On the maximum of module of characteristic functions of probabilistic law (Q2768812)

Let \(\phi(t)=\int_{-\infty}^{\infty}e^{it\sigma}dF(\sigma)\) be the characteristic function of the distribution function \(F(\sigma)\). For \(r<R\) let us denote \(M_{\phi}(r)=\max\{|\phi(z)|:\;| z|=r\}\) and \(\mu_{F}=\sup\{e^{rx}T(x):\;x\in[0,\infty)\}\), \(T(x)\buildrel\text{def}\over=1-F(x)+F(-x)\). The authors prove that if \(\phi\) is an entire function, then for any \(\varepsilon>0\) there exists a set \(E\subset[0;+\infty)\) with finite Lebesgue measure such that for all \(r\in[0;+\infty]\setminus E\) the following inequality holds true: \(M_{\phi}(r)\leq\mu_{F}(r)(\ln\mu_{F}(r))^{1/2+\varepsilon}\).