that this is true also for $$0<\alpha <1/2.$$ 0 < α < 1 / 2 . Here, $$k_\alpha $$ k α is the standard extremal function in the class of normalized convex functions of order $$\alpha $$ α and $$k_0(z)=z/(1-z).$$ k 0 ( z ) = z / ( 1 - z ) . We prove the conjecture and study geometric properties of convex functions of order $$\alpha .$$ α . In particular, we prove that $$(f+g)/2$$ ( f + g ) / 2 is starlike whenever both f and g are convex of order 3 / 5. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Notes on Convex Functions of Order $$\alpha $$ α

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References (12)

Publisher
Springer Journals
Copyright
Copyright © 2015 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/s40315-015-0122-2
Publisher site
See Article on Publisher Site

Abstract

Marx and Strohhäcker showed in 1933 that f(z) / z is subordinate to $$1/(1-z)$$ 1 / ( 1 - z ) for a normalized convex function f on the unit disk $$|z|<1.$$ | z | < 1 . In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to $$k_\alpha (z)/z$$ k α ( z ) / z if f is convex of order $$\alpha $$ α for $$1/2\le \alpha <1$$ 1 / 2 ≤ α < 1 and conjectured that this is true also for $$0<\alpha <1/2.$$ 0 < α < 1 / 2 . Here, $$k_\alpha $$ k α is the standard extremal function in the class of normalized convex functions of order $$\alpha $$ α and $$k_0(z)=z/(1-z).$$ k 0 ( z ) = z / ( 1 - z ) . We prove the conjecture and study geometric properties of convex functions of order $$\alpha .$$ α . In particular, we prove that $$(f+g)/2$$ ( f + g ) / 2 is starlike whenever both f and g are convex of order 3 / 5.

Journal

Computational Methods and Function Theory – Springer Journals

Published: May 30, 2015

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