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Register a new userArestovs theorems on Bernsteins inequality
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Tamas Erdelyi
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We give a simple, elementary, and at least partially new proof of Arestovs famous extension of Bernsteins inequality in $L_p$ to all $p geq 0$. Our crucial observation is that Boyds approach to prove Mahlers inequality for algebraic polynomials $P_n in {mathcal P}_n^c$ can be extended to all trigonometric polynomials $T_n in {mathcal T}_n^c$.
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We establish Bernstein inequalities for functions of general (general-state-space, not necessarily reversible) Markov chains. These inequalities achieve sharp variance proxies and recover the classical Bernsteins inequality under independence. The key analysis lies in upper bounding the operator norm of a perturbed Markov transition kernel by the limiting operator norm of a sequence of finite-rank and perturbed Markov transition kernels. For each finite-rank and perturbed Markov kernel, we bound its norm by the sum of two convex functions. One coincides with what delivers the classical Bernsteins inequality, and the other reflects the influence of the Markov dependence. A convex analysis on conjugates of these two functions then derives our Bernstein inequalities.
We present a new approach to the Marcinkiewicz interpolation inequality for the distribution function of the Hilbert transform, and prove an abstract version of this inequality. The approach uses logarithmic determinants and new estimates of canonical products of genus one.
In this paper we prove two Bloch type theorems for quaternionic slice regular functions. We first discuss the injective and covering properties of some classes of slice regular functions from slice regular Bloch spaces and slice regular Bergman spaces, respectively. And then we show that there exits a universal ball contained in the image of the open unit ball $mathbb{B}$ in quaternions $mathbb{H}$ through the slice regular rotation $widetilde{f}_{u}$ of each slice regular function $f:overline{mathbb{B}}rightarrow mathbb{H}$ with $f(0)=1$ for some $uin partialmathbb{B}$.
In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmunds inequality to the polar derivative of a polynomial.
In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $mathcal F$ in a domain $Dsubset mathbb C,$ and for a positive constant $epsilon$, if for each $fin mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$min{rho(a_f(z),b_f(z)), rho(b_f(z),c_f(z)), rho(c_f(z),a_f(z))}geq epsilon,$$ for all $zin D$, then $mathcal F$ is normal in $D$. Here, $rho$ is the spherical metric in $widehat{mathbb C}$. In this paper, we establish the high-dimension
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