scholarly journals Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators

2020 ◽  
Vol 293 (9) ◽  
pp. 1776-1790
Author(s):  
Wencai Liu
Keyword(s):  
Sharp Bounds ◽  
Embedded Eigenvalues

scholarly journals Sharp Bounds on Davenport-Schinzel Sequences of Every Order

2015 ◽  
Vol 62 (5) ◽  
pp. 1-40 ◽  
Author(s):  
Seth Pettie
Keyword(s):  
Sharp Bounds

The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

2020 ◽  
Vol 70 (4) ◽  
pp. 849-862
Author(s):  
Shagun Banga ◽  
S. Sivaprasad Kumar
Keyword(s):  
Triangle Inequality ◽  
Starlike Functions ◽  
Sharp Bound ◽  
The Novel ◽  
Sharp Bounds ◽  
Hankel Determinants ◽  
Lemniscate Of Bernoulli ◽  
Carathéodory Functions ◽  
The Right

AbstractIn this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

scholarly journals Coefficient inequalities for a subclass of Bazilevič functions

2020 ◽  
Vol 53 (1) ◽  
pp. 27-37
Author(s):  
Sa’adatul Fitri ◽  
Derek K. Thomas ◽  
Ratno Bagus Edy Wibowo ◽  
Keyword(s):  
Recent Work ◽  
Sharp Bounds ◽  
Coefficient Problems ◽  
Coefficient Inequalities ◽  
Bazilevic Functions

AbstractLet f be analytic in {\mathbb{D}}=\{z:|z\mathrm{|\hspace{0.17em}\lt \hspace{0.17em}1\}} with f(z)=z+{\sum }_{n\mathrm{=2}}^{\infty }{a}_{n}{z}^{n}, and for α ≥ 0 and 0 < λ ≤ 1, let { {\mathcal B} }_{1}(\alpha ,\lambda ) denote the subclass of Bazilevič functions satisfying \left|f^{\prime} (z){\left(\frac{z}{f(z)}\right)}^{1-\alpha }-1\right|\lt \lambda for 0 < λ ≤ 1. We give sharp bounds for various coefficient problems when f\in { {\mathcal B} }_{1}(\alpha ,\lambda ), thus extending recent work in the case λ = 1.

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 Bounds of Fractional Metric Dimension and Applications with Grid-Related Networks

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1383
Author(s):  
Ali H. Alkhaldi ◽  
Muhammad Kamran Aslam ◽  
Muhammad Javaid ◽  
Abdulaziz Mohammed Alanazi
Keyword(s):  
Metric Dimension ◽  
Hard Problem ◽  
Np Hard ◽  
Weighted Version ◽  
Sharp Bounds ◽  
Np Hard Problem ◽  
Metric Dimensions

Metric dimension of networks is a distance based parameter that is used to rectify the distance related problems in robotics, navigation and chemical strata. The fractional metric dimension is the latest developed weighted version of metric dimension and a generalization of the concept of local fractional metric dimension. Computing the fractional metric dimension for all the connected networks is an NP-hard problem. In this note, we find the sharp bounds of the fractional metric dimensions of all the connected networks under certain conditions. Moreover, we have calculated the fractional metric dimension of grid-like networks, called triangular and polaroid grids, with the aid of the aforementioned criteria. Moreover, we analyse the bounded and unboundedness of the fractional metric dimensions of the aforesaid networks with the help of 2D as well as 3D plots.

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

scholarly journals A note on magnitude bounds for the mask coefficients of the interpolatory Dubuc–Deslauriers subdivision scheme

2014 ◽  
Vol 17 (1) ◽  
pp. 226-232
Author(s):  
H. E. Bez ◽  
N. Bez
Keyword(s):  
Recent Result ◽  
Operator Norm ◽  
Subdivision Scheme ◽  
Sharp Bounds

AbstractWe analyse the mask associated with the $2n$-point interpolatory Dubuc–Deslauriers subdivision scheme $S_{a^{[n]}}$. Sharp bounds are presented for the magnitude of the coefficients $a^{[n]}_{2i-1}$ of the mask. For scales $i \in [1,\sqrt{n}]$ it is shown that $|a^{[n]}_{2i-1}|$ is comparable to $i^{-1}$, and for larger power scales, exponentially decaying bounds are obtained. Using our bounds, we may precisely analyse the summability of the mask as a function of $n$ by identifying which coefficients of the mask contribute to the essential behaviour in $n$, recovering and refining the recent result of Deng–Hormann–Zhang that the operator norm of $S_{a^{[n]}}$ on $\ell ^\infty $ grows logarithmically in $n$.

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