Displaying similar documents to “On the topology of polynomials with bounded integer coefficients”

L p inequalities for the growth of polynomials with restricted zeros

Nisar A. Rather, Suhail Gulzar, Aijaz A. Bhat (2022)

Archivum Mathematicum

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Let P ( z ) = ν = 0 n a ν z ν be a polynomial of degree at most n which does not vanish in the disk | z | < 1 , then for 1 p < and R > 1 , Boas and Rahman proved P ( R z ) p ( R n + z p / 1 + z p ) P p . In this paper, we improve the above inequality for 0 p < by involving some of the coefficients of the polynomial P ( z ) . Analogous result for the class of polynomials P ( z ) having no zero in | z | > 1 is also given.

On square functions associated to sectorial operators

Christian Le Merdy (2004)

Bulletin de la Société Mathématique de France

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We give new results on square functions x F = 0 F ( t A ) x 2 d t t 1 / 2 p associated to a sectorial operator A on L p for 1 &lt; p &lt; . Under the assumption that A is actually R -sectorial, we prove equivalences of the form K - 1 x G x F K x G for suitable functions F , G . We also show that A has a bounded H functional calculus with respect to . F . Then we apply our results to the study of conditions under which we have an estimate ( 0 | C e - t A ( x ) | 2 d t ) 1 / 2 q M x p , when - A generates a bounded semigroup e - t A on L p and C : D ( A ) L q is a linear mapping.

Geometric rigidity of × m invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let μ be a probability measure on [ 0 , 1 ] which is invariant and ergodic for T a ( x ) = a x 𝚖𝚘𝚍 1 , and 0 < 𝚍𝚒𝚖 μ < 1 . Let f be a local diffeomorphism on some open set. We show that if E and ( f μ ) E μ E , then f ' ( x ) ± a r : r at μ -a.e. point x f - 1 E . In particular, if g is a piecewise-analytic map preserving μ then there is an open g -invariant set U containing supp μ such that g U is piecewise-linear with slopes which are rational powers of a . In a similar vein, for μ as above, if b is another integer and a , b are not powers of a common integer, and if ν is...

Neutral set differential equations

Umber Abbas, Vasile Lupulescu, Donald O&amp;#039;Regan, Awais Younus (2015)

Czechoslovak Mathematical Journal

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The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type D H X ( t ) = F ( t , X t , D H X t ) , X | [ - r , 0 ] = Ψ , where F : [ 0 , b ] × 𝒞 0 × 𝔏 0 1 K c ( E ) is a given function, K c ( E ) is the family of all nonempty compact and convex subsets of a separable Banach space E , 𝒞 0 denotes the space of all continuous set-valued functions X from [ - r , 0 ] into K c ( E ) , 𝔏 0 1 is the space of all integrally bounded set-valued functions X : [ - r , 0 ] K c ( E ) , Ψ 𝒞 0 and D H is the Hukuhara derivative. The continuous dependence of solutions on initial...

A Hardy type inequality for W 0 m , 1 ( Ω ) functions

Hernán Castro, Juan Dávila, Hui Wang (2013)

Journal of the European Mathematical Society

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We consider functions u W 0 m , 1 ( Ω ) , where Ω N is a smooth bounded domain, and m 2 is an integer. For all j 0 , 1 k m - 1 , such that 1 j + k m , we prove that i u ( x ) d ( x ) m - j - k W 0 k , 1 ( Ω ) with k ( i u ( x ) d ( x ) m - j - k ) L 1 ( Ω ) C u W m , 1 ( Ω ) , where d is a smooth positive function which coincides with dist ( x , Ω ) near Ω , and l denotes any partial differential operator of order l .

Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Hongbin Wang (2016)

Czechoslovak Mathematical Journal

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Let Ω L s ( S n - 1 ) for s 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μ Ω and b is defined by [ b , μ Ω ] ( f ) ( x ) = ( 0 | x - y | t Ω ( x - y ) | x - y | n - 1 [ b ( x ) - b ( y ) ] f ( y ) d y | 2 d t t 3 1 / 2 . In this paper, the author proves the ( L p ( · ) ( n ) , L p ( · ) ( n ) ) -boundedness of the Marcinkiewicz integral operator μ Ω and its commutator [ b , μ Ω ] when p ( · ) satisfies some conditions. Moreover, the author obtains the corresponding result about μ Ω and [ b , μ Ω ] on Herz spaces with variable exponent.

A Diophantine inequality with four squares and one k th power of primes

Quanwu Mu, Minhui Zhu, Ping Li (2019)

Czechoslovak Mathematical Journal

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Let k 5 be an odd integer and η be any given real number. We prove that if λ 1 , λ 2 , λ 3 , λ 4 , μ are nonzero real numbers, not all of the same sign, and λ 1 / λ 2 is irrational, then for any real number σ with 0 < σ < 1 / ( 8 ϑ ( k ) ) , the inequality | λ 1 p 1 2 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 2 + μ p 5 k + η | < max 1 j 5 p j - σ has infinitely many solutions in prime variables p 1 , p 2 , , p 5 , where ϑ ( k ) = 3 × 2 ( k - 5 ) / 2 for k = 5 , 7 , 9 and ϑ ( k ) = [ ( k 2 + 2 k + 5 ) / 8 ] for odd integer k with k 11 . This improves a recent result in W. Ge, T. Wang (2018).