Research A rticle
Toeplitz Matrices Whose Elements Are the Coefficients of
Functions with Bounded Boundary Rotation
V. Radhika,
1
S. Sivasubramanian,
2
G. Murugusundaramoorthy,
3
and Jay M. Jahangiri
4
1
Department of Mathematics, Easwari Engineering College, Ramapuram, Chennai 600089, India
2
Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam 604001, India
3
Departmen t of Ma thema tics, Sc hool of Advanced Sciences, VIT U n iversity, Ve llor e 632 014, India
4
Department of Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA
Correspondence should be addressed to Jay M. Jahangiri; jjahangi@kent.edu
Received 3 July 2016; Accepted 3 August 2016
Academic Editor: Yan Xu
Copyright © 2016 V. Radhika et al. is 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.
Let R denote the family of functions ()=+
∑
∞
𝑛=2
𝑛
𝑛
of bounded boundary rotation so that (
())>0in the open unit
disk U ={:||<1}. We obtain sharp bounds for Toeplitz determinants whose elements are the coecients of functions ∈R.
1. Introduction
Let A denote the class of all functions of the form
(
)
=+
∞
𝑛=2
𝑛
𝑛
,
(1)
which are analytic in the open unit disk U ={:||<1}
and let S denote the subclass of A consisting of univalent
functions. Obviously, for functions ∈S,wemusthave
=
0in U.For∈S we consider the family R of functions of
bounded boundary rotation so that (
())>0in U.e
family R is properly contained in the class of close-to-convex
functions (e.g., see Brannan [1], Pinchuk [2], or Duren [3] pp.
269–271.)
Toeplitz matrices are one of the well-studied classes
of structured matrices. ey arise in all branches of pure
and applied mathematics, statistics and probability, image
processing, quantum mechanics, queueing networks, signal
processing, and time series analysis, to name a few (e.g.,
seeYeandLim[4]).Toeplitzmatriceshavesomeofthe
most attractive computational properties and are amenable
to a wide range of disparate algorithms and determinant
computations. Here we consider the symmetric Toeplitz
determinant
𝑞
(
)
=
𝑛
𝑛+1
⋅⋅⋅
𝑛+𝑞−1
𝑛+1
⋅⋅⋅
.
.
.
.
.
.
𝑛+𝑞−1
⋅⋅⋅
𝑛
(2)
and obtain sharp bounds for the coecient body |
𝑞
()|; =
2,3; =1,2,3,wheretheentriesof
𝑞
()are the coecients
of functions of form (1) that are in the family R of functions
of bounded boundary rotation. As far as we are concerned,
the results presented here are new and noble and the only
prior compatible result is published by omas and Halim
[5] for the classes of starlike and close-to-convex functions.
It is worth noticing that the bounds presented here are much
ner than those presented in [5].
2. Main Results
We note that, for the functions of form (1) that are in the
family R of functions of bounded boundary rotation, we can
Hindawi Publishing Corporation
Journal of Complex Analysis
Volume 2016, Article ID 4960704, 4 pages
http://dx.doi.org/10.1155/2016/4960704
2 Journal of Complex Analysis
write
()=(),where∈P,theclassofpositivereal
part function satisfying (())>0for ∈U and is of the
form
(
)
=1+
∞
𝑛=1
𝑛
𝑛
.
(3)
We shall state the following result [6], to prove our main
theorems.
Lemma 1. Let () = 1+
∑
∞
𝑛=1
𝑛
𝑛
∈ P.enforsome
complex valued with || ≤ 1and some complex valued
with ||≤1we have
2
2
=
2
1
+4−
2
1
4
3
=
3
1
+24−
2
1
1
−
1
4−
2
1
2
+24−
2
1
1−
|
|
2
.
(4)
In our rst theorem we obtain a sharp bound for the
coecient body |
2
(2)|.
eorem 2. Let ∈R be given by (1). en we have the sharp
bound
2
(
2
)
=
2
3
−
2
2
≤
5
9
.
(5)
Proo f. Firstnotethatbyequatingthecorrespondingcoe-
cients of
()=(),weobtain
2
=
1
2
(6)
3
=
2
3
(7)
4
=
3
4
.
