Research Article
The Pre-Schwarzian Norm Estimate for
Analytic Concave Functions
Young Jae Sim and Oh Sang Kwon
Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea
Correspondence should be addressed to Oh Sang Kwon; oskwon@ks.ac.kr
Received 6 February 2015; Revised 27 March 2015; Accepted 29 March 2015
Academic Editor: Harvinder S. Sidhu
Copyright © 2015 Y. J. Sim and O. S. Kwon. 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.
Let D denote the open unit disk and let S denote the class of normalized univalent functions which are analytic in D. Let Co() be
the class of concave functions ∈ S, which have the condition that the opening angle of (D) at infnity is less than or equal to
, ∈(1,2]. In this paper, we fnd a sufcient condition for the Gaussian hypergeometric functions to be in the class Co(). And
we defne a class Co(,,), (−1 ≤ < ≤ 1), which is a subclass of Co() and we fnd the set of variabilities for the functional
(1−||
2
)(
()/
()) for ∈ Co(,,). Tis gives sharp upper and lower estimates for the pre-Schwarzian norm of functions
in Co(,,). We also give a characterization for functions in Co(,,) in terms of Hadamard product.
1. Introduction
Let H denote the class of functions analytic in the unit disk
D ={∈ C : || < 1}. We denote the class of locally univalent
functions by LU. Let A denote the class of functions ∈ H
with normalization (0) =
(0) − 1 = 0 and let S be the
class of functions in A that are univalent in D. Also we defne
the subclass K ⊂ S of convex functions whenever (D) is a
convex domain.
A function : D → C is said to belong to the family
Co() if satisfes the following conditions:
(i) is analytic in D with the standard normalization
(0) =
(0)−1 = 0. In addition, it satisfes (1) = ∞.
(ii) maps D conformally onto a set whose complement
with respect to C is convex.
(iii) Te opening angle of (D) at ∞ is less than or equal
to , ∈(1,2].
Te class Co() is referred to as the class of concave
univalent functions. We note that for ∈ Co(), ∈ (1, 2],
the closed set C \(D) is convex and unbounded. We observe
that Co(2) contains the classes Co(), ∈(1,2].
Avkhadiev and Wirths [1] found the analytic characteri-
zation for functions in Co(), ∈ (1, 2]: ∈ Co() if and
only if
Re {
2
−1
{
(+1)
2
1+
1−
−1−
()
()
}}>0, ∈ D.
(1)
For ∈ LU, the pre-Schwarzian derivative
is defned by
=
/
and we defne the norm of
by
= sup
∈D
(1−||
2
)
()
.
(2)
It is well known that ‖
‖≤6 for ∈ S and ‖
‖≤4 for ∈
K. In [2], Bhowmik et al. obtained the estimate of the pre-
Schwarzian norm for functions ∈ Co() as the following:
4≤
≤2+2, ∈ Co (). (3)
For more investigation of concave functions, we may refer to
[3–7].
We say that is subordinate to in D, written as ≺, if
and only if () = (()) for some Schwarz functions (),
(0) = 0, and |()| < 1, ∈ D. If () is univalent in D,
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2015, Article ID 814805, 6 pages
http://dx.doi.org/10.1155/2015/814805