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Register a new userZeros of Normalized Sections of Non Convergent Power Series
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Authors
Alberto Dayan
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A well known result due to Carlson affirms that a power series with finite and positive radius of convergence R has no Ostrowski gaps if and only if the sequence of zeros of its nth sections is asymptotically equidistributed to {|z|=R}. Here we extend this characterization to those power series with null radius of convergence, modulo some necessary normalizations of the sequence of the sections of f.
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Given a regular compact set $E$ in the complex plane, a unit measure $mu$ supported by $partial E,$ a triangular point set $beta := {{beta_{n,k}}_{k=1}^n}_{n=1}^{infty},betasubset partial E$ and a function $f$, holomorphic on $E$, let $pi_{n,m}^{beta,f}$ be the associated multipoint $beta-$ Pade approximant of order $(n,m)$. We show that if the sequence $pi_{n,m}^{beta,f}, ninLambda, m-$ fixed, converges exact maximally to $f$, as $ntoinfty,ninLambda$ inside the maximal domain of $m-$ meromorphic continuability of $f$ relatively to the measure $mu,$ then the points $beta_{n,k}$ are uniformly distributed on $partial E$ with respect to the measure $mu$ as $ ninLambda$. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence $Lambda$ is provided, under the condition that $Lambda$ is dense enough.
An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fej{e}rs theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to recover basic results of complex analysis. Some classical results of function theory are also shown to be consequences of the series expansion.
Let $mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $sum_n x_n$ is divergent. For each $x in mathscr{X}$, let $mathcal{I}_x$ be the collection of all $Asubseteq mathbf{N}$ such that the subseries $sum_{n in A}x_n$ is convergent. Moreover, let $mathscr{A}$ be the set of sequences $x in mathscr{X}$ such that $lim_n x_n=0$ and $mathcal{I}_x eq mathcal{I}_y$ for all sequences $y=(y_n) in mathscr{X}$ with $liminf_n y_{n+1}/y_n>0$. We show that $mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.
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.
In this paper we shall use the boundary Schwarz lemma of Osserman to obtain some generalizations and refinements of some well known results concerning the maximum modulus of the polynomials with restricted zeros due to Turan, Dubinin and others.
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