(8)
Now by (2), (6), and (7) we have
2
(
2
)
=
2
3
3
2
=
2
2
−
2
3
=
2
1
4
−
2
2
9
. (9)
Making use of Lemma 1 to express
2
in terms of
1
,weobtain
2
3
−
2
2
=
2
1
4
−
2
1
+4−
2
1
2
36
.
(10)
Without loss of generality, let 0≤
1
=≤2. Applying
triangle inequality, we get
2
3
−
2
2
≤
1
36
4−
2
2
|
|
2
+2
2
4−
2
|
|
+
2
9−
2
š Φ
(
;
|
|
)
.
(11)
Dierentiating Φ(;||)with respect to we obtain
Φ
(
;
|
|
)
=
18
2
|
|
2
−2
|
|
−1
2
−8
|
|
2
−8
|
|
−9.
(12)
Setting Φ(;||)/=0leads to either =0or
2
=
9+8
|
|
−8
|
|
2
1+2
|
|
−
|
|
2
.
(13)
But (9+8||−8||
2
)/(1+2||−||
2
)>4for ||≤1. erefore
the maximum of |
2
3
−
2
2
|occurs either at =0or =2.
For =0we obtain
2
=0and
3
=2/3which implies
|
2
3
−
2
2
|=|(2/3)
2
|≤4/9.
For =2we obtain
2
=1and
3
=2/3which implies
|
2
3
−
2
2
|=|1−(2/3)
2
|≤5/9.
is bound is sharp and the extremal function is given by
()=(1+)/(1−).
We remark that the sharp bound |
2
3
−
2
2
|≤5/9given by
eorem 2 is much ner than |
2
3
−
2
2
|≤5that was obtained
byomasandHalim[5]fortheclassoffunctionsofform
(1) that are close-to-convex in U.
Next, we determine a sharp bound for the coecient body
|
2
(3)|.
eorem 3. Let ∈R be given by (1). en we have the sharp
bound
2
(
3
)
=
2
4
−
2
3
≤
4
9
.
(14)
Proo f. Notethat,by(2),(7),and(8),wehave
2
(
3
)
=
3
4
4
3
=
2
4
−
2
3
=
2
3
16
−
2
2
9
. (15)
Making use of Lemma 1 and triangle inequality, we obtain
2
4
−
2
3
≤
2
2
256
+
2
64
−
2
64
|
|
4
+
2
2
64
−
2
32
|
|
3
+
2
2
64
−
2
288
+
4
128
−
3
64
+
2
64
+
2
18
⋅
|
|
2
+
4
64
+
2
32
|
|
+
3
64
+
2
64
+
4
36
−
6
256
fl Φ
(
,
|
|
)
,
(16)
where, without loss of generality, we let 0≤
1
=≤2and
=4−
2
.
Dierentiating and using a simple calculus shows that
Φ(,||)/|| ≥ 0for || ∈ [0,1]and xed ∈[0,2].
Journal of Complex Analysis 3
It follows that Φ(,||)is an increasing function of ||.So
Φ(,||)≤Φ(,1).Uponletting||=1,asimplealgebraic
manipulation yields
2
4
−
2
3
≤
64−9
2
144
≤
4
9
.
(17)
is bound is sharp and the extremal function is given by
()=(1+
2
)/(1−
2
).
No bounds for |
2
4
−
2
3
|was obtained b y omas and
Halim [5] for the class of functions of form (1) that are close-
to-convex in U. In our next theorem we determine a sharp
bound for the coecient body |
3
(1)|.
eorem 4. Let ∈R be given by (1). en we have the sharp
bound
3
(
1
)
=
1
2
3
2
1
2
3
2
1
≤
13
9
. (18)
Proo f. Expanding the determinant
3
(1)and letting =4−
2
we obtain
3
(
1
)
=
1+2
2
2
3
−1−
2
3
=
1+
2
1
2
2
3
−1−
2
2
9
=
1+
1
18
4
1
−
1
2
2
1
+
1
36
2
1
−
1
36
2
2
.
(19)
As before, without loss o f generality, we assume that
1
=,
where 0≤≤2. en, by using the triangle inequality and
the fact that ||≤1,weobtain
3
(
1
)
≤
1+
1
18
4
−
1
2
2
+
1
36
2
4−
2
+
1
36
4−
2
2
fl Ψ
(
)
.
(20)
Considering the modulus as positive, we get
Ψ
(
)
=
1
18
4
−11
2
+26
.
