scholarly journals On the Difference of Inverse Coefficients of Univalent Functions

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2040
Author(s):  
Young Jae Sim ◽  
Derek Keith Thomas
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Upper Bound ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Starlike Functions ◽  
Upper And Lower Bounds ◽  
The Difference ◽  
Bazilevic Functions

Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions with f(0)=0, and f′(0)=1. Let F be the inverse function of f, given by F(z)=ω+∑n=2∞Anωn for some |ω|≤r0(f). Let S*⊂S be the subset of starlike functions in D, and C the subset of convex functions in D. We show that −1≤|A3|−|A2|≤3 for f∈S, the upper bound being sharp, and sharp upper and lower bounds for |A3|−|A2| for the more important subclasses of S* and C, and for some related classes of Bazilevič functions.

scholarly journals On the Difference of Coefficients of Starlike and Convex Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1521
Author(s):  
Young Jae Sim ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Convex Functions ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper And Lower Bounds ◽  
Starlike And Convex Functions ◽  
The Difference

Let f be analytic in the unit disk D={z∈C:|z|<1}, and S be the subclass of normalized univalent functions given by f(z)=z+∑n=2∞anzn for z∈D. Let S*⊂S be the subset of starlike functions in D and C⊂S the subset of convex functions in D. We give sharp upper and lower bounds for |a3|−|a2| for some important subclasses of S* and C.

ON THE DIFFERENCE OF COEFFICIENTS OF OZAKI CLOSE-TO-CONVEX FUNCTIONS

2020 ◽  
pp. 1-8
Author(s):  
YOUNG JAE SIM ◽  
DEREK K. THOMAS
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Convex Functions ◽  
Univalent Functions ◽  
Upper And Lower Bounds ◽  
The Difference

Let $f$ be analytic in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and ${\mathcal{S}}$ be the subclass of normalised univalent functions given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ for $z\in \mathbb{D}$ . We give sharp upper and lower bounds for $|a_{3}|-|a_{2}|$ and other related functionals for the subclass ${\mathcal{F}}_{O}(\unicode[STIX]{x1D706})$ of Ozaki close-to-convex functions.

A NOTE ON SPIRALLIKE FUNCTIONS

2021 ◽  
pp. 1-7
Author(s):  
Y. J. SIM ◽  
D. K. THOMAS
Keyword(s):  
Unit Disk ◽  
Lower Bounds ◽  
Open Problem ◽  
Univalent Functions ◽  
Inverse Function ◽  
Upper And Lower Bounds ◽  
Spirallike Functions ◽  
New Inequalities

Abstract Let f be analytic in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1 \}$ and let ${\mathcal S}$ be the subclass of normalised univalent functions with $f(0)=0$ and $f'(0)=1$ , given by $f(z)=z+\sum _{n=2}^{\infty }a_n z^n$ . Let F be the inverse function of f, given by $F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$ for $|\omega |\le r_0(f)$ . Denote by $ \mathcal {S}_p^{* }(\alpha )$ the subset of $ \mathcal {S}$ consisting of the spirallike functions of order $\alpha $ in $\mathbb {D}$ , that is, functions satisfying $$\begin{align*}{\mathrm{Re}} \ \bigg\{e^{-i\gamma}\dfrac{zf'(z)}{f(z)}\bigg\}>\alpha\cos \gamma, \end{align*}$$ for $z\in \mathbb {D}$ , $0\le \alpha <1$ and $\gamma \in (-\pi /2,\pi /2)$ . We give sharp upper and lower bounds for both $ |a_3|-|a_2| $ and $ |A_3|-|A_2| $ when $f\in \mathcal {S}_p^{* }(\alpha )$ , thus solving an open problem and presenting some new inequalities for coefficient differences.

scholarly journals On the difference of inverse coefficients of convex functions

2022 ◽  
Author(s):  
Young Jae Sim ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
The Difference

