{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22708"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22708","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Growth comparisons for certain Nevanlinna theory functionals","abstract":"We show that for all entire f with $\\vert{\\rm f}(0)\\vert\\ge 1$ and all ${\\rm r}>0$, $$\\rm log\\ M(r,f)\\le d\\sb\\alpha T(r,f)\\sp{1\\over 2}T(\\alpha r,f)\\sp{1\\over 2},\\leqno(*)$$and$$\\rm m\\sbsp{p}{+}(r,f)\\le d\\sbsp{\\alpha}{p-1\\over p}\\ T(r,f)\\sp{p+1\\over 2p}\\ T(\\alpha r,f)\\sp{p-1\\over 2p},$$where $\\alpha > 1,$ $\\rm m\\sbsp{p}{+}(r,f)$ is the L$\\sb{\\rm p}$ norm of $\\rm log\\sp+\\vert f(re\\sp{i\\theta})\\vert,$ and$$\\rm d\\sb\\alpha = {4\\sqrt{3}\\ \\alpha\\sp{1\\over 2}(\\alpha\\sp{1\\over 2} + 1)\\over \\alpha - 1}.$$The inequality $(*)$ improves the well-known inequality $\\rm log\\ M(r,f)\\le {R + r\\over R - r}\\ T(R,f),$ $\\rm 0 0.$$Using a technique introduced by W. K. Hayman, we show all these inequalities are sharp, and in particular that $(*)$ does not hold in general for any d$\\sb\\alpha$ for which$$\\rm d\\sb\\alpha = o\\left({1\\over \\alpha - 1}\\right),\\quad \\alpha\\to 1.$$","abstract_html":"We show that for all entire f with $\\vert{\\rm f}(0)\\vert\\ge 1$ and all ${\\rm r}&gt;0$, $<span class=\"etd-inline-math\">\\rm log M(r,f)\\le d\\sb&alpha; T(r,f)\\sp{1\\over 2}T(&alpha; r,f)\\sp{1\\over 2},\\leqno(*)</span>$and$<span class=\"etd-inline-math\">\\rm m\\sbsp{p}{+}(r,f)\\le d\\sbsp{&alpha;}{p-1\\over p} T(r,f)\\sp{p+1\\over 2p} T(&alpha; r,f)\\sp{p-1\\over 2p},</span>$where $\\alpha &gt; 1,$ $\\rm m\\sbsp{p}{+}(r,f)$ is the L$\\sb{\\rm p}$ norm of $\\rm log\\sp+\\vert f(re\\sp{i\\theta})\\vert,$ and$<span class=\"etd-inline-math\">\\rm d\\sb&alpha; = {4\\sqrt{3} &alpha;\\sp{1\\over 2}(&alpha;\\sp{1\\over 2} + 1)\\over &alpha; - 1}.</span>$The inequality $(*)$ improves the well-known inequality $\\rm log\\ M(r,f)\\le {R + r\\over R - r}\\ T(R,f),$ $\\rm 0 0.$$Using a technique introduced by W. K. Hayman, we show all these inequalities are sharp, and in particular that $(*)$ does not hold in general for any d$\\sb\\alpha$ for which$<span class=\"etd-inline-math\">\\rm d\\sb&alpha; = o\\left({1\\over &alpha; - 1}\\right),\\quad &alpha;\\to 1.</span>$","abstract_has_math":true,"creators":["Kwon, Ki-Ho"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Miles, Joseph B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:48:49Z","date_published":"2011-05-07T13:48:49Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1991 Kwon, Ki-Ho"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9210882","(UMI)AAI9210882"],"render_values":[{"text":"AAI9210882","href":null,"code":true},{"text":"(UMI)AAI9210882","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22708","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Miles, Joseph B."]