Local properties of entire functions of bounded index in a frame

Abstract

UDC 517.555

We extend the concept of entire functions of bounded index in a variable direction  to the case where the variable direction is a continuous vector-valued function. The previous investigations of this class of functions assumed that the variable direction is an entire vector-valued function. An entire function $F\colon \mathbb{C}^n\to \mathbb{C}$ is called a function of bounded index in a frame $\mathbf{b}(z),$ if~there exists $m_{0} \in\mathbb{Z}_{+}$ such that, for every $m \in\mathbb{Z}_{+},$ for all $z\in \mathbb{C}^{n},$ and for all $t\in\mathbb{C},$ one has $\dfrac{\big|{\partial^{m}_{\mathbf{b}(z)}F(z + t\mathbf{b}(z))}\big|}{m!}\leq\max_{0\leq k \leq m_{0}} \dfrac{\big|{\partial^{k}_{\mathbf{b}(z)}F(z + t\mathbf{b}(z))}\big|}{k!},$  where $\partial^{0}_{\mathbf{b}(z)}F(z + t\mathbf{b}(z)) = F(z + t\mathbf{b}(z)),$  $\partial^{k}_{\mathbf{b}(z)}F(z + t\mathbf{b}(z)) := \dfrac{k!}{2\pi i}\displaystyle\int\limits_{|\tau| = r}\dfrac{F(z + t\mathbf{b}(z) + \tau \mathbf{b}(z))}{\tau^{k + 1}} d\tau$ and $\mathbf{b}\colon \mathbb{C}^n\to\mathbb{C}^n$ is a vector-valued continuous function. Here we investigate some properties of these functions.  The obtained results are counterparts of the known statements obtained for entire functions of bounded index in a fixed direction.  These results describe the local behavior of the modulus $\partial_{\mathbf{b}(z)}^kF(z + t\mathbf{b} + \tau\mathbf{b}(z))$ in the disc $|\tau| = \eta.$  We give some estimates for this expression by means of the values $\partial_{\mathbf{b}(z)}^kF(z + t\mathbf{b}).$