Abstract: Publication date: Available online 13 December 2017 Source:Comptes Rendus Mathematique Author(s): Edmond W.H. Lee Over the years, several finite semigroups have been found to generate varieties with continuum many subvarieties. However, finite involution semigroups that generate varieties with continuum many subvarieties seem much rarer; in fact, only one example—an inverse semigroup of order 165—has so far been published. Nevertheless, it is shown in the present article that there are many smaller examples among involution semigroups that are unstable in the sense that the varieties they generate contain some involution semilattice with nontrivial unary operation. The most prominent examples are the unstable finite involution semigroups that are inherently non-finitely based, the smallest ones of which are of order six. It follows that the join of two finitely generated varieties of involution semigroups with finitely many subvarieties can contain continuum many subvarieties.

Abstract: Publication date: Available online 12 December 2017 Source:Comptes Rendus Mathematique Author(s): Davide Barbieri, Carlos Cabrelli, Eugenio Hernández, Peter Luthy, Ursula Molter, Carolina Mosquera In this note, we investigate the existence of frames of exponentials for L 2 ( Ω ) in the setting of LCA groups. Our main result shows that sub-multitiling properties of Ω ⊂ G ˆ with respect to a uniform lattice Γ of G ˆ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of Γ. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.

Abstract: Publication date: Available online 8 December 2017 Source:Comptes Rendus Mathematique Author(s): Andrei K. Lerner We show that the L 2 ( w ) operator norm of the composition M ∘ T Ω , where M is the maximal operator and T Ω is a rough homogeneous singular integral with angular part Ω ∈ L ∞ ( S n − 1 ) , depends quadratically on [ w ] A 2 , and that this dependence is sharp.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Yu Yang In the present paper, we study the ordinariness of coverings of stable curves. Let f : Y → X be a morphism of stable curves over a discrete valuation ring R with algebraically closed residue field of characteristic p > 0 . Write S for Spec R and η (resp. s) for the generic point (resp. closed point) of S. Suppose that the generic fiber X η of X is smooth over η, that the morphism f η : Y η → X η over η on the generic fiber induced by f is a Galois étale covering (hence Y η is smooth over η too) whose Galois group is a solvable group G, that the genus of the normalization of each irreducible component of the special fiber X s is ≥2, and that Y s is ordinary. Then we have that the morphism f s : Y s → X s over s induced by f is an admissible covering. This result extends a result of M. Raynaud concerning the ordinariness of coverings to the case where X s is a stable curve. If, moreover, we suppose that G is a p-group, and that the p-rank of the normalization of each irreducible component of X s is ≥2, we can give a numerical criterion for the admissibility of f s .

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Quan-Hui Yang, Qing-Qing Zhao Let q be a positive integer. Recently, Niu and Liu proved that, if n ≥ max { q , 1198 − q } , then the product ( 1 3 + q 3 ) ( 2 3 + q 3 ) ⋯ ( n 3 + q 3 ) is not a powerful number. In this note, we prove (1) that, for any odd prime power ℓ and n ≥ max { q , 11 − q } , the product ( 1 ℓ + q ℓ ) ( 2 ℓ + q ℓ ) ⋯ ( n ℓ + q ℓ ) is not a powerful number, and (2) that, for any positive odd integer ℓ, there exists an integer N q , ℓ such that, for any positive integer n ≥ N q , ℓ , the product ( 1 ℓ + q ℓ ) ( 2 ℓ + q ℓ ) ⋯ ( n ℓ + q ℓ ) is not a powerful number.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Anshul Adve, Alexander Yong We discuss implications of the following statement about representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation and every nonnegative integer appears infinitely often as a Littlewood–Richardson coefficient and as a Kronecker coefficient.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Tim Seynnaeve Motivated by the symmetric version of matrix multiplication we study the plethysm S k ( sl n ) of the adjoint representation sl n of the Lie group S L n . In particular, we describe the decomposition of this representation into irreducible components for k = 3 , and find highest-weight vectors for all irreducible components. Relations to fast matrix multiplication, in particular the Coppersmith–Winograd tensor, are presented.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Ivica Martinjak, Riste Škrekovski We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a consequence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Cornelius Greither, Toufik Zaïmi Let K be a noncyclotomic CM field. We show that the field K ∩ R has a reciprocal unit-primitive element when K does. Also, we prove some related conditions that make the converse of this assertion true.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Hsuan-Yi Liao, Mathieu Stiénon, Ping Xu We establish a formality theorem for smooth dg manifolds. More precisely, we prove that, for any finite-dimensional dg manifold ( M , Q ) , there exists an L ∞ quasi-isomorphism of dglas from ( T poly • ⊕ ( M ) , [ Q , − ] , [ − , − ] ) to ( D poly • ⊕ ( M ) , 〚 m + Q , − 〛 , 〚 − , − 〛 ) whose first Taylor coefficient (1) is equal to the composition hkr ∘ ( td ( M , Q ) ∇ ) 1 2 : T poly • ⊕ ( M ) → D poly • ⊕ ( M ) of the action of ( td ( M , Q ) ∇ ) 1 2 ∈ ∏ k ≥ 0 ( Ω k ( M ) ) k on T poly • ⊕ ( M ) (by contraction) with the Hochschild–Kostant–Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we prove the Kontsevich–Shoikhet conjecture: a Kontsevich–Duflo-type theorem holds for all finite-dimensional smooth dg manifolds.

Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Huibin Chen, Zhiqi Chen, Shaoqiang Deng In this article, we prove that the compact simple Lie groups S U ( n ) for n ≥ 6 , S O ( n ) for n ≥ 7 , S p ( n ) for n ≥ 3 , E 6 , E 7 , E 8 , and F 4 admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Farid Bencherif, Benali Benzaghou, Schehrazade Zerroukhat Dans cet article, nous établissons une identité pour des polynômes d'Appell généralisant des formules explicites pour les nombres et polynômes de Bernoulli généralisés. In this paper, we establish an identity for some Appell polynomials generalizing explicit formulas for generalized Bernoulli numbers and polynomials.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Marco Antonio Armenta, Bernhard Keller We prove the derived invariance of the cap product for associative algebras projective over a commutative ring.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Nadia Drisi, Brahim Es-sebbar We obtain sufficient conditions for the existence and uniqueness of a positive compact almost automorphic solution to a logistic equation with discrete and continuous delay. Moreover, we provide a counterexample to some results in literature which deal with the uniqueness of almost periodic solutions to logistic type equations.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Armand Koenig We are interested in the exact null controllability of the equation ∂ t f − ∂ x 2 f − x 2 ∂ y 2 f = 1 ω u , with control u supported on ω. We show that, when ω does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an L 2 estimate on polynomials, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator − ∂ x 2 + ( n x ) 2 with Dirichlet conditions on ( − 1 , 1 ) , and we show a quite precise estimation it satisfies, even when n is in some complex domain.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Abbas Moameni A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Dana Bartošová, Jordi Lopez-Abad, Martino Lupini, Brice Mbombo We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group Aut ( G ) of surjective linear isometries of the Gurarij space G is extremely amenable, and that the canonical action Aut ( P ) ↷ P is the universal minimal flow of the group Aut ( P ) of affine homeomorphisms of the Poulsen simplex P . This answers questions of Melleray–Tsankov and Conley–Törnquist.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Sander Cornelis Hille, Tomasz Szarek, Maria Aleksandra Ziemlańska The relation between the equicontinuity – the so-called e-property – and the stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is non-empty satisfies the e-property.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Gengsheng Wang, Donghui Yang, Yubiao Zhang In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Gianfranco Casnati, Yeongrak Kim We deal with the behaviour of Ulrich bundles with respect to push-forward and pull-back via blowing-up points. We also correct a wrong statement in [11].

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Arnaud Duran, Jean-Paul Vila, Rémy Baraille We present a semi-implicit scheme for a two-dimensional multilayer shallow water system with density stratification, formulated on general staggered meshes. The main result of the present note concerns the control of the mechanical energy at the discrete level, principally based on advective fluxes implying a diffusion term expressed in terms of the gradient pressure. The scheme is also designed to capture the dynamics of low-Froude-number regimes and offers interesting positivity and well-balancing results. A numerical test is proposed to highlight the scheme's efficiency in the one-layer case.

Abstract: Publication date: December 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 12 Author(s): Christoph Schumacher, Ivan Veselić We prove a Lifshitz tail bound on the integrated density of states of random breather Schrödinger operators. The potential is composed of translated single-site potentials. The single-site potential is an indicator function of the set tA where t is from the unit interval and A is a measurable set contained in the unit cell. The challenges of this model are that, since A is not assumed to be star-shaped, the dependence of the potential on the parameter t is not monotone. It is also non-linear and not differentiable.

