{"docId":108,"paperId":108,"url":"https:\/\/hrj.episciences.org\/108","doi":"10.46298\/hrj.1989.108","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":23,"name":"Volume 12 - 1989"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01104337","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-01104337v1","dateSubmitted":"2015-03-03 16:13:41","dateAccepted":"2015-06-12 16:05:23","datePublished":"1989-01-01 08:00:00","titles":{"en":"A Lemma in complex function theory I"},"authors":["Balasubramanian, R","Ramachandra, K"],"abstracts":{"en":"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."},"keywords":[["analytic function"],["semi-circular portion"],"[MATH] Mathematics [math]"]}