Research Article
Some Applications of Ordinary Differential Operator to
Certain Multivalent Functions
Müfit Fan,
1
Hüseyin Irmak,
1
and Ayhan Ferbetçi
2
1
Department of Mathematics, Faculty of Science, C ¸ ankırı Karatekin University, 18100 C ¸ ankırı, Turkey
2
Department of Mathematics, Faculty of Science, Ankara University, 06100 Ankara, Turkey
Correspondence should be addressed to H¨ useyin Irmak; hisimya@yahoo.com
Received 1 July 2015; Accepted 11 August 2015
Academic Editor: Viliam Makis
Copyright © 2015 M¨ uft S ¸an 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.
Te aim of this paper is to apply the well-known ordinary diferential operator to certain multivalent functions which are analytic
in the certain domains of the complex plane and then to determine some criteria concerning analytic and geometric properties of
the related complex functions.
1. Introduction, Notations, and Definitions
Let A(;) denote the class of functions () of the following
form:
()=
+
+1
+1
+
+2
+2
+⋅⋅⋅
(
∈ C;∈ Z ={±1,±2,±3,...}; ∈ N
={1,2,3,...}, Z
−
= Z − N),
(1)
which are analytic and multivalent in the domain
Δ=
{
{
{
U, when ∈ N
D, when ∈ Z
−
,
(2)
where C is the set of complex numbers. As is known, the
domains U and D are known as unit open disk and punctured
open unit disk, respectively. Also let M() := A(−;) and
T() := A(;) when ∈ N.
By diferentiating both sides of the function () in the
form (1), -times with respect to complex variable , one can
easily derive the following (ordinary) diferential operator:
()
() =
{
{
{
{
{
{
{
{
{
{
{
!
( − )!
−
+
∞
∑
=+1
!
( − )!
−
, if ∈ T ()
(+−1)!
( − 1)!
(−1)
−−
+
∞
∑
=+1
!
( − )!
−
, if ∈ M (),
(3)
where ≥, ∈ N, and ∈ N
0
:= N ∪ {0}.
In this investigation, by applying the diferential operator,
defned by (3), to certain analytic functions which are
multivalent in U or meromorphic multivalent in D, several
criteria, which also include both analytic and geometric
properties of univalent functions (see [1, 2]), for functions
() in the classes M() and T(), are then determined. In
the literature, by using certain operators, several researchers
obtained some results concerning functions belonging to
the general class A(;). In this paper, we also determined
many results which include starlikeness, convexity, close-to-
convexity, and close-to-starlikeness of analytic functions in
Hindawi Publishing Corporation
Journal of Mathematics
Volume 2015, Article ID 825903, 4 pages
http://dx.doi.org/10.1155/2015/825903