The three-lines theorem

Letf(z) be an entire function represented by a Dirichlet series which is absolutely convergent in the finite plane and whose exponents _k 0; let M(x) be the exact supremum of ¦f (z)¦ on {z: 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) >- 0. In the paper, this theorem is supplemented by the assertion that for x + the upper limit of F(x) is larger than a positive constant which depends only on {_k}. In the case of positive coefficients of the series, the obtained bound cannot be improved.Translated from Matematicheskie Zametki, Vol. 15, No. 1, pp. 45–53, January, 1974.In conclusion the authors express their gratitude to I. V. Ostrovskii for valuable advice and assistance.