Research Article
Convexity of Certain -Integral Operators of -Valent Functions
K. A. Selvakumaran,
1
S. D. Purohit,
2
Aydin Secer,
3
and Mustafa Bayram
3
1
Department of Mathematics, RMK College of Engineering and Technology, Puduvoyal, Tamil Nadu 601206, India
2
Department of Basic Sciences (Mathematics), College of Technology and Engineering, M. P. University of Agriculture and Technology,
Udaipur, Rajasthan 313001, India
3
Department of Mathematical Engineering, Yildiz Technical University, Davutpasa, 34210 Istanbul, Turkey
Correspondence should be addressed to Aydin Secer; asecer@yildiz.edu.tr
Received 1 March 2014; Accepted 3 May 2014; Published 12 May 2014
Academic Editor: Guotao Wang
Copyright © 2014 K. A. Selvakumaran et al. his 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.
By applying the concept (and theory) of fractional -calculus, we irst deine and introduce two new -integral operators for certain
analytic functions deined in the unit disc U. Convexity properties of these -integral operators on some classes of analytic functions
deined by a linear multiplier fractional -diferintegral operator are studied. Special cases of the main results are also mentioned.
1. Introduction and Preliminaries
he subject of fractional calculus has gained noticeable
importance and popularity due to its established applica-
tions in many ields of science and engineering during the
past three decades or so. Much of the theory of fractional
calculus is based upon the familiar Riemann-Liouville frac-
tional derivative (or integral). he fractional -calculus is
the extension of the ordinary fractional calculus in the -
theory. Recently, there was a signiicant increase of activity in
the area of the -calculus due to applications of the -calculus
in mathematics, statistics, and physics. For more details, one
may refer to the books [1–4] on the subject. Recently, Purohit
and Raina [5–7] have added one more dimension to this study
by introducing certain subclasses of functions which are
analytic in the open disk U, by using fractional -calculus.
Purohit [8] also studied similar work and considered new
classes of multivalently analytic functions in the open unit
disk.
he aim of this paper is to consider a linear multiplier
fractional -diferintegral operator and to deine certain new
subclasses of functions which are -valent and analytic in
the open unit disk. he results derived include convexity
properties of these -integral operators on some classes of
analytic functions. Special cases of the main results are also
mentioned.
Let A
denote the class of functions () of the form
()=
+
∞
∑
=+1
, (∈ N ={1,2,3,...}),
(1)
which are analytic and -valent in the open unit disk U =
{∈ C :||<1}. A function ∈ A
is said to be -valently
starlike of order (0≤<) if and only if
R {
()
()
}>, (∈ U). (2)
We denote by S
∗
() the class of all such functions. On the
other hand, a function ∈ A
is said to be in the class C
()
of -valently convex of order (0≤<) if and only if
R {1+
()
()
}>, (∈ U). (3)
Note that S
∗
(0)= S
∗
and C
(0)=
are, respectively, the
classes of -valently starlike and -valently convex functions
in U. Also, we note that S
∗
1
(0) = S
∗
and C
1
(0) = C are,
respectively, the usual classes of starlike and convex functions
in U. A function ∈ A
is said to be in the class US
(,)
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 925902, 7 pages
http://dx.doi.org/10.1155/2014/925902