Some coefficient bounds associated with transforms of bounded turning functions
UDC 517.5 We present the derivation of an upper bound for the Hankel determinants of certain orders linked with the $k$th-root transform $[f(z ^k )] ^{\frac{1}{k}}$ of the holomorphic mapping $f(z)$ whose derivative has a positive real part with normalization, namely, $f(0)=0$ and $f'(0)=1....
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2023
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512496275685376 |
|---|---|
| author | Vani, N. Krishna, D. Vamshee Shalini, D. Vani, N. Krishna, D. Vamshee Shalini, D. |
| author_facet | Vani, N. Krishna, D. Vamshee Shalini, D. Vani, N. Krishna, D. Vamshee Shalini, D. |
| author_sort | Vani, N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-01-23T14:02:45Z |
| description | UDC 517.5
We present the derivation of an upper bound for the Hankel determinants of certain orders linked with the $k$th-root transform $[f(z ^k )] ^{\frac{1}{k}}$ of the holomorphic mapping $f(z)$ whose derivative has a positive real part with normalization, namely, $f(0)=0$ and $f'(0)=1.$ |
| doi_str_mv | 10.37863/umzh.v74i12.6671 |
| first_indexed | 2026-03-24T03:29:43Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i12.6671
UDC 517.5
N. Vani (Gitam Inst. Sci., GITAM Univ., Visakhapatnam, India),
D. Vamshee Krishna1 (Gitam Sci., GITAM Univ., Visakhapatnam, India),
D. Shalini (Dr. B. R. Ambedkar Univ., Srikakulam, India)
SOME COEFFICIENT BOUNDS ASSOCIATED WITH TRANSFORMS
OF BOUNDED TURNING FUNCTIONS
ДЕЯКI ОЦIНКИ КОЕФIЦIЄНТIВ, ПОВ’ЯЗАНI З ПЕРЕТВОРЕННЯМИ
ОБМЕЖЕНИХ ПОВОРОТНИХ ФУНКЦIЙ
We present the derivation of an upper bound for the Hankel determinants of certain orders linked with the k\mathrm{t}\mathrm{h}-root transform
[f(zk)]
1
k of the holomorphic mapping f(z) whose derivative has a positive real part with normalization, namely, f(0) = 0
and f \prime (0) = 1.
Отримано верхню межу для визначникiв Ганкеля певних порядкiв, що пов’язанi з k-м кореневим перетворенням
[f(zk)]
1
k голоморфного вiдображення f(z), похiдна якого має додатну дiйсну частину з нормалiзацiєю, а саме
f(0) = 0 i f \prime (0) = 1.
1. Preliminaries. Let \scrA represent group of mappings f of the type
f(z) = z +
\infty \sum
n=2
anz
n (1.1)
in \scrU d = \{ z \in \scrC : | z| < 1\} , denotes the open unit disc and S is the subfamily of \scrA , possessing
univalent (schlicht) mappings. The k\mathrm{t}\mathrm{h}-root transform for the mapping f in (1.1) is
G(z) :=
\bigl[
f(zk)
\bigr] 1
k = z +
\infty \sum
n=1
bkn+1z
kn+1. (1.2)
At this sequel, we have introduced and interpreted the concept of Hankel determinant for G(z) for
f (1.1) with q, t, k \in \BbbN = \{ 1, 2, 3, . . .\} as
Hq,k(t) =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
bk(t - 1)+1 bkt+1 . . . bk(t+q - 2)+1
bkt+1 bk(t+1)+1 . . . bk(t+q - 1)+1
. . . . . . . . . . . .
bk(t+q - 2)+1 bk(t+q - 1)+1 . . . bk[t+2(q - 1) - 1]+1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
, b1 = 1. (1.3)
In particular, if k = 1 in (1.3) it reduces to the Hankel determinant Hq,k(t) = Hq(t), given by
Pommerenke [10], investigated by several authors. In specific for q = 2, t \in \{ 1, 2, 3\} and q = 3,
t = 1, the Hankel determinant (1.3) has been simplified respectively to
H2,k(1) =
\bigm| \bigm| \bigm| \bigm| \bigm| b1 bk+1
bk+1 b2k+1
\bigm| \bigm| \bigm| \bigm| \bigm| = b2k+1 - b2k+1, (1.4)