Abstract: Publication date: Available online 24 November 2017 Source:Comptes Rendus Mathematique Author(s): Caiyun Fang, Yan Xu Let A > 1 be a constant, and let F be a family of meromorphic functions in a domain D. If, for every function f ∈ F , f has only zeros of multiplicity at least 2 and satisfies the following conditions: (1) f ( z ) = 0 ⇒ f ″ ( z ) ≤ A z , (2) f ″ ( z ) ≠ z , (3) all poles of f have multiplicity at least 4, then F is normal in D. In this paper, we first give an example to show that condition (3) is sharp, and prove that our counterexample is unique in some sense.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Sumire Sawada, Simona Settepanella, So Yamagata We show that points in specific degree-2 hypersurfaces in the Grassmannian G r ( 3 , n ) correspond to generic arrangements of n hyperplanes in C 3 with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Benjamin Linowitz, D.B. McReynolds, Paul Pollack, Lola Thompson In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any k ≥ 2 , we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Guoyou Qian Let { a i } i = 1 ∞ be a strictly increasing sequence of positive integers ( a i < a j if i < j ). In 1978, Borwein showed that for any positive integer n, we have ∑ i = 1 n 1 lcm ( a i , a i + 1 ) ≤ 1 − 1 2 n , with equality occurring if and only if a i = 2 i − 1 for 1 ≤ i ≤ n + 1 . Let 3 ≤ r ≤ 7 be an integer. In this paper, we investigate the sum ∑ i = 1 n 1 lcm ( a i , . . . , a i + r − 1 ) and show that ∑ i = 1 n 1 lcm ( a i , . . . , a i + r − 1 ) ≤ U r ( n ) for any positive integer n, where U r ( n ) is a constant depending on r and n. Further, for any integer n ≥ 2 , we also give a characterization of the sequence { a i } i = 1 ∞ such that the equality ∑ i = 1 n 1 lcm ( a i , . . . , a i + r − 1 ) = U r ( n ) holds.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Masoud Hassani In this paper, we study the irreducible representation of PSL ( 2 , R ) in PSL ( 5 , R ) . This action preserves a quadratic form with signature ( 2 , 3 ) . Thus, it acts conformally on the 3-dimensional Einstein universe E in 1 , 2 . We describe the orbits induced in E in 1 , 2 and its complement in R P 4 . This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on E in 1 , 2 [5].

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Anton Alekseev, Florian Naef For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space C π 1 ( Σ ) / [ C π 1 ( Σ ) , C π 1 ( Σ ) ] carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in H 1 ( Σ ) and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem. In this note, we give an elementary proof of this isomorphism over C . It uses the Knizhnik–Zamolodchikov connection on C \ { z 1 , … z n } . We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets. Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Karam Allali The existence and the uniqueness of solutions to a problem of miscible liquids are investigated in this note. The model consists of Navier–Stokes equations with Korteweg stress terms coupled with the reaction–diffusion equation for the concentration. We assume that the fluid is incompressible and the Boussinesq approximation is adopted. The global existence and uniqueness of solutions is established for some optimal conditions on the reaction source term and the external force functions.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Félix del Teso, Jørgen Endal, Espen R. Jakobsen We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of (0.1) ∂ t u − L σ , μ [ φ ( u ) ] = g ( x , t ) in R N × ( 0 , T ) , where φ is merely continuous and nondecreasing, and L σ , μ is the generator of a general symmetric Lévy process. This means that L σ , μ can have both local and nonlocal parts like, e.g., L σ , μ = Δ − ( − Δ ) 1 2 . New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for L σ , μ . Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Anthony T. Patera, Masayuki Yano We extend the linear program empirical quadrature procedure proposed in [9] and subsequently [3] to the case in which the functions to be integrated are associated with a parametric manifold. We pose a discretized linear semi-infinite program: we minimize as objective the sum of the (positive) quadrature weights, an ℓ 1 norm that yields sparse solutions and furthermore ensures stability; we require as inequality constraints that the integrals of J functions sampled from the parametric manifold are evaluated to accuracy δ ¯ . We provide an a priori error estimate and numerical results that demonstrate that under suitable regularity conditions, the integral of any function from the parametric manifold is evaluated by the empirical quadrature rule to accuracy δ ¯ as J → ∞ . We present two numerical examples: an inverse Laplace transform; reduced-basis treatment of a nonlinear partial differential equation.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Mohammad Eslamian In this paper, we introduce an explicit parallel algorithm for finding common solutions to a system of variational inequalities over the set of common fixed points of a finite family of demi-contractive operators. Under suitable assumptions, we prove the strong convergence of this algorithm in the framework of a Hilbert space. The results obtained in this paper extend and improve the results of Tian and Jiang (2017), of Censor, Gibali and Reich (2012), and of many others.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Francesco D'Andrea, Thomas Weber We present a simple no-go theorem for the existence of a deformation quantization of a homogeneous space M induced by a Drinfel'd twist: we argue that equivariant line bundles on M with non-trivial Chern class and symplectic twist star products cannot both exist on the same manifold M. This implies, for example, that there is no symplectic star product on the projective space CP n − 1 induced by a twist based on U ( gl n ( C ) ) 〚 h 〛 or any sub-bialgebra, for every n ≥ 2 .

