Abstract
By means of a differential operator, we introduce and investigate some new subclasses of 
-valently analytic functions with negative coefficients, which are starlike or convex of complex order. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.
Similar content being viewed by others
1. Introduction
Let 
denote the class of functions of the following form:
which are analytic and multivalent in the open unit dics
Let
denote the 
th-order ordinary differential operator for a function 
, that is,
where 
; 
Next, we define the differential operator 
as
In view of (1.3), it is clear that
If we take 
and 
for 
, then 
become the differential operator defined by S
l
gean [1].
Finally, in terms of a differential operator
defined by (1.3) above, let
denote the subclass of
consisting of functions
which satisfy the following inequality:
where 
, 
, 
, 
; 
; 
For 
, 
, and 
, we define the next subclasses of 
where 
; 
; 
; 
Remark 1.1.
(
) 
was studied by Nasr and Aouf [2] (also see Bulboac
et al. [3]).
(
) 
and 
, 
were introduced by Srivastava et al. [4].
(
) 
and 
, 
were introduced by Silverman [5].
(
) 
and 
were introduced by Parvathan and Ponnusanny [6, pages 163-164].
(
) For 
and 
, the classes 
, 
, and 
are closely related with 
, 
, and 
which are defined by Owa and Sălăgean in [7].
In this paper we give relationships between the classes of 
, 
, and 
In the particular case when 
and 
, 
, and 
, we obtain the same results as in [8].
2. Main Results
Our main results are contained in
Theorem 2.1.
Let 
, 
and let 
; then
(1)
(2)
(3)if 
, then
(4)if 
, then 
;
(5)if 
, then 
Proof.
(
) Let 
We prove that
If 
has the series expansion
then
We use the fact that 
for 
and
these imply
for 
From (2.4) and (2.5), we deduce
By using the definition of 
from this last inequality we, obtain (2.2) which implies
hence 
(
) Let 
be in 
Then (2.7) holds and, by using (2.3), this is equivalent to
For 
if 
, from (2.8) we obtain
which is equivalent to
Then multiplying the relation last inequality with 
we obtain 
(
) if 
is a real positive number, then the definitions of 
and 
are equivalent, hence 
By using (
) and (
) from this theorem, we obtain (
).
(
) We have the following two cases.
Case 1.
Let
be defined by
and let 
We have
or
and then
(see the definition of 
).
Let now
Then, by a simple computation and by using the fact that
we obtain
where
, 
, and
For 
we, have 
where 
is the disc with the center
and the radius
We have 
where 
and we deduce that 
for all 
does not hold.
We have obtained that for 
but 
and in this case 
Case 2.
We consider the function
defined by (2.11) for 
In this case, the inequality (2.13) holds too and this implies that 
We also obtain that
like in Case 1.
(
) Let 
be given by (2.11), where
and
Then
which implies that
We have
where 
is given by (2.17).
From 
where 
and
are given by (2.18) and (2.19), we obtain
If
and
, then
and if
then (2.24) also holds. By combining (2.24) with (2.23) and the definition of 
, we obtain that
References
Sălăgean GŠ: Subclasses of univalent functions. In Complex Analysis—Fifth Romanian-Finnish Seminar Part 1 (Bucharest, 1981), 1983, Lecture Notes in Mathematics. Volume 1013. Springer; 362–372.
Nasr MA, Aouf MK: Starlike function of complex order. The Journal of Natural Sciences and Mathematics 1985, 25(1):1–12.
Bulboacă T, Nasr MA, Sălăgean GŞ: Function with negative coefficients
-starlike of complex order. Universitatis Babeş-Bolyai. Studia Mathematica 1991, 36(2):7–12.Srivastava HM, Owa S, Chatterjea SK: A note on certain classes of starlike functions. Rendiconti del Seminario Matematico della Università di Padova 1987, 77: 115–124.
Silverman H: Univalent functions with negative coefficients. Proceedings of the American Mathematical Society 1975, 51: 109–116. 10.1090/S0002-9939-1975-0369678-0
R. Parvathan and S. Ponnusanny, Eds., “Open problems,” World Scientific, 1990, Conference on New Trends in Geometric Function Theory and Applications.
Owa S, Sălăgean GS: Starlike or convex of complex order functions with negative coefficients. Sūrikaisekikenkyūsho Kōkyūroku 1998, (1062):77–83.
Owa S, Sălăgean GS: On an open problem of S. Owa. Journal of Mathematical Analysis and Applications 1998, 218(2):453–457. 10.1006/jmaa.1997.5808
-starlike of complex order. Universitatis Babeş-Bolyai. Studia Mathematica 1991, 36(2):7–12.Author information
Authors and Affiliations
Corresponding author
Appendix
In this paper, we discuss the class 
of analytic functions with negative coefficients. Let us consider the functions 
given by
. For such a function 
, we say that 
if it satisfies
with 
.If we define the function 
for 
by
is analytic in 
, 
, and 
. Thus 
is the Carathéodory function. Since the extremal function for the Carathéodory function 
is given by
defined by the above is the extremal function for the class 
. But our class 
is defined with analytic functions 
with negative coefficients. Thus we do not know how we can consider the extremal function for this class.Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Uyanik, N., Deniz, E., Kadioǧlu, E. et al. On Certain Multivalent Starlike or Convex Functions with Negative Coefficients. J Inequal Appl 2010, 751721 (2010). https://doi.org/10.1155/2010/751721
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/751721