1 Corresponding author, e-mail: vamsheekrishna1972@gmail.com.
c\bigcirc N. VANI, D. VAMSHEE KRISHNA, D. SHALINI, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12 1673
1674 N. VANI, D. VAMSHEE KRISHNA, D. SHALINI
H2,k(2) =
\bigm| \bigm| \bigm| \bigm| \bigm| bk+1 b2k+1
b2k+1 b3k+1
\bigm| \bigm| \bigm| \bigm| \bigm| = bk+1b3k+1 - b22k+1, (1.5)
H2,k(3) =
\bigm| \bigm| \bigm| \bigm| \bigm| b2k+1 b3k+1
b3k+1 b4k+1
\bigm| \bigm| \bigm| \bigm| \bigm| = b2k+1b4k+1 - b23k+1, (1.6)
and
H3,k(1) =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b1 bk+1 b2k+1
bk+1 b2k+1 b3k+1
b2k+1 b3k+1 b4k+1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| . (1.7)
Expanding the determinant in H3,k(1), we have
H3,k(1) =
\bigl[
b4k+1(b2k+1 - b2k+1) + b3k+1(bk+1b2k+1 - b3k+1)+
+b2k+1(bk+1b3k+1 - b22k+1)
\bigr]
. (1.8)
Ali et al. [1] derived exact estimates for | b2k+1 - \mu b2k+1| , represents the generalized Fekete –
Szegö functional related to the function G(z), when f is a member of specific subfamilies of S.
We mention H2,k(2), H2,k(3) and H3,k(1) respectively the 2nd and 3rd order Hankel determinants
for the function f given in (1.2).
In Section 2, motivated by the results obtained by earlier authors, we estimate an upper bound to
H2,k(3) and H3,k(1) for the family of bounded turning functions, denoted by \Re , defined below.
Definition 1.1. The mapping f (1.1) belongs to \Re , if
\mathrm{R}\mathrm{e}f \prime (z) > 0, z \in \scrU d.
A. E. Livingston [7] established the subfamily \Re of S, and later MacGregor [8] carried a consis-
tent study about the properties of functions belongs to this class.
In deriving our results, the required sharp estimates specified below, given in the form of lemmas,
which holds suitable for functions possessing positive real part.
The collection \scrP of all functions g, each one called as Carathéodory function [4] of the form
g(z) = 1 +
\infty \sum
t=1
ctz
t, (1.9)
holomorphic in \scrU d and \mathrm{R}\mathrm{e}g(z) > 0 for z \in \scrU d .
Lemma 1.1 [5]. If g \in \scrP , then | ci - \mu cjci - j | \leq 2 satisfies for the values i, j \in \BbbN with i > j
and \mu \in [0, 1].
Lemma 1.2 [7]. If g \in \scrP , then | ci - cjci - j | \leq 2 holds for the values i, j \in \BbbN with i > j.
Lemma 1.3 [9]. If g \in \scrP , then | ct| \leq 2, for t \in \BbbN , equality occurs for the function h(z) =
=
1 + z
1 - z
, z \in \scrU d.
Lemma 1.4 [11]. If g \in \scrP , then
\bigm| \bigm| c2c4 - c23
\bigm| \bigm| \leq 4 - 1
2
| c2| 2 +
1
4
| c2| 3.
To procure our results, we adopt the procedure framed through Libera and Zlotkiewicz [6].
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SOME COEFFICIENT BOUNDS ASSOCIATED WITH TRANSFORMS . . . 1675
2. Important outcomes.
Theorem 2.1. If f \in \Re and G given in (1.2) is the kth-root transformation of f, then
\bigm| \bigm| H2,k(3)
\bigm| \bigm| \leq \biggl[
542k3 - 383k2 - 30k + 15
540k5
\biggr]
and the result is sharp for k = 1.
Proof. For f \in \Re , according to the Definition 1.1,
f \prime (z) = g(z), z \in \scrU d. (2.1)
Substitute the values for f and g in (2.1), it simplifies to
an+1 =
cn
n+ 1
, n \in \BbbN . (2.2)
For the mapping f (1.1), a calculation gives
\bigl[
f(zk)
\bigr] 1
k =
\biggl[
zk +
\infty \sum
n=2
anz
nk
\biggr] 1
k
=
=
\biggl[
z +
1
k
a2z
k+1 +
\biggl\{
1
k
a3 +
1 - k
2k2
a22
\biggr\}
z2k+1+
+
\biggl\{
1
k
a4 +
1 - k
k2
a2a3 +
(1 - k)(1 - 2k)
6k3
a32
\biggr\}
z3k+1+
+
\biggl\{
1
k
a5 +
1 - k
2k2
(a23 + 2a2a4) +
(1 - k)(1 - 2k)
2k3
a22a3+
+
(1 - k)(1 - 2k)(1 - 3k)
24k4
a42
\biggr\}
z4k+1 + . . .
