scholarly journals Sharp Bounds of the Hermitian Toeplitz Determinants for Some Classes of Close-to-Convex Functions

2021 ◽  
Author(s):  
Adam Lecko ◽  
Barbara Śmiarowska
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Upper And Lower Bounds ◽  
Toeplitz Determinants ◽  
Sharp Bounds

AbstractSharp upper and lower bounds of the Hermitian Toeplitz determinants of the second and third orders are found for various subclasses of close-to-convex functions.

scholarly journals Euclidean linear invariance and uniform local convexity

1992 ◽  
Vol 52 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Wancang Ma ◽  
David Minda
Keyword(s):  
Lower Bounds ◽  
Convex Subset ◽  
Convex Functions ◽  
Convex Set ◽  
Holomorphic Functions ◽  
Geometric Interpretation ◽  
Upper And Lower Bounds ◽  
Local Convexity ◽  
Related Matter ◽  
The Family

AbstractLet S(p) be the family of holomorphic functions f defined on the unit disk D, normalized by f(0) = f1(0) – 1 = 0 and univalent in every hyperbolic disk of radius p. Let C(p) be the subfamily consisting of those functions which are convex univalent in every hyperbolic disk of radius p. For p = ∞ these become the classical families S and C of normalized univalent and convex functions, respectively. These families are linearly invariant in the sense of Pommerenke; a natural problem is to calculate the order of these linearly invariant families. More precisely, we give a geometrie proof that C(p) is the universal linearly invariant family of all normalized locally schlicht functions of order at most coth(2p). This gives a purely geometric interpretation for the order of a linearly invariant family. In a related matter, we characterize those locally schlicht functions which map each hyperbolically k-convex subset of D onto a euclidean convex set. Finally, we give upper and lower bounds on the order of the linearly invariant family S(p) and prove that this class is not equal to the universal linearly invariant family of any order.

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.

scholarly journals Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Wei Gao ◽  
Muhammad Kamran Jamil ◽  
Aisha Javed ◽  
Mohammad Reza Farahani ◽  
Shaohui Wang ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Graph Transformations ◽  
Mathematical Properties ◽  
Bicyclic Graphs ◽  
Important Branch

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as∑uv∈E(G)‍(d(u)+d(v))2, whered(v)is the degree of the vertexvin a graphG=(V(G),E(G)). In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied. Using these nice mathematical properties, the extremal graphs amongn-vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index. Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.

scholarly journals Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions

2020 ◽  
Vol 114 (4) ◽  
Author(s):  
Piotr Jastrzȩbski ◽  
Bogumiła Kowalczyk ◽  
Oh Sang Kwon ◽  
Adam Lecko ◽  
Young Jae Sim
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Third Order ◽  
Toeplitz Determinants

Abstract For some subclasses of close-to-star functions the sharp upper and lower bounds of the second and third-order Hermitian Toeplitz determinants are computed.

scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

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.

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 Derivation of bounds of several kinds of operators via $(s,m)$-convexity

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Young Chel Kwun ◽  
Ghulam Farid ◽  
Shin Min Kang ◽  
Babar Khan Bangash ◽  
Saleem Ullah
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Integral Operators ◽  
Upper And Lower Bounds ◽  
Hadamard Inequality

AbstractThe objective of this paper is to derive the bounds of fractional and conformable integral operators for $(s,m)$(s,m)-convex functions in a unified form. Further, the upper and lower bounds of these operators are obtained in the form of a Hadamard inequality, and their various fractional versions are presented. Some connections with already known results are obtained.

scholarly journals Derivation of Bounds of an Integral Operator via Exponentially Convex Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hong Ye ◽  
Ghulam Farid ◽  
Babar Khan Bangash ◽  
Lulu Cai
Keyword(s):  
Integral Operator ◽  
Lower Bounds ◽  
Differentiable Function ◽  
Convex Functions ◽  
Integral Operators ◽  
Compact Form ◽  
Upper And Lower Bounds ◽  
Hadamard Inequality ◽  
Absolute Value ◽  
Exponentially Convex Functions

In this paper, bounds of fractional and conformable integral operators are established in a compact form. By using exponentially convex functions, certain bounds of these operators are derived and further used to prove their boundedness and continuity. A modulus inequality is established for a differentiable function whose derivative in absolute value is exponentially convex. Upper and lower bounds of these operators are obtained in the form of a Hadamard inequality. Some particular cases of main results are also studied.

scholarly journals Harmonic Index and Harmonic Polynomial on Graph Operations

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 456 ◽  
Author(s):  
Juan Hernández-Gómez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José M.  Sigarreta
Keyword(s):  
Lower Bounds ◽  
Convex Functions ◽  
Topological Index ◽  
Cartesian Product ◽  
Harmonic Polynomial ◽  
Upper And Lower Bounds ◽  
Lexicographic Product ◽  
Harmonic Index ◽  
Graph Operations ◽  
Corona Product

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O ( n 2 ) .

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