Displaying similar documents to “A sharp bound for a sine polynomial”

Inequalities for two sine polynomials

Horst Alzer, Stamatis Koumandos (2006)

Colloquium Mathematicae

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We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have α j = 1 n - 1 1 / ( n ² - j ² ) s i n ( j x ) β , with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality 0 < j = 1 n - 1 ( n ² - j ² ) s i n ( j x ) 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].

Polynomial quotients: Interpolation, value sets and Waring's problem

Zhixiong Chen, Arne Winterhof (2015)

Acta Arithmetica

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For an odd prime p and an integer w ≥ 1, polynomial quotients q p , w ( u ) are defined by q p , w ( u ) ( u w - u w p ) / p m o d p with 0 q p , w ( u ) p - 1 , u ≥ 0, which are generalizations of Fermat quotients q p , p - 1 ( u ) . First, we estimate the number of elements 1 u < N p for which f ( u ) q p , w ( u ) m o d p 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...

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&amp;#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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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 | f | C | f | ϱ 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 ( 3 d - 3 ) n - 1 .

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

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Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality p ' [ - 1 , 1 ] 1 2 p [ - 1 , 1 ] for a constrained polynomial p of degree at most n , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval ( - 1 , 1 ) and establish a new asymptotically sharp inequality. ...

On the distribution of the roots of polynomial z k - z k - 1 - - z - 1

Carlos A. Gómez, Florian Luca (2021)

Commentationes Mathematicae Universitatis Carolinae

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We consider the polynomial f k ( z ) = z k - z k - 1 - - z - 1 for k 2 which arises as the characteristic polynomial of the k -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of f k ( z ) which lie inside the unit disk.

Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du, Carlos Martins da Fonseca (2023)

Czechoslovak Mathematical Journal

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We analyse the roots of the polynomial x n - p x n - 1 - q x - 1 for p q 1 . This is the characteristic polynomial of the recurrence relation F k , p , q ( n ) = p F k , p , q ( n - 1 ) + q F k , p , q ( n - k + 1 ) + F k , p , q ( n - k ) for n k , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

An alternative polynomial Daugavet property

Elisa R. Santos (2014)

Studia Mathematica

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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 m a x ω | | I d + ω P | | = 1 + | | P | | . We study the stability of the APDP by c₀-, - and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely L ( μ , X ) and C(K,X), where X has the APDP.

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

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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...

Moser-Trudinger and logarithmic HLS inequalities for systems

Itai Shafrir, Gershon Wolansky (2005)

Journal of the European Mathematical Society

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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 Ω 2 and on all of 2 . In some cases we also address the question of existence of minimizers.

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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We show that for a polynomial mapping F = ( f , . . . , f ) : n m the Łojasiewicz exponent ( F ) of F is attained on the set z n : f ( z ) · . . . · f ( z ) = 0 .

Weak polynomial identities and their applications

Vesselin Drensky (2021)

Communications in Mathematics

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Let R be an associative algebra over a field K generated by a vector subspace V . The polynomial f ( x 1 , ... , x n ) of the free associative algebra K x 1 , x 2 , ... is a weak polynomial identity for the pair ( R , V ) if it vanishes in R when evaluated on V . We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of...

Lower bounds for Schrödinger operators in H¹(ℝ)

Ronan Pouliquen (1999)

Studia Mathematica

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We prove trace inequalities of type | | u ' | | L 2 2 + j k j | u ( a j ) | 2 λ | | u | | L 2 2 where u H 1 ( ) , under suitable hypotheses on the sequences a j j and k j j , with the first sequence increasing and the second bounded.

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study the extensibility of piecewise polynomial functions defined on closed subsets of 2 to all of 2 . The compact subsets of 2 on which every piecewise polynomial function is extensible to 2 can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of . 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...

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations h ( x ) = = h r ( x ) = 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 ( x ) = i = 1 r h ² i ( x ) σ i ( x ) , where σ i are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

Polynomials with values which are powers of integers

Rachid Boumahdi, Jesse Larone (2018)

Archivum Mathematicum

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Let P be a polynomial with integral coefficients. Shapiro showed that if the values of P at infinitely many blocks of consecutive integers are of the form Q ( m ) , where Q is a polynomial with integral coefficients, then P ( x ) = Q ( R ( x ) ) for some polynomial R . In this paper, we show that if the values of P at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form m q where q is an integer greater than 1, then P ( x ) = ( R ( x ) ) q for some polynomial R ( x ) .

On Bernstein inequalities for multivariate trigonometric polynomials in L p , 0 p

Laiyi Zhu, Xingjun Zhao (2022)

Czechoslovak Mathematical Journal

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Let 𝕋 n be the space of all trigonometric polynomials of degree not greater than n with complex coefficients. Arestov extended the result of Bernstein and others and proved that ( 1 / n ) T n ' p T n p for 0 p and T n 𝕋 n . We derive the multivariate version of the result of Golitschek and Lorentz T n cos α + 1 n T n sin α l ( m ) p T n p , 0 p for all trigonometric polynomials (with complex coeffcients) in m variables of degree at most n .

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space Λ 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 Λ p splits as the direct sum d * ( Λ p + 1 ) δ * ( Λ p - 1 ) H p , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial p ( z ) = c n z n + j = μ n c n - j z n - j , 1 μ n , having all its zeros on | z | = k , k 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),...

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

Lieb–Thirring inequalities on the half-line with critical exponent

Tomas Ekholm, Rupert Frank (2008)

Journal of the European Mathematical Society

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We consider the operator - d 2 / d r 2 - V in L 2 ( + ) with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound tr ( - d 2 / d r 2 - V ) - γ C γ , α + ( V ( r ) - 1 / ( 4 r 2 ) ) + γ + ( 1 + α ) / 2 r α d r for any α [ 0 , 1 ) and γ ( 1 - α ) / 2 . This includes a Lieb-Thirring inequality in the critical endpoint case.

Entire functions of exponential type not vanishing in the half-plane z > k , where k > 0

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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