About Non-tangential limits of an analytic function

Question

There is a function which has non-tangential limits at no point of $ \partial \mathbb{D}$ . Where $\mathbb{D}$ is unit open disk in $\mathbb{C}$ ? Can anybody give me that example.

Answer 1

In a paper of 1962, G. R. Maclane gave an example of a holomorphic function $F(z)$ on the unit disc such that for any $\theta$ we have

$$\limsup_{r\to 1} |F(re^{i\theta})| = +\infty, \,\,\, \liminf_{r\to 1} |F(re^{i\theta})| = 0$$

This would ensure that $F(z)$ cannot have a nontangential limit at any point of the unit circle, as well as having no radial limits at any point of the boundary. Interestingly, the function he constructs has no zeros inside the disc and satisfies a growth condition there.

The full pdf file of Maclane's paper can be downloaded from the link above.

Edit:

My first impulse when seeing this question was to look at lacunary series such as $f(z) = z + z^2 + z^4 + z^8 + z^{16} + \cdots$, and it seems that these do indeed have the required properties.

In a paper by J. Marshall Ash (but he seems to be quoting results by Binmore) he states:

If $\{\lambda_n\}$ is a lacunary sequence of integers (i.e. there is a constant $C$ such that $\lambda_{k+1}/\lambda_{k} \ge C > 1$) and the function $$f(z) = \sum_{n=1}^\infty a_n z^{\lambda_n}$$ is analytic on $|z| < 1$ and has coefficients satisfying $\limsup_{n \to \infty} |a_n| > 0$, then $f$ has radial limits at no point of the boundary $|z|=1$. ... In particular $f(z) = 1 + z + z^2 + z^4 + z^8 + z^{16} + \cdots$ is analytic in $|z|<1$, but has radial limits at no point of the boundary $|z|=1$.

Clearly, since this function has no (finite) radial limits at any point of the unit circle, it cannot have a (finite) nontangential limit at any point of the circle either.

The papers seem to be generally available (from my PC they seem to be available via Google Docs) and a Google search for "binmore Analytic functions with Hadamard gaps" should do the trick.