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Edson de Faria, Peter Hazard, Charles Tresser In 1980, Yano showed that on smooth compact manifolds, for endomorphisms in dimension one or above and homeomorphisms in dimensions greater than one, topological entropy is generically infinite. It had earlier been shown that, for Lipschitz endomorphisms on such spaces, topological entropy is always finite. In this article, we investigate what occurs between C 0 -regularity and Lipschitz regularity, focussing on two cases: Hölder mappings and Sobolev mappings.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Emmanuel Rio In this note, we give normal approximation results for the conditional value at risk (CVaR) of partial sums of random variables satisfying moment assumptions. These results are based on Berry–Esseen-type bounds for transport costs in the central limit theorem and extensions of Cantelli's inequalities to the CVaR.

Abstract: Publication date: November 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 11 Author(s): Maria Malin, Cristinel Mardare A sufficiently regular hypersurface immersed in the ( n + 1 ) -dimensional Euclidean space is determined up to a proper isometry of R n + 1 by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. We establish here stronger versions of these uniqueness results by means of inequalities showing that an appropriate distance between two immersions from a domain ω of R n into R n + 1 is bounded by the L p -norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion.

Abstract: Publication date: October 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 10 Author(s): Eusebio Gardella, Martino Lupini We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris.

Abstract: Publication date: October 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 10 Author(s): Quy Thuong Lê In this note, using Cluckers–Loeser's theory of motivic integration, we prove the integral identity conjecture with framework a localized Grothendieck ring of varieties over an arbitrary base field of characteristic zero.

Abstract: Publication date: October 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 10 Author(s): Hoai-Minh Nguyen We sharpen an estimate of [4] for the topological degree of continuous maps from a sphere S d into itself in the case d ≥ 2 . This provides the answer for d ≥ 2 to a question raised by Brezis. The problem is still open for d = 1 .

Abstract: Publication date: October 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 10 Author(s): Dorina Răducanu, Paweł Zaprawa So far, the sharp bound of the expression a 2 a 4 − a 3 2 for the class C of close-to-convex functions has remained unknown. In this paper, we obtain the estimation of this expression, called the second Hankel determinant, for C 0 , i.e. the subset of C consisting of functions f that satisfy in the unit disk the inequality Re ( z f ′ ( z ) / g ( z ) ) > 0 with a starlike function g. Moreover, some remarks on the second Hankel determinant for the class S of univalent functions are made. It is proven that max { a 2 a 4 − a 3 2 : f ∈ S } is greater than 1.

Abstract: Publication date: Available online 9 October 2017 Source:Comptes Rendus Mathematique Author(s): Rafał Czyż, Van Thien Nguyen In this note, we prove that the constant D ( p , m ) in the energy estimate, for m-subharmonic function with bounded p-energy, is strictly bigger than 1, for p > 0 , p ≠ 1 .

Abstract: Publication date: Available online 9 October 2017 Source:Comptes Rendus Mathematique Author(s): Abdullah Mir, Imtiaz Hussain Let P ( z ) be a polynomial of degree n and for any complex number α, let D α P ( z ) : = n P ( z ) + ( α − z ) P ′ ( z ) denote the polar derivative of P ( z ) with respect to α. In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem.