Research Article
Certain Geometric Properties of Normalized Wright Functions
Mohsan Raza,
1
Muhey U Din,
1
and Sarfraz Nawaz Malik
2
1
Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan
2
Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan
Correspondence should be addressed to Sarfraz Nawaz Malik; snmalik110@yahoo.com
Received 15 October 2016; Accepted 4 December 2016
Academic Editor: Adrian Petrusel
Copyright © 2016 Mohsan Raza 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.
In this article, we fnd some geometric properties like starlikeness, convexity of order , close-to-convexity of order (1+)/2,
and close-to-convexity of normalized Wright functions with respect to the certain functions. Te sufcient conditions for the
normalized Wright functions belonging to the classes T
() and L
() are the part of our investigations. We also obtain the
conditions on normalized Wright function to belong to the Hardy space H
.
1. Introduction and Preliminaries
Let H denote the class of all analytic functions in the open
unit disk U ={:||<1} and H
∞
denote the space of all
bounded functions on H. Tis is Banach algebra with respect
to the norm
∞
= sup
∈U
()
.
(1)
We denote H
, 0<<∞, for the space of all functions
∈ H such that ||
admits a harmonic majorant. H
is
a Banach space if the norm of is defned to be th root of
the least harmonic majorant of ||
for some fxed ∈ U.
Another equivalent defnition of norm is given as follows: let
∈ H, and set
(,)
=
{
{
{
{
{
(
1
2
∫
2
0
(
)
)
1/
, 0<<∞,
max {
()
:||≤}, =∞.
(2)
Ten the function ∈ H
if
(,) is bounded for all ∈
[0,1). It is clear that
H
∞
⊂ H
⊂ H
, 0<<<∞. (3)
For some details, see [1]. It is also known [1] that Re{
()}>
0 in U, and then
∈ H
, <1,
∈ H
/(1−)
, 0<<1.
(4)
Let A be the class of functions of the form
()=+
∞
∑
=2
, (5)
analytic in the open unit disc U ={:||<1}, and S denote
the class of all functions in A which are univalent in U. Let
S
∗
(), C(), and K() denote the classes of starlike, convex,
and close-to-convex functions of order , respectively, and
they are defned as
S
∗
()={:∈ A, Re (
()
()
)>,∈ U,
∈[0,1)},
C ()={:∈ A, Re (1+
()
()
)>,
∈ U,∈[0,1)},
Hindawi Publishing Corporation
Journal of Function Spaces
Volume 2016, Article ID 1896154, 8 pages
http://dx.doi.org/10.1155/2016/1896154