{"docId":103,"paperId":103,"url":"https:\/\/hrj.episciences.org\/103","doi":"10.46298\/hrj.1988.103","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":22,"name":"Volume 11 - 1988"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01104306","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01104306v1","dateSubmitted":"2015-03-03 16:13:39","dateAccepted":"2015-06-12 16:05:20","datePublished":"1988-01-01 08:00:00","titles":{"fr":"Some local-convexity theorems for the zeta-function-like analytic functions"},"authors":["Balasubramanian, R","Ramachandra, K"],"abstracts":{"en":"In this paper we investigate lower bounds for $$I(\\sigma)= \\int^H_{-H}\\vert f(\\sigma+it_0+iv)\\vert^kdv,$$ where $f(s)$ is analytic for $s=\\sigma+it$ in $\\mathcal{R}=\\{a\\leq\\sigma\\leq b, t_0-H\\leq t\\leq t_0+H\\}$ with $\\vert f(s)\\vert\\leq M$ for $s\\in\\mathcal{R}$. Our method rests on a convexity technique, involving averaging with the exponential function. We prove a general lower bound result for $I(\\sigma)$ and give an application concerning the Riemann zeta-function $\\zeta(s)$. We also use our methods to prove that large values of $\\vert\\zeta(s)\\vert$ are ``rare'' in a certain sense."},"keywords":[{"en":"functional equation"},{"en":"analytic functions"},{"en":"local-convexity"},"[MATH] Mathematics [math]"]}