Sharp Estimates Research Articles

Sharp estimate for starlikeness related to a tangent domain

In the recent years, the study of the Hankel determinant problems have been widely investigated by many researchers. We were essentially motivated by the recent research going on with the Hankel determinant and other coefficient bounds problems. In this research article, we first considered the subclass of analytic starlike functions connected with the domain of the tangent function. We then derived the initial four sharp coefficient bounds, the sharp Fekete-Szegö inequality, and the sharp second and third order Hankel determinant for the defined class. Also, we derived sharp estimates like sharp coefficient bounds, Fekete-Szegö estimate, and sharp second order Hankel determinant for the functions having logarithmic coefficient and for the inverse coefficient, respectively, for the defined functions class.

Localized Phase for the Erdős–Rényi Graph

We analyse the eigenvectors of the adjacency matrix of the Erdős–Rényi graph G(N,d/N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {G}}(N,d/N)$$\\end{document} for logN≪d≲logN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{\\log N} \\ll d \\lesssim \\log N$$\\end{document}. We show the existence of a localized phase, where each eigenvector is exponentially localized around a single vertex of the graph. This complements the completely delocalized phase previously established in Alt et al. (Commun Math Phys 388(1):507–579, 2021). For large enough d, we establish a mobility edge by showing that the localized phase extends up to the boundary of the delocalized phase. We derive explicit asymptotics for the localization length up to the mobility edge and characterize its divergence near the phase boundary. The proof is based on a rigorous verification of Mott’s criterion for localization, comparing the tunnelling amplitude between localization centres with the eigenvalue spacing. The first main ingredient is a new family of global approximate eigenvectors, for which sharp enough estimates on the tunnelling amplitude can be established. The second main ingredient is a lower bound on the spacing of approximate eigenvalues. It follows from an anticoncentration result for these discrete random variables, obtained by a recursive application of a self-improving anticoncentration estimate due to Kesten.

Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

On the Fourier asymptotics of absolutely continuous measures with power-law singularities

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon–Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.

Sharp Estimates for the Zalcman Conjecture and Second Order Hankel Determinant

In this work, we found sharp estimates for the Zalcman conjecture and second order Hankel determinant for the inverse function when it belongs to the class of starlike functions with respect to symmetric points, denoted by . These results are new.

Новый метод оценки модулей начальных тейлоровских коэффициентов на классе ограниченных не обращающихся в нуль функций

The task of obtaining the sharp estimate of the modulus of the n-th Taylor coefficient on the class B of bounded non-vanishing functions has been reduced to the problem of estimating a functional over the class of normalized bounded functions, which in turn has been reduced to the problem of finding the constrained maximum of a non-negative objective function of 2n-3 real arguments with constraints of the inequality type, that allows us to apply the standard numerical methods of finding constrained extrema. Analytical expressions of the first six objective functions have been obtained and their Lipschitz continuity has been proved. Based on the Lipschitz continuity of the objective function with number n, a method for the sharp estimating of the modulus of the n-th Taylor coefficient on the class B is rigorously proven. An algorithm of finding the global constrained maximum of the objective function is being discussed. The first step of this algorithm involves a brute-force search with a relatively large step. The second step of the algorithm uses a method for finding a local maximum with the initial points obtained at the previous step. The results of the numerical calculations are presented graphically and confirm the Krzyz conjecture for n=1,…,6. Based on these calculations, as well as on so-called asymptotic estimates, a sharp estimate of the moduli of the first six Taylor coefficients on the class B is derived. The obtained results are compared with previously known estimates of the moduli of initial Taylor coefficients on the class B and its subclasses B_t, t≥0. The extremals for B_t subclasses are discussed and the Krzyz hypothesis is updated for B_t subclasses. A brief historical overview of research of the estimations of moduli of initial Taylor coefficients on the class B is provided.

Handy formulas for binomial moments

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package. The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results.

