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Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications.…

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $\Phi(s, z, c)$ is obtained from the Lerch zeta function…

We prove that a function, which is defined on a union of lines $\mathbb{C} E$ through the origin in $\mathbb{C}^n$ with direction vectors in $E\subset \mathbb{C}^n$ and is holomorphic of fixed finite order and finite type along each line,…

Let $\phi$ be a real-valued smooth function on $\mathbf{C}$ satisfying $0 \le \Delta \phi \le M$ for some $M \ge 0$. We consider the space of all holomorphic functions which are square-integrable with respect to the measure $e^{-\phi(z)}…

Let $M$ be a subharmonic function with Riesz measure $\nu_M$ in a domain $D$ in the $n$-dimensional complex Euclidean space $\mathbb C^n$, and let $f$ be a nonzero function that is holomorphic in $D$, vanishes on a set ${\sf Z}\subset D$,…

A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding…

We announce a generalization of Zimmer's cocycle superrigidity theorem proven using harmonic map techniques. This allows us to generalize many results concerning higher rank lattices to all lattices in semisimple groups with property $(T)$.…

We introduce a class of functions near zero on the logarithmic cover of the complex plane that have convergent expansions into generalized power series. The construction covers cases where non-integer powers of $z$ and also terms containing…

Towards the Hilbert-P\'olya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes…

Let $\mathbb D$ be the unit disc in $\mathbb C$ and let $f:\mathbb D \to \mathbb C$ be a Riemann map, $\Delta=f(\mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a…

In this article, we consider the family of functions $f$ meromorphic in the unit disk $\ID=\{z :\,|z| < 1\}$ with a pole at the point $z=p$, a Taylor expansion \[f(z)= z+\sum_{k=2}^{\infty} a_kz^k, \quad |z|<p, \] and satisfying the…

Let $\alpha$ \in (0; 1). We show that any $\alpha$-H\"older homeomorphism from the unit circle in the plane to the plane can be extended to an $\alpha$-H\"{o}lder homeomorphism from the whole unit disc.

Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the condition $${\real} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2} ~\mbox{ for…

Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint…

In a previous paper we considered a positive function f, uniquely determined for s>0 by the requirements f(1)=1, log(1/f) is convex and the functional equation f(s)=psi(f(s+1)) with psi(s)=s-1/s. We prove that the meromorphic extension of f…

For each $n > 0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by $z \to z^n$'. We show that these families are free of relations, which determines the…

Let $A,B$ be C*-algebras, $B_A(0;r)$ the open ball in $A$ centered at $0$ with radius $r>0$, and $H:B_A(0;r)\to B$ an orthogonally additive holomorphic map. If $H$ is zero product preserving on positive elements in $B_A(0;r)$, we show, in…

We construct, for every \(0<k<1\), a bounded globally univalent harmonic mapping \[ f=h+\overline g \colon \D\to\C \] such that \[ |g'(z)|\le k|h'(z)|,\qquad z\in\D, \] while the analytic part \(h\) is unbounded. The construction is based…

For an analytic function $f(z)$ on the unit disk $|z|<1$ with $f(0)=f'(0)-1=0$ and $f(z)\ne0, 0<|z|<1,$ we consider the power deformation $f_c(z)=z(f(z)/z)^c$ for a complex number $c.$ We determine those values $c$ for which the operator…

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…