\biggr]
. (2.3)
Comparing the coefficients of zk+1, z2k+1, z3k+1 and z4k+1 in the expressions (1.2) and (2.3), we
obtain
bk+1 =
1
k
a2, b2k+1 =
1
k
a3 +
1 - k
2k2
a22,
b3k+1 =
\biggl[
1
k
a4 +
1 - k
k2
a2a3 +
(1 - k)(1 - 2k)
6k3
a32
\biggr]
, (2.4)
b4k+1 =
\biggl[
1
k
a5 +
1 - k
2k2
(a23 + 2a2a4) +
(1 - k)(1 - 2k)
2k3
a22a3+
+
(1 - k)(1 - 2k)(1 - 3k)
24k4
a42
\biggr]
.
Simplifying the expressions in (2.2) and (2.4), we get
bk+1 =
c1
2k
, b2k+1 =
c2
3k
- k - 1
8k2
c21,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1676 N. VANI, D. VAMSHEE KRISHNA, D. SHALINI
b3k+1 =
\biggl[
c3
4k
- k - 1
6k2
c1c2 +
(k - 1)(2k - 1)
48k3
c31
\biggr]
, (2.5)
b4k+1 =
\biggl[
c4
5k
- k - 1
18k2
c22 -
k - 1
8k2
c1c3 +
(k - 1)(2k - 1)
24k3
c21c2 -
- (k - 1)(2k - 1)(3k - 1)
384k4
c41
\biggr]
.
Substituting the values namely b2k+1, b3k+1 and b4k+1 from (2.5) in the functional given in (1.6),
we have
H2,k(3) =
\biggl[
1
15k2
c2c4 -
1
16k2
c23 -
k - 1
54k3
c32 +
k - 1
24k3
c1c2c3 +
k2 - 1
144k4
c21c
2
2 -
- k - 1
40k3
c21c4 -
k2 - 1
192k4
c31c3 -
(k2 - 1)(2k - 1)
1152k5
c41c2 +
(k2 - 1)(k - 1)(2k - 1)
9216k6
c61
\biggr]
. (2.6)
On grouping the terms in (2.6), to apply lemmas mentioned in this paper, we get
H2,k(3) =
\biggl[
1
16k2
\{ c2c4 - c23\} +
1
240k2
c2c4 -
k - 1
54k3
c32 -
k - 1
40k3
c21
\biggl\{
c4 -
5(k + 1)
18k
c22
\biggr\}
-
- (k2 - 1)(2k - 1)
1152k5
c41
\biggl\{
c2 -
2k - 1
8k
c21
\biggr\}
+
k - 1
24k3
c1c3
\biggl\{
c2 -
k + 1
8k
c21
\biggr\} \biggr]
. (2.7)
Applying the triangle inequality in (2.7), we obtain
| H2,k(3)| \leq
\biggl[
1
16k2
\bigm| \bigm| c2c4 - c23
\bigm| \bigm| + 1
240k2
| c2| | c4| +
k - 1
54k3
| c2| 3+
+
k - 1
40k3
| c1| 2
\bigm| \bigm| \bigm| \bigm| c4 - 5(k + 1)
18k
c22
\bigm| \bigm| \bigm| \bigm| + (k2 - 1)(2k - 1)
1152k5
| c1| 4
\bigm| \bigm| \bigm| \bigm| c2 - 2k - 1
8k
c21
\bigm| \bigm| \bigm| \bigm| +
+
k - 1
24k3
| c1| | c3|
\bigm| \bigm| \bigm| \bigm| c2 - k + 1
8k
c21
\bigm| \bigm| \bigm| \bigm| \biggr] . (2.8)
By using Lemmas 1.1, 1.3 and 1.4 in the inequality (2.8), we have
\bigm| \bigm| H2,k(3)
\bigm| \bigm| \leq \biggl[
542k3 - 383k2 - 30k + 15
540k5
\biggr]
. (2.9)
Remark 2.1. In particular, k = 1 in the expression (2.9), then the result coincides with the
development obtained by Zaprawa [11] and the respective function, for which the equality holds
f(z) =
\bigl[
\mathrm{l}\mathrm{o}\mathrm{g}(1 + z) - \mathrm{l}\mathrm{o}\mathrm{g}(1 - z)
\bigr]
= 2z +
2
3
z3 +
2
5
z5 + . . . .