Problems involving combinations of coefficients for the inverse of some complex-valued analytical functions

<p>Inequalities are essential in solving mathematical problems in many different areas of mathematics. Among these, problems involving coefficient combinations that occurred in the Taylor–Maclaurin series of the inverse of complex-valued analytic functions are the challenging ones to solve. In the current article, our aim is to study certain coefficient-related problems that construct from coefficients of the inverse of specific analytic functions. These problems include the Zalcman and Fekete–Szegö inequalities, as well as sharp estimates of the second and third-order Hankel determinants with inverse function coefficients. Also, one of the obtained results gives an improvement of the problem that has been recently published in the journal "AIMS Mathematics".</p>

Sharp coefficient problems of functions with bounded turning subordinated to the domain of cosine hyperbolic function

<abstract><p>In the current article, we consider a class of bounded turning functions associated with the cosine hyperbolic function and give some results containing coefficient functionals using the familiar Carathéodory functions. An improvement on the bound of the third-order Hankel determinant for functions in this class is provided. Furthermore, we obtain sharp estimates of the Fekete-Szegö, Krushkal, and Zalcman functionals with logarithmic coefficients as entries. All the findings are proved to be sharp.</p></abstract>

Rate of propagation for the Fisher-KPP equation with nonlocal diffusion and free boundaries

In this paper, we obtain sharp estimates for the rate of propagation of the Fisher-KPP equation with nonlocal diffusion and free boundaries. The nonlocal diffusion operator is given by \int_{\R}J(x-y)u(t,y)\,dy-u(t,x) , and our estimates hold for some typical classes of kernel functions J(x) . For example, if for |x|\gg 1 the kernel function satisfies J(x)\sim |x|^{-\gamma} with \gamma>1 , then it follows from [Y. Du et al., J. Math. Pures Appl. 154, 30–66 (2021)] that there is a finite spreading speed when \gamma>2 , namely the free boundary x=h(t) satisfies \lim_{t\to\infty}h(t)/t=c_{0} for some uniquely determined positive constant c_{0} depending on J , and when \gamma\in (1, 2] , \lim_{t\to\infty}h(t)/t=\infty ; the estimates in the current paper imply that, for t\gg 1 , c_{0}t-h(t)\sim \begin{cases}1&\text{ when $\gamma>3$,}\\ \ln t&\text{ when }\gamma=3,\\ t^{3-\gamma}&\text{ when }\gamma\in (2, 3),\\ \end{cases} \qquad h(t)\sim \begin{cases} t\ln t&\text{ when }\gamma =2,\\ t^{1/(\gamma-1)}&\text{ when }\gamma\in (1,2). \end{cases} Our approach is based on subtle integral estimates and constructions of upper and lower solutions, which rely crucially on guessing correctly the order of growth of the term to be estimated. The techniques developed here lay the groundwork for extensions to more general situations.

Minimal [formula omitted]-solutions to singular sublinear elliptic problems

We solve the existence problem for the minimal positive solutions u∈Lp(Ω,dx) to the Dirichlet problems for sublinear elliptic equations of the form Lu=σuq+μinΩ,lim infx→yu(x)=0y∈∂∞Ω,where 0<q<1 and Lu≔−div(A(x)∇u) is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient σ and data μ are nonnegative Radon measures on an arbitrary domain Ω⊂Rn with a positive Green function associated with L. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.

Analytic Functions Related to a Balloon-Shaped Domain

One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a2,a3, and a4, and for three second-order and third-order Hankel determinants, H2,1X,H2,2X, and H3,1X. The optimality of each obtained estimate is given as well.

Existence of unattainable states for Schrödinger type flows on the half-line

Abstract We prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of $L^2$ boundary controls with data spaces $H^{-1}(\mathbb{R}_+)$ and $H^{-2}(\mathbb{R}_+)$ for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.

Sharp decay estimate for solutions of general Choquard equations

Consider the general Choquard equation{Δpu+(∫RN|u(y)|pα⁎|x−y|αdy)|u|pα⁎−2u=0,inRN,u∈D1,p(RN), where 1<p<N,0<α<N,p<pα⁎=(N−α2)pN−p and Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator. By using the Wolff potential theory, we prove that for any solution u of the equation, there exists a constant c such that|u(x)|≤c(1+|x|)−N−pp−1, for x∈RN. The decay estimate is sharp in the sense that the positive solution u satisfiesu(x)≥c(1+|x|)−N−pp−1.

