The second Hankel determinant for alpha-convex functions

The second Hankel determinant for alpha-convex functions DOI 10.1007/s10986-018-9397-0 Lithuanian Mathematical Journal a b Janusz Sokól and Derek K. Thomas Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom (e-mail: jsokol@ur.edu.pl; d.k.thomas@swansea.ac.uk) Received March 5, 2018; revised May 16, 2018 Abstract. Let S denote the class of analytic univalent functions in D := {z ∈ C: |z| < 1} normalized so that f(z)= n ∗ z + a z .Let C and S be the subclasses of S consisting of convex and starlike functions, respectively. For real α, n=2 the class M of alpha-convex functions f ∈S defined by zf (z) zf (z) Re (1 − α) + α +1 > 0,z ∈ D, f(z) f (z) ∗ 2 is well known, so that M = C and M = S . We give bounds for the second Hankel determinant H (2) = |a a − a | 1 0 2 2 4 when f ∈M and α  0, thus extending the well-known results in the cases α =0 and α =1. We also give bounds for a wider class of functions. MSC: 30C45, 30C55 Keywords: http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Lithuanian Mathematical Journal Springer Journals

The second Hankel determinant for alpha-convex functions

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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Mathematics, general; Ordinary Differential Equations; Actuarial Sciences; Number Theory; Probability Theory and Stochastic Processes
ISSN
0363-1672
eISSN
1573-8825
D.O.I.
10.1007/s10986-018-9397-0
Publisher site
See Article on Publisher Site

Abstract

DOI 10.1007/s10986-018-9397-0 Lithuanian Mathematical Journal a b Janusz Sokól and Derek K. Thomas Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom (e-mail: jsokol@ur.edu.pl; d.k.thomas@swansea.ac.uk) Received March 5, 2018; revised May 16, 2018 Abstract. Let S denote the class of analytic univalent functions in D := {z ∈ C: |z| < 1} normalized so that f(z)= n ∗ z + a z .Let C and S be the subclasses of S consisting of convex and starlike functions, respectively. For real α, n=2 the class M of alpha-convex functions f ∈S defined by zf (z) zf (z) Re (1 − α) + α +1 > 0,z ∈ D, f(z) f (z) ∗ 2 is well known, so that M = C and M = S . We give bounds for the second Hankel determinant H (2) = |a a − a | 1 0 2 2 4 when f ∈M and α  0, thus extending the well-known results in the cases α =0 and α =1. We also give bounds for a wider class of functions. MSC: 30C45, 30C55 Keywords:

Journal

Lithuanian Mathematical JournalSpringer Journals

Published: Jun 2, 2018

References

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