Theorem 2.1 is proved.
Theorem 2.2. If f \in \Re and G given in (1.2) is the kth-root transformation of f, then
\bigm| \bigm| H3,k(1)
\bigm| \bigm| \leq \biggl[
90k4 + 414k3 + 15k2 - 135k - 15
540k5
\biggr]
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
SOME COEFFICIENT BOUNDS ASSOCIATED WITH TRANSFORMS . . . 1677
Proof. Substituting bk+1, b2k+1, b3k+1 and b4k+1 values from (2.5) in (1.8),
it simplifies to
H3,k(1) =
\biggl[
1
15k2
c2c4 -
k + 1
54k3
c32 -
1
16k2
c23 -
k + 1
40k3
c21c4+
+
k + 1
24k3
c1c2c3 +
k2 - 1
144k4
c21c
2
2 -
(k2 - 1)(2k + 1)
1152k5
c41c2 -
- k2 - 1
192k4
c31c3 +
(k2 - 1)(k + 1)(2k + 1)
9216k6
c61
\biggr]
. (2.10)
On grouping the terms in (2.10), to apply lemmas given in this paper, then
H3,k(1) =
\biggl[
1
20k2
c4
\biggl\{
c2 -
20k2(k + 1)
40k3
c21
\biggr\}
- 1
16k2
c3
\biggl\{
c3 -
16k2
16k3
c1c2
\biggr\}
+
+
1
27k2
c2
\biggl\{
c4 -
27k2(k + 1)
54k3
c22
\biggr\}
- 1
48k
c2
\biggl\{
c4 -
48k(2k - 1)
48k3
c1c3
\biggr\}
+
+
(k2 - 1)(2k + 1)
1152k5
c41
\biggl\{
c2 -
1152k5(k + 1)
9216k6
c21
\biggr\}
- k2 - 1
192k4
c31c3 +
k2 - 1
144k4
c21c
2
2+
+
\biggl\{
- 1
20k2
- 1
27k2
+
1
48k
+
1
15k2
\biggr\}
c2c4
\biggr]
.
Further, we have
H3,k(1) =
\biggl[
1
20k2
c4
\biggl\{
c2 -
k + 1
2k
c21
\biggr\}
- 1
16k2
c3
\biggl\{
c3 -
1
k
c1c2
\biggr\}
+
+
1
27k2
c2
\biggl\{
c4 -
k + 1
2k
c22
\biggr\}
- 1
48k
c2
\biggl\{
c4 -
2k - 1
k2
c1c3
\biggr\}
+
+
(k2 - 1)(2k + 1)
1152k5
c41
\biggl\{
c2 -
k + 1
8k
c21
\biggr\}
- k2 - 1
192k4
c31c3 +
k2 - 1
144k4
c21c
2
2+
+
\biggl\{
45k - 44
2160k2
\biggr\}
c2c4
\biggr]
. (2.11)
Applying the triangle inequality in the expression (2.11), we obtain
| H3,k(1)| \leq
\biggl[
1
20k2
| c4|
\bigm| \bigm| \bigm| \bigm| c2 - k + 1
2k
c21
\bigm| \bigm| \bigm| \bigm| + 1
16k2
| c3|
\bigm| \bigm| \bigm| \bigm| c3 - 1
k
c1c2
\bigm| \bigm| \bigm| \bigm| +
+
1
27k2
| c2|
\bigm| \bigm| \bigm| \bigm| c4 - k + 1
2k
c22
\bigm| \bigm| \bigm| \bigm| + 1
48k
| c2|
\bigm| \bigm| \bigm| \bigm| c4 - 2k - 1
k2
c1c3
\bigm| \bigm| \bigm| \bigm| +
+
(k2 - 1)(2k + 1)
1152k5
| c41|
\bigm| \bigm| \bigm| \bigm| c2 - k + 1
8k
c21
\bigm| \bigm| \bigm| \bigm| +
+
k2 - 1
192k4
| c31| | c3| +
k2 - 1
144k4
| c21| | c22| +
\biggl\{
45k - 44
2160k2
\biggr\}
| c2| | c4|
\biggr]
. (2.12)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1678 N. VANI, D. VAMSHEE KRISHNA, D. SHALINI
Upon applying Lemmas 1.1, 1.2 and 1.3 in the inequality (2.12), it simplifies to
\bigm| \bigm| H3,k(1)
\bigm| \bigm| \leq \biggl[
90k4 + 414k3 + 15k2 - 135k - 15
540k5
\biggr]
. (2.13)
Remark 2.2. In particular, if k = 1 in the expression (2.13), then the result coincides with the
result of Zaprawa [12].
