Sharp Upper Bound Research Articles

Area Problem for Univalent Functions in the Unit Disk with Quasiconformal Extension to the Plane

Let $$\Delta (r,f)$$ denote the area of the image of the subdisk $$|z|<r,\,0<r\le 1,$$ under an analytic function f in the unit disk $$|z|<1$$ . Without loss of generality, in this context, we consider only the analytic functions f in the unit disk with the normalization $$f(0)=0=f'(0)-1$$ . We set $$F_f(z)=z/f(z)$$ . Our objective in this paper is to obtain a sharp upper bound of $$\Delta (r,F_f),$$ when f varies over the class of normalized analytic univalent functions in the unit disk with quasiconformal extension to the entire complex plane.

Upper bounds on the locating chromatic number of trees

In this paper, we focus on a special chromatic invariant of graphs, so-called the locating chromatic number. We give a sharp upper bound of the locating chromatic number of trees by using the number of leaves.

Prime lattice points in ovals

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.

Upper bounds for strictly concave distortion risk measures on moment spaces

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces

Let $(M,\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue $\lambda_1$ of the Kohn-Laplacian $\Box_b$ on $(M,\theta)$. In the present paper, we give a sharp upper bound for $\lambda_1$, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when $M$ is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for $\lambda_1$ when the pseudohermitian structure $\theta$ is volume-normalized.

Some trace inequalities for exponential and logarithmic functions

Consider a function F(X, Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute $$F(X,Y)= X^pY^q.$$ Our first main result gives conditions on F such that $$\mathrm{Tr}[ X \log (F(Z,Y))] \le \mathrm{Tr}[X(p\log X + q \log Y)]$$ for all X, Y, Z such that $$\mathrm{Tr}Z =\mathrm{Tr}X$$ . (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X, Y, Z instead of just X, Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $$D(X||Y) = \mathrm{Tr}[X(\log X-\log Y])$$ , and two others, the Donald relative entropy $$D_D(X||Y)$$ , and the Belavkin–Stasewski relative entropy $$D_{BS}(X||Y)$$ . They are known to satisfy $$D_D(X||Y) \le D(X||Y)\le D_{BS}(X||Y)$$ . We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on $$\mathrm{Tr}[ X \log (F(Z,Y))]$$ in a number of cases in which F(Z, Y) is homogeneous of degree 1 in Z and −1 in Y. We also investigate the Legendre transforms in X of $$D_D(X||Y)$$ and $$D_{BS}(X||Y)$$ , and show how our results about these Legendre transforms lead to new refinements of the Golden–Thompson inequality.

The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in higher dimensions

This paper is concerned with the lifespan estimate of classical solutions with small initial data to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent. We establish the following sharp upper bound of the lifespanT(ε)≤exp⁡(Cε−2n) in higher dimensions (n≥4), by using the heat kernel as the test function. Then, together with the previous results, a complete result on the sharp lower and upper bound estimates is obtained in this case.

Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term (DW). The result is the sharp upper bound of lifespan of solution with respect to the small parameter $\ep$ when $p_F(N)\leq p\leq p_0(N+\mu)$, where $p_F(N)$ denotes the Fujita exponent for the nonlinear heat equations and $p_0(n)$ denotes the Strauss exponent for nonlinear wave equation in $n$-dimension with $\mu=0$. Consequently, by connecting the result of D'Abbicco--Lucente--Reissig 2015, our result clarifies the threshold exponent $p_0(N+\mu)$ for dividing blowup phenomena and global existence of small solutions when $N=3$. The crucial idea is to construct suitable test functions satisfying the conjugate linear equation $\pa_t^2\Phi-\Delta \Phi-\pa_t(\frac{\mu}{1+t}\Phi)=0$ of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou--Han in 2014.

Further results on non-self-centrality measures of graphs

For indicating the non-self-centrality extent of graphs, two new eccentricity-based measures namely third Zagreb eccentricity index E3(G) and non-self-centrality number N(G) of a connected graph G have recently been introduced as E3(G) = ?uv?E(G)|?G(u)-?G(v)| and N(G) = ? {u,v}?V(G) |?G(u)-?G(v)|, where ?G(u) denotes the eccentricity of a vertex u in G. In this paper, we find relation between the third Zagreb eccentricity index of graphs with some eccentricity-based invariants such as second Zagreb eccentricity index and second eccentric connectivity index. We also give sharp upper and lower bounds on the nonself-centrality number of graphs in terms of some structural parameters and relate it to two well-known eccentricity-based invariants namely total eccentricity and first Zagreb eccentricity index. Furthermore, we present exact expressions or sharp upper bounds on the third Zagreb eccentricity index and non-selfcentrality number of several graph operations such as join, disjunction, symmetric difference, lexicographic product, strong product, and generalized hierarchical product. The formulae for Cartesian product and rooted product as two important special cases of generalized hierarchical product and the formulae for corona product as a special case of rooted product are also given.

