The three-lines theorem

Let $f(z)$ be an entire function represented by a Dirichlet series which is absolutely convergent in the finite plane and whose exponents $\lambda_k\ge0$; let $M(x)$ be the exact supremum of $|f(z)|$ on $\{z:\operatorname{Re}z=x\}$. If we assume that $F(x)=\ln M(x)$ has a continuous second derivative, the three-lines theorem asserts that $F''(x)\ge0$. In the paper, this theorem is supplemented by the assertion that for $x\to+\infty$ the upper limit of $F''(x)\ge0$ is larger than a positive constant which depends only on $\{\lambda_k\}$. In the case of positive coefficients of the series, the obtained bound cannot be improved.