Theorem 2.2 is proved.
References
1. R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramaniam, The Fekete – Szegö coefficient functional for transforms of
analytic functions, Bull. Iranian Math. Soc., 35, № 2, 119 – 142 (2009).
2. A. K. Bakhtin, I. V. Denega, Extremal decomposition of the complex plane with free poles, J. Math. Sci., 246, № 1,
1 – 17 (2020).
3. I. Denega, Extremal decomposition of the complex plane for n-radial system of points, Azerbaijan J. Math., Special
Issue Dedicated to the 67th Birth Anniversary of Prof. M. Mursaleen, 64 – 74 (2021).
4. P. L. Duren, Univalent functions, Grundlehren Math. Wiss., vol. 259, Springer, New York (1983).
5. T. Hayami, S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal., 4, № 52, 2573 – 2585
(2010).
6. R. J. Libera, E. J. Zlotkiewicz, Early coefficients of theinverse of a regular convex function, Proc. Amer. Math. Soc.,
85, № 2, 225 – 230 (1982).
7. A. E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc., 21, № 3, 545 – 552
(1969).
8. T. H. MacGregor, Functions whose derivative have a positive real part, Trans. Amer. Math. Soc., 104, № 3, 532 – 537
(1962).
9. Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Göttingen (1975).
10. Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. (1), 41,
111 – 122 (1966).
11. P. Zaprawa, On Hankel determinant H2(3) for univalent functions, Results Math., 73, № 3, Article 89 (2018).
12. P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math., 14, № 1, 1 – 10
(2017).
Received 04.04.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
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| id | umjimathkievua-article-6671 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:29:43Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/c1/2399a39134b60fd6bbf162be0e83f1c1.pdf |
| spelling | umjimathkievua-article-66712023-01-23T14:02:45Z Some coefficient bounds associated with transforms of bounded turning functions Some coefficient bounds associated with transforms of bounded turning functions Vani, N. Krishna, D. Vamshee Shalini, D. Vani, N. Krishna, D. Vamshee Shalini, D. holomorphic bounded turning function, upper bound, functionals connected with Hankel determinant, positive real part function. 2010 MSC: 30C45; 30C50 UDC 517.5 We present the derivation of an upper bound for the Hankel determinants of certain orders linked with the $k$th-root transform $[f(z ^k )] ^{\frac{1}{k}}$ of the holomorphic mapping $f(z)$ whose derivative has a positive real part with normalization, namely, $f(0)=0$ and $f'(0)=1.$ УДК 517.5 Деякі оцінки коефіцієнтів, пов'язані з перетвореннями обмежених поворотних функцій Отримано верхню межу для визначників Ганкеля певних порядків, що пов’язані з $k$-м кореневим перетворенням $[f(z ^k )] ^{\frac{1}{k}}$&nbsp; голоморфного відображення $f(z),$ похідна якого має додатну дійсну частину з нормалізацією, а саме $f(0)=0$ і $f'(0)=1.$ Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6671 10.37863/umzh.v74i12.6671 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1673 - 1678 Український математичний журнал; Том 74 № 12 (2022); 1673 - 1678 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6671/9343 Copyright (c) 2023 VAMSHEE KRISHNA DEEKONDA, Shalini .Dasumahanthi, Vani Nallamothu |
| spellingShingle | Vani, N. Krishna, D. Vamshee Shalini, D. Vani, N. Krishna, D. Vamshee Shalini, D. Some coefficient bounds associated with transforms of bounded turning functions |
| title | Some coefficient bounds associated with transforms of bounded turning functions |
| title_alt | Some coefficient bounds associated with transforms of bounded turning functions |
| title_full | Some coefficient bounds associated with transforms of bounded turning functions |
| title_fullStr | Some coefficient bounds associated with transforms of bounded turning functions |
| title_full_unstemmed | Some coefficient bounds associated with transforms of bounded turning functions |
| title_short | Some coefficient bounds associated with transforms of bounded turning functions |
| title_sort | some coefficient bounds associated with transforms of bounded turning functions |
| topic_facet | holomorphic bounded turning function upper bound functionals connected with Hankel determinant positive real part function. 2010 MSC: 30C45 30C50 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6671 |
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