Can an entire non-constant function be bounded on only a finite set of lines through the origin? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:30:03Z http://mathoverflow.net/feeds/question/29734 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29734/can-an-entire-non-constant-function-be-bounded-on-only-a-finite-set-of-lines-thro Can an entire non-constant function be bounded on only a finite set of lines through the origin? Andreas Rüdinger 2010-06-27T21:41:47Z 2010-06-29T23:07:54Z <p>If an entire function is bounded for all $z \in \mathbb{C}$, than it's a constant by Liouville's theorem. Of course an entire function can be bounded on lines through the origin $z=r \exp(i \phi), \phi= const., r \in \mathbb{R}$ without being constant (e.g. $\cos(z^n)$ is bounded on $n$ lines). </p> <p>What is the maximum cardinality of the set of "directions" $\phi$ for which an entire function can be bounded without being constant? </p> <p>From intuition I would expect only finitely many directions. Is this correct? </p> <p>(Picard's second theorem says that in any open set containing $\infty$ every value with possibly a single exception is taken infinitely often by an entire non-constant function. Here I'm asking a somehow "orthogonal" question, looking for lines through $\infty$ where an entire non-constant function is bounded.) </p> http://mathoverflow.net/questions/29734/can-an-entire-non-constant-function-be-bounded-on-only-a-finite-set-of-lines-thro/29735#29735 Answer by Jonas Meyer for Can an entire non-constant function be bounded on only a finite set of lines through the origin? Jonas Meyer 2010-06-27T22:01:22Z 2010-06-29T23:07:54Z <p>Newman gave an example in 1976 of a non-constant entire function bounded on each line through the origin in "<a href="http://www.jstor.org/stable/2977024" rel="nofollow">An entire function bounded in every direction</a>". </p> <p>I like the second sentence of the article:</p> <blockquote> <p>This is exactly what is needed to confuse students who have just struggled to comprehend the meaning of Liouville's theorem.</p> </blockquote> <p>Armitage gave examples in 2007 of non-constant entire functions that go to zero in every direction in "Entire functions that tend to zero on every line". For this I have only seen the <a href="http://www.ams.org/mathscinet-getitem?mr=2290290" rel="nofollow">MR review</a>. (If you don't have MathSciNet access, the link should still give you the publication information to find the article.)</p> <hr> <p><strong>Update:</strong> I just decided to take a look at the Armitage paper, and the introduction was enlightening:</p> <blockquote> <p>Although every bounded entire (holomorphic) function on $\mathbb{C}$ is constant (Liouville’s theorem), it has been known for more than a hundred years that there exist nonconstant entire functions $f$ such that $f(z) → 0$ as $z →∞$ along every line through 0 (see, for example, Lindelöf’s book [10, pp. 119–122] of 1905). And it has been known for more than eighty years that such functions can tend to 0 along any line whatsoever (see Mittag-Leffler [11], Grandjot [8], and Bohr [4]). Further references to related work are given in Burckel’s review [5] of Newman’s note [12]. Entire functions with radial decay are used by Beardon and Minda [3] and Ullrich [14] in studies of pointwise convergent sequences of entire functions.</p> </blockquote> <p>Armitage goes on to mention that Mittag-Leffler and Grandjot also gave explicit constructions, but states, "The examples given in what follows may nevertheless be of some interest because of their comparative simplicity." The examples are $$F(z)=\exp\left(-\int_0^\infty t^{-t}\cosh(tz^2)dt\right) - \exp\left(-\int_0^\infty t^{-t}\cosh(2tz^2)dt\right)$$ and $$G(z)=\int_0^\infty e^{i\pi t}t^{-t}\cosh(t\sqrt{z})dt\int_0^\infty e^{i\pi t}t^{-t}\cos(t\sqrt{z})dt .$$</p> http://mathoverflow.net/questions/29734/can-an-entire-non-constant-function-be-bounded-on-only-a-finite-set-of-lines-thro/29736#29736 Answer by Andrey Rekalo for Can an entire non-constant function be bounded on only a finite set of lines through the origin? Andrey Rekalo 2010-06-27T22:09:06Z 2010-06-27T22:41:00Z <p><a href="http://en.wikipedia.org/wiki/Mittag-Leffler_function" rel="nofollow">The Mittag-Leffler function</a> $E_{\alpha,1}$, $\alpha>0$, is bounded in the sector $$\frac{\alpha\pi}{2}&lt; \arg z&lt;2\pi-\frac{\alpha\pi}{2}.$$</p> <p>In particular, $e^z=E_{1,1}(z)$ is bounded in $$\frac{\pi}{2}&lt; \arg z&lt;\frac{3\pi}{2}.$$</p>