On sharp estimates for Schrödinger groups of fractional powers of nonnegative self-adjoint operators

Let L be a non negative, selfadjoint operator on L2(X), where X is a metric space endowed with a doubling measure. Consider the Schrödinger group for fractional powers of L. If the heat flow e−tL satisfies suitable conditions of Davies–Gaffney type, we obtain the following estimate in Hardy spaces associated to L:‖(I+L)−β/2eiτLγ/2f‖HLp(X)≤C(1+|τ|)nsp‖f‖HLp(X) where p∈(0,1], γ∈(0,1], β/γ=n|12−1p|=nsp and τ∈R.If in addition e−tL satisfies a localized Lp0→L2 polynomial estimate for some p0∈[1,2), we obtain‖(I+L)−β/2eiτLγ/2f‖p0,∞≤C(1+|τ|)nsp0‖f‖p0,∀τ∈R, provided 0<γ≠1, β/γ=n|12−1p|=nsp and τ∈R. By interpolation, the second estimate implies also, for all p∈(p0,p0′), the strong (p,p) type estimate‖(I+L)−β/2eiτLγ/2f‖p≤C(1+|τ|)nsp‖f‖p. The applications of our theory span a diverse spectrum, ranging from the Schrödinger operator with an inverse square potential to the Dirichlet Laplacian on open domains. It showcases the effectiveness of our theory across various settings.

Sharp Instability Estimates for Bidisperse Convection with Local Thermal Non-equilibrium

We analyse a theory for thermal convection in a Darcy porous material where the skeletal structure is one with macropores, but also cracks or fissures, giving rise to a series of micropores. This is thus thermal convection in a bidisperse, or double porosity, porous body. The theory allows for non-equilibrium thermal conditions in that the temperature of the solid skeleton is allowed to be different from that of the fluid in the macro- or micropores. The model does, however, allow for independent velocities and pressures of the fluid in the macro- and micropores. The threshold for linear instability is shown to be the same as that for global nonlinear stability. This is a key result because it shows that one may employ linearized theory to ensure that the key physics of the thermal convection problem has been captured. It is important to realize that this has not been shown for other theories of bidisperse media where the temperatures in the macro- and micropores may be different. An analytical expression is obtained for the critical Rayleigh number and numerical results are presented employing realistic parameters for the physical values which arise.Article HighlightsA two-temperature regime for a bidisperse Darcy porous medium is proposed to study the thermal convection problem.The optimal result of coincidence between the linear instability and nonlinear stability critical thresholds is proven.Numerical analysis enhances that the scaled heat transfer coefficient between the fluid and solid and the porosity-weighted conductivity ratio stabilize the problem significantly.

Growth of hypercyclic entire functions for some non-convolution operators

Abstract A continuous linear operator T T defined on a Fréchet space X X is said to be hypercyclic if there exists f ∈ X f\in X such that, the orbit { T n f } \left\{{T}^{n}f\right\} is dense in X X . In this article, we consider the operators introduced by Aron and Markose, defined on the space of entire functions by T λ , b f ( z ) = f ′ ( λ z + b ) {T}_{\lambda ,b}f\left(z)=f^{\prime} \left(\lambda z+b) , λ ∈ C \ { 1 } \lambda \in {\mathbb{C}}\setminus \left\{1\right\} , b ∈ C b\in {\mathbb{C}} , and we aimed to explore the rate of growth of hypercyclic vectors for T λ , b {T}_{\lambda ,b} . We discover that T λ , b {T}_{\lambda ,b} is a weighted backward shift with respect to some basis and this fact allows us to find sharp estimates of the growth of T λ , b {T}_{\lambda ,b} -hypercyclic vectors. When ∣ λ ∣ = 1 | \lambda | =1 , the T λ , b {T}_{\lambda ,b} -hypercyclic function growth is similar to the D D -hypercyclic functions ( D D is the differentiation operator), and when ∣ λ ∣ &gt; 1 | \lambda | \gt 1 , the T λ , b {T}_{\lambda ,b} -hypercyclic functions can grow very slowly but not arbitrarily slowly. A lower bound of this growth is found in terms of the W W -Lambert function. Finally, partial results are obtained for T λ , b {T}_{\lambda ,b} -frequently hypercyclic functions.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

Some Analysis of the Coefficient-Related Problems for Functions of Bounded Turning Associated with a Symmetric Image Domain

In the last few years, numerous subfamilies of univalent functions, whether directly or indirectly associated with exponential functions, have been introduced and thoroughly investigated. Among these, the families Se*, Ce and Re defined by subordination to ez have been intensively investigated. While the coefficient problem on the class Se* and Ce has been solved in many cases, in this paper, we mainly intend to compute the sharp estimates of some initial coefficients, the Feketo–Szegö inequality, and the sharp bounds of second- and third-order Hankel determinants for functions belonging to the class Re. This work has the potential to significantly enrich and enhance the exploration of univalent functions in conjunction with exponential functions, making the field more comprehensive and robust.

Sharp boundary regularity for some degenerate-singular Monge-Ampère equations on k-convex domain

We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity we construct sub-solutions for the Dirichlet problem of some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.