We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $\alpha \le {\sum }_{j=1}^{n-1}1/(n²-j²)sin\left(jx\right)\le \beta$, with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0<{\sum }_{j=1}^{n-1}(n²-j²)sin\left(jx\right)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p,w}\left(u\right)$ are defined by $q_{p,w}\left(u\right)\equiv (u^w-u^{wp})/pmodp$ with $0\le q_{p,w}\left(u\right)\le p-1$, u ≥ 0, which are generalizations of Fermat quotients $q_{p,p-1}\left(u\right)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f\left(u\right)\equiv q_{p,w}\left(u\right)modp$ for a given polynomial f(x) over the finite field ${}_p$. In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...

We extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in $L_p$, p > -2.

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $ma{x}_{\omega \in }\left|\right|Id+\omega P\left|\right|=1+\left|\right|P\left|\right|$. We study the stability of the APDP by c₀-, ${\ell }_{\infty }$- and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty }(\mu ,X)$ and C(K,X), where X has the APDP.

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of...

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.

We show that for a polynomial mapping $F=(f₁,...,fₘ):{ℂ}^n\to {ℂ}^m$ the Łojasiewicz exponent ${}_{\infty }\left(F\right)$ of F is attained on the set $z\in {ℂ}^n:f₁\left(z\right)·...·fₘ\left(z\right)=0$.

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.

We study the extensibility of piecewise polynomial functions defined on closed subsets of ${ℝ}^2$ to all of ${ℝ}^2$. The compact subsets of ${ℝ}^2$ on which every piecewise polynomial function is extensible to ${ℝ}^2$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁\left(x\right)=\cdots =h_r\left(x\right)=0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f|}_V$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h\left(x\right)={\sum }_{i=1}^{r}h{²}_i\left(x\right){\sigma }_i\left(x\right)$, where ${\sigma }_i$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

In the space ${\Lambda }^p$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^p$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that ${\Lambda }^p$ splits as the direct sum $d*\left({\Lambda }^{p+1}\right)\oplus \delta *\left({\Lambda }^{p-1}\right)\oplus H^p$, where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

In this paper we obtain certain results for the polar derivative of a polynomial $p\left(z\right)=c_n{z}^n+{\sum }_{j=\mu }^n{c}_{n-j}{z}^{n-j}$, $1\le \mu \le n$, having all its zeros on $\left|z\right|=k$, $k\le 1$, which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

We consider the operator $-d^2/d{r}^2-V$ in $L_2\left({ℝ}_{+}\right)$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $\mathrm{tr}{(-d^2/d{r}^2-V)}_{-}^{\gamma }\le C_{\gamma ,\alpha }{\int }_{{ℝ}_{+}}{(V\left(r\right)-1/\left(4r^2\right))}_{+}^{\gamma +(1+\alpha )/2}{r}^{\alpha }dr$ for any $\alpha \in [0,1)$ and $\gamma \ge (1-\alpha )/2$. This includes a Lieb-Thirring inequality in the critical endpoint case.

Let $P\left(z\right)$ be a polynomial of degree $n$ having no zeros in $\left|z\right|<k$, $k\le 1$, and let $Q\left(z\right):=z^n\overline{P(1/\overline{z})}$. It was shown by Govil that if ${max}_{\left|z\right|=1}\left|P^{\text{'}}\left(z\right)\right|$ and ${max}_{\left|z\right|=1}\left|Q^{\text{'}}\left(z\right)\right|$ are attained at the same point of the unit circle $\left|z\right|=1$, then $$\underset{\left|z\right|=1}{max}|P^{\text{'}}\left(z\right)|\le \frac{n}{1+k^n}\underset{\left|z\right|=1}{max}\left|P\left(z\right)\right|.$$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

We consider the space ${}^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on ${}^{ℳ}$. A description of the space ${(}^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Let $f=ax+b{x}^q+x^{2q-1}{\in }_q\left[x\right]$. We find explicit conditions on a and b that are necessary and sufficient for f to be a permutation polynomial of ${}_{q²}$. This result allows us to solve a related problem: Let $g_{n,q}{\in }_p\left[x\right]$ (n ≥ 0, $p=cha{r}_q$) be the polynomial defined by the functional equation ${\sum }_{c{\in }_q}{(x+c)}^n=g_{n,q}(x^q-x)$. We determine all n of the form $n=q^{\alpha }-q^{\beta }-1$, α > β ≥ 0, for which $g_{n,q}$ is a permutation polynomial of ${}_{q²}$.

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $\left(J_{\infty }\right)$: There does not exist a sequence $(p_k,q_k)\subset ℂ²$ of complex singular points of F such that the imaginary parts $(\Im \left(p_k\right),\Im \left(q_k\right))$ tend to (0,0), the real parts $(\Re \left(p_k\right),\Re \left(q_k\right))$ tend to ∞ and $F(\Re \left(p_k\right),\Re \left(q_k\right)))\to a\in ℝ²$. It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $\left(J_{\infty }\right)$ and if, in addition, the restriction of F to every real level set $P^{-1}\left(c\right)$ is proper for values of |c| large enough.

Let $X={\sum }_{i=1}^k{a}_i{U}_i$, $Y={\sum }_{i=1}^k{b}_i{U}_i$, where the $U_i$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_i,b_i$ are real constants. We prove that if $b{²}_i$ is majorized by $a{²}_i$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^p$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

Let $f_c(x,y)\equiv {\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }{a}_{jk}(1-cosjx)(1-cosky)$ with $a_{jk}\ge 0$ for all j,k ≥ 1. We estimate the integral ${\int }_0^{\pi }{\int }_0^{\pi }{x}^{\alpha -1}{y}^{\beta -1}\varphi \left(f_c(x,y)\right)dxdy$ in terms of the coefficients $a_{jk}$, where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].

We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f\left(x\right),g\left(x\right){\in }_p\left[x\right]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^{1/E(1+\kappa /(1-\kappa )}>M>p^{\epsilon }$, where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $\left|f\left([0,M]\right)\cap g\left([0,M]\right)\right|\lesssim M^{1-\epsilon }$.

We establish several mixed $A_p-A_{\infty }$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_{\infty }$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily...

We determine the optimal constants $C_{p,q}$ in the moment inequalities ${\left|\right|g\left|\right|}_p\le C_{p,q}{\left|\right|f\left|\right|}_q$, 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials $u₀,...,u_{m-1}\in K[X₀,...,X_{m-1}]$ such that $f\left({\sum }_{j=0}^{m-1}{\xi }^j{X}_j\right)={\sum }_{j=0}^{m-1}{\xi }^j{u}_j$. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u₀,...,u_{m-1}$ have no common divisor in $K[X₀,...,X_{m-1}]$ of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several...