An extremal problem for certain subharmonic functions in the plane
Let \(\delta\in (0,1)\) and \(K\subset {\mathbb{R}}\) be a closed set which is \(\delta\)-dense in the sense the \(| K\cap I| \geq \delta\) for every interval \(I\subset {\mathbb{R}}\) with length \((I)=1\). There is a function \(u=u(\cdot,K): {\mathbb{C}}\to [0,\infty)\) which is harmonic in \({\mathbb{C}}\setminus K\), vanishes on K, is symmetric with respect to \({\mathbb{R}}\) and satisfies \(\lim_{y\to \infty}y^{-1}u(x+iy)=1,\) \(\forall x\in {\mathbb{R}}\). V. I. Levin posed the problem of maximizing \(\sup_{x\in {\mathbb{R}}}u(x,K)\) when K runs over all \(\delta\)-dense sets. He conjectured that the maximum should be achieved when \(K=K_ 0=\cup^{\infty}_{n=- \infty}[n-(\delta /2),n+(\delta /2)].\) In this paper we prove Levin's conjecture in stronger form by showing that for each convex nondecreasing function \(\Phi\), each \(y\in {\mathbb{R}}\), each interval \(I\subset {\mathbb{R}}\) of length one, and each \(\delta\)-dense K, we have \[ \int_{I}\Phi (u(x+iy),K)dx\leq \int_{I}\Phi (u(x+iy),K_ 0)dx. \] Moreover, if \(\Phi (0)=0\) and if equality holds for some \(\Phi\) and y, then either K is a translation of \(K_ 0\) or both sides are zero. The theorems about the functions u are limiting cases of analogous theorems about certain harmonic measures. To prove these theorems, one introduces variants of ``star-functions'', which are shown to have appropriate subharmonicity properties. It turns out Levin's conjecture, in the strong form stated above, but without discussion of equality, had been proved earlier by \textit{A. E. Fryntov} [Dokl. Akad. Nauk SSSR 300, No.4, 819-820 (1988; Zbl 0698.31001)], using similar methods.
scientific article; zbMATH DE number 4146344