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Register a new userSharp Estimates for Blowing Down Functions in a Denjoy-Carleman Class
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If F is an infinitely differentiable function whose composition with a blowing-up belongs to a Denjoy-Carleman class C_M (determined by a log convex sequence M=(M_k)), then F, in general, belongs to a larger shifted class C_N, where N_k = M_2k; i.e., there is a loss of regularity. We show that this loss of regularity is sharp. In particular, loss of regularity of Denjoy-Carleman classes is intrinsic to arguments involving resolution of singularities.
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In this article, we establish a sharp effectiveness result of Demaillys strong openness conjecture. We also establish a sharp effectiveness result related to a conjecture posed by Demailly and Kollar.
The Bohr radius for a class $mathcal{G}$ consisting of analytic functions $f(z)=sum_{n=0}^{infty}a_nz^n$ in unit disc $mathbb{D}={zinmathbb{C}:|z|<1}$ is the largest $r^*$ such that every function $f$ in the class $mathcal{G}$ satisfies the inequality begin{equation*} dleft(sum_{n=0}^{infty}|a_nz^n|, |f(0)|right) = sum_{n=1}^{infty}|a_nz^n|leq d(f(0), partial f(mathbb{D})) end{equation*} for all $|z|=r leq r^*$, where $d$ is the Euclidean distance. In this paper, our aim is to determine the Bohr radius for the classes of analytic functions $f$ satisfying differential subordination relations $zf(z)/f(z) prec h(z)$ and $f(z)+beta z f(z)+gamma z^2 f(z)prec h(z)$, where $h$ is the Janowski function. Analogous results are obtained for the classes of $alpha$-convex functions and typically real functions, respectively. All obtained results are sharp.
Inspired by the recent works of Srivastava et al. (HMS-AKM-PG), Frasin and Aouf (BAF-MKA) and others (Ali-Ravi-Ma-Mina-class,Caglar-Orhan,Goyal-Goswami,Xu-HMS-AML,Xu-HMS-AMC), we propose to investigate the coefficient estimates for a general class of analytic and bi-univalent functions. Also, we obtain estimates on the coefficients |a2| and |a3| for functions in this new class. Some interesting remarks, corollaries and applications of the results presented here are also discussed.
Let $phi$ be a normalized convex function defined on open unit disk $mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f(z)+ alpha z f(z) prec phi(z)$ for all $zin mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.
For two types of moderate growth representations of $(mathbb{R}^d,+)$ on sequentially complete locally convex Hausdorff spaces (including F-representations [J. Funct. Anal. 262 (2012), 667-681], we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [Conjecture 6.; J. Funct. Anal. 262 (2012), 667-681] for analytic vectors of representations of $G =(mathbb{R}^d,+)$. As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.
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