Research Article
A Subclass of Analytic Functions Related to -Uniformly
Convex and Starlike Functions
Saqib Hussain,
1
Akhter Rasheed,
2
Muhammad Asad Zaighum,
2
and Maslina Darus
3
1
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan
2
Department of Mathematics & Statistics, Riphah International University, Islamabad, Pakistan
3
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi, Selangor, Malaysia
Correspondence should be addressed to Saqib Hussain; saqib_math@yahoo.com
Received 26 January 2017; Accepted 20 April 2017; Published 23 May 2017
Academic Editor: Maria Alessandra Ragusa
Copyright © 2017 Saqib Hussain 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.
We investigate some subclasses of -uniformly convex and -uniformly starlike functions in open unit disc, which is generalization
of class of convex and starlike functions. Some coefcient inequalities, a distortion theorem, the radii of close-to-convexity, and
starlikeness and convexity for these classes of functions are studied. Te behavior of these classes under a certain modifed
convolution operator is also discussed.
1. Introduction
Let A be the class of all analytic functions in open unit disc
Δ={:||<1}, normalized by (0) = 0 and
(0) = 1.
Tus, any ∈ A has the following Maclaurin’s series:
()=+
∞
∑
=2
. (1)
A function is said to be univalent if it never takes same
value twice. By S we mean the subclass of A which is com-
posed of univalent functions. By ST and CV we mean the
well-known subclasses of A that are, respectively, starlike and
convex.
In 1991, Goodman [1, 2] introduced the classes UCV and
UST of uniformly convex and uniformly starlike functions,
respectively. A function ∈ is uniformly convex if ()
maps every circular arc contained in Δ with center ∈Δ
onto a convex arc. Te function ∈ is uniformly starlike
if () maps every circular arc contained in Δ with center
∈Δ onto a starlike arc with respect to (). A more useful
representation of UCV and UST was given in [3–6] as
∈ UCV ⇐⇒ ∈ A,
Re (
(
())
()
)>
()
()
,
∈Δ.
∈ UST ⇐⇒ ∈ A,
Re (
()
()
)>
()
()
−1
,
∈Δ.
(2)
In 1999, for ≥0, Kanas and Wisniowska [7] introduced
the class − UCV and − UST as
∈− UCV ⇐⇒
∈− UST ⇐⇒ ∈ A,
Re (
(
())
()
)>
()
()
,
∈Δ.
(3)
Observe that 0− UCV ≡ CV, 0− UST ≡ UST and
1− UCV ≡ UCV, 1− UST ≡ UST.
For fxed ≥0, these classes have a nice geometrical
representation; for detail see [7–9].
Hindawi
Journal of Function Spaces
Volume 2017, Article ID 9010964, 7 pages
https://doi.org/10.1155/2017/9010964