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${\mathcal {S}}$$ S be the subclass of normalised univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . Let F be the inverse function of f defined in some set $$|\omega |\le r_{0}(f)$$ | ω | ≤ r 0 ( f ) , and be given by $$F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$$ F ( ω ) = ω + ∑ n = 2 ∞ A n ω n . We prove the sharp inequalities $$-1/3 \le |A_4|-|A_3| \le 1/4$$ - 1 / 3 ≤ | A 4 | - | A 3 | ≤ 1 / 4 for the class $${\mathcal {K}}\subset {\mathcal {S}}$$ K ⊂ S of convex functions, thus providing an analogue to the known sharp inequalities $$-1/3 \le |a_4|-|a_3| \le 1/4$$ - 1 / 3 ≤ | a 4 | - | a 3 | ≤ 1 / 4 , and giving another example of an invariance property amongst coefficient functionals of convex functions.

scholarly journals The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

2021 ◽  
Author(s):  
Young Jae Sim ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

scholarly journals On the Difference of Coefficients of Bazilevič Functions

2019 ◽  
Vol 19 (4) ◽  
pp. 671-685 ◽  
Author(s):  
Nak Eun Cho ◽  
Young Jae Sim ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
The Difference ◽  
Bazilevic Functions

Abstract Let f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$D={z∈C:|z|<1}, and $${\mathcal {S}}$$S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$f(z)=z+∑n=2∞anzn for $$z\in {\mathbb {D}}$$z∈D. We give bounds for $$| |a_3|-|a_2| | $$||a3|-|a2|| for the subclass $${\mathcal B}(\alpha ,i \beta )$$B(α,iβ) of generalized Bazilevič functions when $$\alpha \ge 0$$α≥0, and $$\beta $$β is real.

THE TSP AND THE SUM OF ITS MARGINAL VALUES

2006 ◽  
Vol 16 (04) ◽  
pp. 333-343
Author(s):  
MOSHE DROR ◽  
YUSIN LEE ◽  
JAMES B. ORLIN ◽  
VALENTIN POLISHCHUK
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Traveling Salesperson Problem ◽  
Marginal Value ◽  
Marginal Values ◽  
The Difference

This paper introduces a new notion related to the traveling salesperson problem (TSP) — the notion of the TSP ratio. The TSP ratio of a TSP instance I is the sum of the marginal values of the nodes of I divided by the length of the optimal TSP tour on I, where the marginal value of a node i ∈ I is the difference between the length of the optimal tour on I and the length of the optimal tour on I\i. We consider the problem of establishing exact upper and lower bounds on the TSP ratio. To our knowledge, this problem has not been studied previously. We present a number of cases for which the ratio is never greater than 1. We establish a tight upper bound of 2 on the TSP ratio of any metric TSP. For the TSP on six nodes, we determine the maximum ratio of 1.5 in general, 1.2 for the case of metric TSP, and 10/9 for the geometric TSP in the L1 metric. We also compute the TSP ratio experimentally for a large number of random TSP instances on small number of points.

Localization of Discrete Spectrum of Multiparticle Schrödinger Operators

1985 ◽  
Vol 40 (10) ◽  
pp. 1052-1058 ◽  
Author(s):  
Heinz K. H. Siedentop
Keyword(s):  
Discrete Spectrum ◽  
Lower Bounds ◽  
Upper Bound ◽  
Schrödinger Operators ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators

An upper bound on the dimension of eigenspaces of multiparticle Schrödinger operators is given. Its relation to upper and lower bounds on the eigenvalues is discussed.

scholarly journals On the Fekete-Szegö problem

2000 ◽  
Vol 24 (9) ◽  
pp. 577-581 ◽  
Author(s):  
B. A. Frasin ◽  
Maslina Darus
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Upper Bound ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Strongly Starlike Functions ◽  
Strongly Starlike

Letf(z)=z+a2z2+a3z3+⋯be an analytic function in the open unit disk. A sharp upper bound is obtained for|a3−μa22|by using the classes of strongly starlike functions of orderβand typeαwhenμ≥1.

scholarly journals Second Hankel determinant for certain class of bi-univalent functions defined by Chebyshev polynomials

2019 ◽  
Vol 12 (02) ◽  
pp. 1950017
Author(s):  
H. Orhan ◽  
N. Magesh ◽  
V. K. Balaji
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Chebyshev Polynomials ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
Upper Bound Estimate

In this work, we obtain an upper bound estimate for the second Hankel determinant of a subclass [Formula: see text] of analytic bi-univalent function class [Formula: see text] which is associated with Chebyshev polynomials in the open unit disk.

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