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We present a characterization of one-component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by J. Cima and R. Mortini. In particular we prove that for any inner function $Theta$ whose singular set has measure zero, one can find a Blaschke product $B$ such that $Theta B$ is one-component. We also obtain a characterization of one-component singular inner functions which is used to produce examples of discrete and continuous one-component singular inner functions.
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We
A one-component inner function $Theta$ is an inner function whose level set $$Omega_{Theta}(varepsilon)={zin mathbb{D}:|Theta(z)|<varepsilon}$$ is connected for some $varepsilonin (0,1)$. We give a sufficient condition for a Blaschke product with z
We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then f = g up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational
Let $Pin Bbb Q_p[x,y]$, $sin Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=frac{1}{1-p^{-1}}int_{Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(Bbb Z_p)$. We show that if $P$ is homogeneous then for each charact
In the present investigation our main aim is to give lower bounds for the ratio of some normalized $q$-Bessel functions and their sequences of partial sums. Especially, we consider Jacksons second and third $q$-Bessel functions and we apply one normalization for each of them.