A Lemma in complex function theory I

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