(21)
One can apply an elementary calculus to show that Ψ()
attains its maximum value of 13/9on [0,2]when =0.
Similarly, considering the modulus as negative, we obtain
Ψ
(
)
=
1
18
−
4
+7
2
−10
.
(22)
Again, using an elementary calculus argument shows that this
expression has a maximum value of 1/8on [0,2]when =
7/2.esharpbound|
3
(1)|=13/9is achieved for
1
=0
and
2
=2.
We remark that the sharp bound |
3
(1)| ≤ 13/9given
by eorem 4 is much ner than |
3
(1)| ≤ 8obtained by
omas and Halim [5] fo r the class of functions of form (1)
that are close-to-convex in U.Finally,anupperboundforthe
coecient body |
3
(2)|is presented in the following.
eorem 5. Let ∈R be given by (1). en we have the upper
bound
3
(
2
)
=
2
3
4
3
2
3
4
3
2
≤
4
9
. (23)
Proo f. Write
3
(
2
)
=
2
−
4
2
2
−2
2
3
+
2
4
. (24)
Using the same techniques as above, one can obtain with
simple computations that |
2
−
4
|≤1/2. us we need to
show that |
2
2
−2
2
3
+
2
4
|≤8/9. In v iew of (6), (7), and (8),
a simple computation leads to
2
2
−2
2
3
+
2
4
=
2
1
4
−
2
2
2
9
+
1
3
8
.
(25)
Expressing
2
and
3
in terms of
1
as earlier and using
Lemma 1 with =4−
2
1
and =(1−||
2
),weobtain
2
2
−2
2
3
+
2
4
=
2
1
4
−
4
1
18
−
2
2
18
−
7
144
2
1
+
1
32
4
1
−
1
32
2
1
2
+
1
16
1
.
(26)
Applying the triangle inequality and assuming that 0≤
1
=
≤2,weobtain
2
2
−2
2
3
+
2
4
≤
2
4
−
7
288
4
+
1
18
|
|
2
4−
2
2
+
7
144
2
4−
2
|
|
+
1
32
2
4−
2
|
|
2
+
1
16
4−
2
1−
|
|
2
fl
(
,
|
|
)
.
(27)
We need to nd the maximum value of (,||)on [0,2]×
[0,1]. First, assume that there is a maximum at an interior
point (,|
0
|)of [0,2]×[0,1]. en dierentiating (
0
,||)
with respect to ||andequalingitto0wouldimplythat
0
=2, which is a contradiction. us to nd the maximum
of (,||), we need only to consider the end points of [0,2]×
[0,1].
When =0, (0,||)=(16/18)||
2
≤8/9.
When =2, (2,||)=11/18.
When ||=0, (,0)=|
2
/4−(7/288)
4
|+(1/16)(4−
2
),
which has maximum value 11/18on [0,2].
When ||=1, (,1)=|
2
/4−(7/288)
4
|+(1/18)(4−
2
)
2
+(7/144)
2
(4−
2
)+(1/32)
2
(4−
2
),whichhasmaximum
value 8/9on [0,2].
No bounds for |
3
(2)|were obtained by omas and
Halim[5]fortheclassoffunctionsofform(1)thatareclose-
to-convex in U.
4 Journal of Complex Analysis
Competing Interests
e authors declare that there are no competing interests
regarding the publication of this paper.
Acknowledgments
e work of the second author is supported by a grant from
Department of Science and Technology, Government of India
vide ref: SR/FTP/MS-022/2012 under fast track scheme.
References
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[2] B. Pinchuk, “A variational method for f unctions of bounded
boundary rotation,” Transactions of the American Mathematical
Society,vol.138,pp.107–113,1969.
[3] P. L. Duren, Univalent Functions,vol.259ofGrundlehren der
Mathematischen Wissenschaen, Springer, New York, NY, USA,
1983.
[4] K. Ye and L.-H. Lim, “Every matrix is a product of Toeplitz
matrices,” Foundations of Computational Mathematics,vol.16,
no . 3, pp. 577–598, 2016.
[5]D.K.omasandS.A.Halim,“Toeplitzmatriceswhose
elements are the coecients of starlike and close-to-convex
functions,” Bulletin of the Malaysian Mathematical Sciences
Society,2016.
[6] R. J. Libera and E. J. Zloktiewicz, “Coecient bounds for the
inverse of a function with derivative in P,” Proceedings of the
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