Acta Mathematica Hungarica最新文献

Finding the sharp estimate of (max_{|z|=1} |p'(z)|) in terms of (max_{|z|=1} |p(z)|) for the class of polynomials p(z) satisfying (p(z) equiv z^n p(1/z)) has been a well-known open problem for a long time and many papers in this direction have appeared. The earliest result is due to Govil, Jain and Labelle [9] who proved that for polynomials p(z) satisfying (p(z) equiv z^n p(1/z)) and having all the zeros either in left half or right half-plane, the inequality (max_{|z|=1} |p'(z)| le frac{n}{sqrt{2}} max_{|z|=1} |p(z)|) holds. A question was posed whether this inequality is sharp. In this paper, we answer this question in the negative by obtaining a bound sharper than (frac{n}{sqrt{2}}). We also conjecture that for such polynomials

and provide evidence in support of this conjecture.

We introduce some new weighted maximal operators of the Fejér means of the Walsh–Fourier series. We prove that for some "optimal" weights these new operators are bounded from the martingale Hardy space (H_{p}(G)) to the space (text{weak-}L_{p}(G)) , for (0<p<1/2). Moreover, we also prove sharpness of this result. As a consequence we obtain some new and well-known results.

We establish a Hardy–Sobolev inequality for Riesz potentials of functions in Orlicz spaces.

Let (G) be a finite subgroup of ( rm PGL_3(mathbb C)), and let (sigma) be the generatorof Gal((mathbb C/ mathbb R)). We say that (G) has a real field of moduli if (sigma G) and (G) are( rm PGL_3(mathbb C))-conjugates. Furthermore, we say that (mathbb R) is a field of definition for (G) orthat (G) is definable over (mathbb R) if (G) is (textrm{PGL}_3(mathbb C))-conjugate to some (acute{G} ,subset , PGL_3(mathbb R)). Inthis situation, we call (acute {G}) a model for (G) over (mathbb R). On the other hand, if (G) has areal field of moduli but is not definable over (mathbb R), then we call (G) pseudo-real.

In this paper, we first show that any finite cyclic subgroup (G = mathbb Z / n mathbb Z) in( rm PGL_3(mathbb C)) has a real field of moduli and we provide a necessary and sufficient conditionfor (G = mathbb Z / n mathbb Z) to be definable over (mathbb R); see Theorems 2.1, 2.2, and 2.3. Wealso prove that any dihedral group (D_2n) with (n geq 3) in ( rm PGL_3(mathbb C)) is definable over (mathbb R);see Theorem 2.4. Furthermore, we study all other classes of finite subgroups of( rm PGL_3(mathbb C)), and show that all of them except (A_4n), (A_5n) and (S_4n) are pseudo-real; seeTheorems 2.5 and 2.6. Finally, we explore the connection of these notions in grouptheory with their analogues in arithmetic geometry; see Theorem 2.7 and Example2.8. As a result, we can say that if (G) is definable over (mathbb R), then its Jordanconstant (J(G)) = 1, 2, 3, 6 or 60.

Let (G) be a (p) -group in which every centralizer is either finite or of finite index. It is shown that if the size of the (FC) -center of (G) is infinite and (G) is not an (FC) -group, then (G) is abelian-by-finite.

Quasi-uniform entropy (h_{QU}(psi)) is defined for a uniformlycontinuous self-map (psi) on a (T_0) quasi-uniform space((X,mathcal{U})). Basic properties are proved about this entropy,and it is shown that the quasi-uniform entropy (h_{QU}(psi ,mathcal{U})) is less than or equal to the uniform entropy (h_U(psi, mathcal{U}^s)) of (psi) considered as a uniformly continuousself-map of the uniform space ((X,mathcal{U}^s)), where(mathcal{U}^s) is the uniformity associated with thequasi-uniformity (mathcal{U}). Finally, we prove that thecompletion theorem for quasi-uniform entropy holds in the class ofall join-compact (T_0) quasi-uniform spaces, that is forjoin-compact (T_0) quasi-uniform spaces the entropy of a uniformlycontinuous self-map coincides with the entropy of its extension tothe bicompletion.

We consider Fourier multipliers in ( mathbb{R} ^2) with singularities on certaincurves, which are closely related to the Bochner–Riesz Fourier multipliers.We prove weighted inequalities and vector valued inequalities for the Fourier multiplieroperators which generalize some known results.

Numbers of the form (kcdot p^n+1) with the restriction (k < p^n) are called generalized Proth numbers. For a fixed prime p we denote them by (mathcal{T}_p). The underlying structure of (mathcal{T}_2) (Proth numbers) was investigated in [2]. In this paper the authors extend their results to all primes. An efficiently computable upper bound for the reciprocal sum of primes in (mathcal{T}_p) is presented.All formulae were implemented and verified by the PARI/GP computer algebra system. We show that the asymptotic density of ( bigcup_{pin mathcal{P}} mathcal{T}_p) is (log 2).