Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 682413, 12 pages
http://dx.doi.org/10.1155/2013/682413
Research Article
On the Largest Disc Mapped by Sum of Convex and
Starlike Functions
Rosihan M. Ali,
1
Naveen Kumar Jain,
2
and V. Ravichandran
2
1
School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia
2
Department of Mathematics, University of Delhi, Delhi 110007, India
Correspondence should be addressed to Rosihan M. Ali; rosihan@cs.usm.my
Received 5 July 2013; Accepted 17 October 2013
Academic Editor: Ferhan M. Atici
Copyright © 2013 Rosihan M. Ali et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For a normalized analytic function defned on the unit disc D, let (,
,
; ) be a function of positive real part in D,
(,
,
; ) need not have that property in D, and =+. For certain choices of and , a sharp radius constant is
determined, 0<<1, so that ()/ maps D onto a specifed region in the right half-plane.
1. Introduction
Let A be the class of functions analytic in D = { ∈
C : || < 1} and normalized by (0) = 0 =
(0) − 1.
Let S be its subclass consisting of univalent functions. For
two analytic functions and , the function is subordi-
nate to , written () ≺ (), if there is an analytic self-
map : D → D with (0) = 0 satisfying () = (()).
Given an analytic function with (0) = 1 and Re () >
0 in D, denote by ST() and CV() the subclasses of
A consisting, respectively, of satisfying
()/() ≺
() and 1 +
()/
() ≺ ().
For various choices of , these classes reduce to well-
known subclasses of starlike and convex functions. For
instance, with () = (1 + (1 − 2))/(1 − ), 0≤<
1, then ST() and CV() are, respectively, the subclasses
consisting of starlike functions of order and convex func-
tions of order . Te classes ST := ST(0) and CV :=
CV(0) are the familiar subclasses of S of starlike and con-
vex functions. For () = (1 + (1 − 2))/(1 − ), >1,
M() = ST() is the class of functions ∈ A satisfying
M () := { ∈ A : Re (
()
()
) < ( ∈ D)} (1)
studied by Uralegaddi et al. [1]. Various subclasses
of M() have been investigated in [2–5]. For () =
((1 + )/(1 − ))
, 0<≤1, the class SST() := ST() is
the class of strongly starlike functions of order . Te
class S
L
:= ST(
√
1 + ) introduced by Sok´ oł and
Stankiewicz [6] consists of functions ∈ A satisfying
(
()
()
)
2
−1
<1 ( ∈ D). (2)
Tus, a function ∈ A is in the class S
L
if
()/() lies in the region bounded by the right-half of
the lemniscate of Bernoulli given by |
2
−1| < 1. Results
related to the class S
L
can be found in [3, 7–9].
In investigating the class UCV of uniformly convex
functions, Rønning [10] introduced a class S
P
of parabolic
starlike functions. Tese are functions ∈ A satisfying
Re (
()
()
)>
()
()
−1
( ∈ D). (3)
It is important to keep in mind that the qualifer
“parabolic” refers to the geometry of the image of D under
the map
()/(); that is, the domain necessarily lies
in a parabolic region of the -plane. It does not convey
the interpretation that the function maps the disk D onto
a parabolic region. Tis terminology of parabolic starlike
functions is however widely accepted and used by authors. Ali