Monatsh Math (2019) 188:653–666
https://doi.org/10.1007/s00605-018-1163-1
The radii of fully starlikeness and fully convexity
of a harmonic operator
Nirupam Ghosh
1
· A. Vasudevarao
2
Received: 3 January 2018 / Accepted: 31 January 2018 / Published online: 8 February 2018
© Springer-Verlag GmbH Austria, part of Springer Nature 2018
Abstract Let f = h + g be a normalized harmonic mapping in the unit disk D :=
{z ∈ C :|z | < 1}. In this paper, we study the radius of fully starlikeness and the radius
of fully convexity of the following harmonic operator
0,1
[ f ]=
z
0
h (ξ)
ξ
d ξ +
z
0
g(ξ)
ξ
d ξ,
where the coefficients of the analytic functions h and g satisfy the conditions of
the harmonic Bieberbach coefficient conjecture. We also study the radius of uniform
starlikeness, and uniform convexity of harmonic mappings.
Keywords Analytic · Univalent · Starlike · Convex · Close-to-convex · Harmonic
functions · Operator
Mathematics Subject Classification Primary 30C45 · 30C50
Communicated by A. Constantin.
B A. Vasudevarao
alluvasudevarao@gmail.com
Nirupam Ghosh
nirupamghoshmath@gmail.com
1
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur,
West Bengal 721 302, India
2
NFA-18, IIT Campus, Indian Institute of Technology Kharagpur, Kharagpur,
West Bengal 721 302, India
123