Relations between upper bounds of absolute values of functions and their higher derivatives

In this paper we consider functionsf(t), – < t < , which are n times continuously differentiable with a given convex modulus of continuity of the n-th derivative. For a certain class of periodic functions we establish a relationship between upper bounds of the absolute values of a function and its n-th derivative.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 329–338, September, 1973.     

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Let C( \mathbbR^m ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbR^m ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm

Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities

m·inf{‖f‖_L^∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖_L^∞(S):f(A)=p(A)}     

Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rⁿ, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rⁿ) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N(r)?A_nrⁿ+o(rⁿ) with an explicit A_n. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.     

Sharp upper bounds on the distance spectral radius of a graph

Let M=(_mij)$M=(m_{ij})$ be a nonnegative irreducible n×n$n\times n$ matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M  , denoted by ρ(M)$\rho (M)$. In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph.     

Sharp upper bounds on the second largest eigenvalues of connected graphs

Let ${\lambda }_2$ be the second largest eigenvalue of a graph. Powers (1988) [4] gave some upper bounds of ${\lambda }_2$ for general graphs and bipartite graphs, respectively. Considering that these bounds are not always attainable for connected graphs, we present sharp upper bounds of ${\lambda }_2$ for connected graphs and connected bipartite graphs in this paper. Moreover, the extremal graphs are completely characterized.     

Sharp upper bounds of the spectral radius of a graph

Let $G$ be a simple connected graph with $n$ vertices and $m$ edges. The spectral radius $\rho \left(G\right)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of $\rho \left(G\right)$ which improves some known bounds: If $\frac{(k?2)(k?3)}{2}\le m?n\le \frac{k(k?3)}{2}$, where $k(3\le k\le n)$ is an integer, then $\rho \left(G\right)\le \sqrt{2m?n?k+\frac{5}{2}+\sqrt{2m?2n+\frac{9}{4}}}.$The equality holds if and only if $G$ is a complete graph $K_n$ or $K_4?e$, where $K_4?e$ is the graph obtained from $K_4$ by deleting some edge $e$.     

Lower and upper bounds for characteristic functions

In the paper, several known inequalities for characteristic functions are sharpened and several new ones are derived. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

Order estimates of the norms of derivatives of functions with zero values on linear functionals and their applications

For a given finite set of linear functionals we construct functions vanishing on them and give order estimates of their derivatives. We also give their different applications.     

Exact upper bounds for the number of the polynomials with given discriminants

We generalize the result of Davenport on the sum of absolute values of discriminants of integer polynomials of degree three. For the first time, we find the exact upper bound for the number of polynomials with given discriminant in the class of cubic polynomials of bounded height for the whole interval of values of discriminants.