A derivation of the Cartwright–Levinson theorem from the theorem of Kolmogorov

P. Kusis

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 76-86

Résumé

An entire function $f$ of exponential type is said to belong to the Cartwright class $C$ if $$ \int_{-\infty}^{+\infty}\frac{\log^+|f(x)|}{1+x^2}\,dx<+\infty. $$ Let $N_+(r)(N_-(r))$ denote the number of zeros of $f$ in $|z|\leqslant R$ with $\operatorname{Re}z\geqslant0$ ($\operatorname{Re}z<0$ respectively). A simple proof, based on the weak type (1.1) Kolmogorov inequality, of the following important result is given. Theorem. Let $f\in C$ и $\displaystyle\varlimsup_{y\to+\infty}\frac{\log |f(iy)|}y=\varlimsup_{y\to-\infty}\frac{\log |f(iy)|}{|y|}=a$. Then $$ \lim_{r\to+\infty}\frac{N_+(r)}r=\lim_{r\to+\infty}\frac{N_-(r)}r=\frac a\pi. $$