An upper bound for third Hankel determinant of starlike functions connected with $$k-$$ k - Fibonacci numbers

In this paper, we investigate the third Hankel determinant problem in some classes of analytic functions in the open unit disc connected with k-Fibonacci numbers $$F_{k,n}$$ $$(k>0)$$ . For this, first, we prove a conjecture, posed in Guney et al. (2017), for sharp upper bound of the second Hankel determinant. In the sequel, we obtain another sharp coefficient bound which we apply in solving the problem of the third Hankel determinant for these functions. Finally, we give an upper bound for the third Hankel determinant in this class. The results presented in the present paper have been shown to generalize and improve some recent work of Sokol et al. (2017).

The Laughlin liquid in an external potential

We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are N-particle wave functions that have the form of a product of Laughlin states and analytic functions of the N variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large N limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes.

Upper Bounds for Concave Distortion Risk Measures on Moment Spaces

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

A maximum problem of S.-T. Yau for variational p-capacity

Abstract Using the semidiameter (in connection to the mean radius and surface radius) of a convex closed hypersurface in ℝ n , n ≥ 2, as a sharp upper bound of the variational p-capacity radius where p ∈ (1,n), this paper settles a restriction or variant of S.-T. Yau’s Problem 59 in [66] from the surface area to the variational p-capacity whose limit as p → 1 actually induces the surface area.

On logarithmic coefficients of some close-to-convex functions

The logarithmic coefficients γ n \gamma _n of an analytic and univalent function f f in the unit disk D = { z ∈ C : | z | &gt; 1 } \mathbb {D}=\{z\in \mathbb {C}:|z|&gt;1\} with the normalization f ( 0 ) = 0 = f ′ ( 0 ) − 1 f(0)=0=f’(0)-1 are defined by log ⁡ f ( z ) z = 2 ∑ n = 1 ∞ γ n z n \log \frac {f(z)}{z}= 2\sum _{n=1}^{\infty } \gamma _n z^n . Recently, D. K. Thomas [Proc. Amer. Math. Soc. 144 (2016), 1681–1687] proved that | γ 3 | ≤ 7 12 |\gamma _3|\le \frac {7}{12} for functions in a subclass of close-to-convex functions (with argument 0 0 ) and claimed that the estimate is sharp by providing a form of an extremal function. In the present paper, we point out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument 0 0 ). We also determine a sharp upper bound of | γ 3 | |\gamma _3| for close-to-convex functions (with argument 0 0 ) with respect to the Koebe function.

Optimal Benchmark for Evaluating Drug-Combination Dose-Finding Clinical Trials

Numerous dose-finding methods have been proposed for drug-combination trials. A head-to-head comparison of the performance of these designs is difficult and often not very meaningful because different designs use different models and decision rules that often require judicious calibration to obtain good small-sample performance. It is desirable to have a general benchmark that can be used to evaluate the absolute performance of combination dose-finding designs. In this article, we propose an optimal nonparametric benchmark for evaluating drug-combination dose-finding methods, which provides an upper bound of accuracy beyond which further improvements are generally not achievable without making parametric assumptions of the dose-toxicity relationship. Our method is based on a new concept called critical information, which provides an upper bound on the information that we could possibly learn from patients while explicitly accounting for the partial order of the dose combinations, a fundamental feature of drug-combination trials. Our numerical study shows that the proposed benchmark provides a sharp upper bound that is useful for evaluating the performance of combination dose-finding designs.

Vertex-Addition Strategy for Domination-Like Invariants

In [J. Graph Theory 13 (1989) 749—762], McCuaig and Shepherd gave an upper bound of the domination number for connected graphs with minimum degree at least two. In this paper, we propose a simple strategy which, together with the McCuaig-Shepherd theorem, gives a sharp upper bound of the domination number via the number of leaves. We also apply the same strategy to other domination-like invariants, and find a relationship between such invariants and the number of leaves.

On $t$-Common List-Colorings

In this paper, we introduce a new variation of list-colorings. For a graph $G$ and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) = \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

On the First Strong Stability Eigenvalue of Closed Submanifolds in the Unit Sphere

We extend a sharp upper bound of the first strong stability eigenvalue recently due to Chen and Cheng (2016) for the context of closed submanifold $$M^n$$ immersed with nonzero parallel mean curvature vector field in the Euclidean unit sphere $$\mathbb S^{n+p}$$ and, as a consequence of this estimate, we obtain new characterizations for certain Clifford tori.

Generalized Zalcman conjecture for some classes of analytic functions

For functions f(z)=z+a2z2+a3z3+⋯ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional ϕ(f,n,m;λ):=|λanam−an+m−1|. For all real parameters λ and β<1, we provide the sharp upper bound of ϕ(f,n,m;λ) for functions f satisfying Ref′(z)>β and hence settle the open problem of estimating ϕ(f,n,m;λ) recently proposed by Agrawal and Sahoo (2016) [1]. For all real values of λ, the estimations of ϕ(f,n,m;λ) are provided for starlike and convex functions of order α(α<1) which are sharp for λ≤0 or for certain positive values of λ. Moreover, for certain positive λ, the sharp estimation of ϕ(f,n,m;λ) is given when f is a typically real function or a univalent function with real coefficients or is in some subclasses of close-to-convex functions.

Sharp Upper Bounds for Multiplicative Degree Distance of Graph Operations

In this paper, multiplicative version of degree distance of a graph is defined and tight upper bounds of the graph operations have been found.