A Lemma in complex function theory I

R Balasubramanian (1) , K Ramachandra (1)

1 TIFR - Tata Institute for Fundamental Research

Continuing our earlier work on the same topic published in the same journal last year we prove the following result in this paper: If $f(z)$ is analytic in the closed disc $\vert z\vert\leq r$ where $\vert f(z)\vert\leq M$ holds, and $A\geq1$, then $\vert f(0)\vert\leq(24A\log M) (\frac{1}{2r}\int_{-r}^r \vert f(iy)\vert\,dy)+M^{-A}.$ Proof uses an averaging technique involving the use of the exponential function and has many applications to Dirichlet series and the Riemann zeta function.

Mots clés: analytic function; semi-circular portion

Domaines: Mathématiques [math]

R Balasubramanian, K Ramachandra. A Lemma in complex function theory I. Hardy-Ramanujan Journal, 1989, Volume 12 - 1989, pp.1 - 5. ⟨10.46298/hrj.1989.108⟩. ⟨hal-01104337⟩