},{"key":"dc:creator","label":"Author","values":["Kwon, Ki-Ho"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:48:49Z","10000-01-01","1991"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1991 Kwon, Ki-Ho"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9210882","(UMI)AAI9210882","http://hdl.handle.net/2142/22708"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We show that for all entire f with $\\vert{\\rm f}(0)\\vert\\ge 1$ and all ${\\rm r}>0$, $$\\rm log\\ M(r,f)\\le d\\sb\\alpha T(r,f)\\sp{1\\over 2}T(\\alpha r,f)\\sp{1\\over 2},\\leqno(*)$$and$$\\rm m\\sbsp{p}{+}(r,f)\\le d\\sbsp{\\alpha}{p-1\\over p}\\ T(r,f)\\sp{p+1\\over 2p}\\ T(\\alpha r,f)\\sp{p-1\\over 2p},$$where $\\alpha > 1,$ $\\rm m\\sbsp{p}{+}(r,f)$ is the L$\\sb{\\rm p}$ norm of $\\rm log\\sp+\\vert f(re\\sp{i\\theta})\\vert,$ and$$\\rm d\\sb\\alpha = {4\\sqrt{3}\\ \\alpha\\sp{1\\over 2}(\\alpha\\sp{1\\over 2} + 1)\\over \\alpha - 1}.$$The inequality $(*)$ improves the well-known inequality $\\rm log\\ M(r,f)\\le {R + r\\over R - r}\\ T(R,f),$ $\\rm 0 0.$$Using a technique introduced by W. K. Hayman, we show all these inequalities are sharp, and in particular that $(*)$ does not hold in general for any d$\\sb\\alpha$ for which$$\\rm d\\sb\\alpha = o\\left({1\\over \\alpha - 1}\\right),\\quad \\alpha\\to 1.$$","Suppose now that f(z) is a nonconstant meromorphic function in the plane, that m$\\sb2$(r,f) is the L$\\sb2$ norm of $\\rm log\\vert f(re\\sp{i\\theta})\\vert,$ and that $\\varphi$(x) is a positive increasing function satisfying $\\rm \\int\\sp\\infty{dx\\over \\varphi(x)} < \\infty.$ Then we prove that there exists a set F with finite logarithmic measure such that$$\\rm{\\lim\\limits\\sb{r\\to\\infty\\atop r\\notin F}}\\ {m\\sb2 (r,f) \\over T(r,f) \\lbrack\\varphi (log\\ T(r,f))\\rbrack\\sp{1\\over 2}} = 0.\\leqno(**)$$The relation $(**)$ is shown to be sharp. We also prove several other theorems of $\\rm {m\\sb2(r,f)\\over T(r,f)},$ and study the upper bounds for $\\rm log\\ M(r,f)\\over T(r,f)$ for analytic functions on the unit disc.","Made available in DSpace on 2011-05-07T13:48:49Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210882.pdf: 1555747 bytes, checksum: aba4d56126f329024b515f8e99739c98 (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:59:27Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Growth comparisons for certain Nevanlinna theory functionals"]}]}],"canonical_facts":{"dc:contributor":["Miles, Joseph B."],"dc:creator":["Kwon, Ki-Ho"],"dc:date":["2011-05-07T13:48:49Z","10000-01-01","1991"],"dc:description":["We show that for all entire f with $\\vert{\\rm f}(0)\\vert\\ge 1$ and all ${\\rm r}>0$, $$\\rm log\\ M(r,f)\\le d\\sb\\alpha T(r,f)\\sp{1\\over 2}T(\\alpha r,f)\\sp{1\\over 2},\\leqno(*)$$and$$\\rm m\\sbsp{p}{+}(r,f)\\le d\\sbsp{\\alpha}{p-1\\over p}\\ T(r,f)\\sp{p+1\\over 2p}\\ T(\\alpha r,f)\\sp{p-1\\over 2p},$$where $\\alpha > 1,$ $\\rm m\\sbsp{p}{+}(r,f)$ is the L$\\sb{\\rm p}$ norm of $\\rm log\\sp+\\vert f(re\\sp{i\\theta})\\vert,$ and$$\\rm d\\sb\\alpha = {4\\sqrt{3}\\ \\alpha\\sp{1\\over 2}(\\alpha\\sp{1\\over 2} + 1)\\over \\alpha - 1}.$$The inequality $(*)$ improves the well-known inequality $\\rm log\\ M(r,f)\\le {R + r\\over R - r}\\ T(R,f),$ $\\rm 0 0.$$Using a technique introduced by W. K. Hayman, we show all these inequalities are sharp, and in particular that $(*)$ does not hold in general for any d$\\sb\\alpha$ for which$$\\rm d\\sb\\alpha = o\\left({1\\over \\alpha - 1}\\right),\\quad \\alpha\\to 1.$$","Suppose now that f(z) is a nonconstant meromorphic function in the plane, that m$\\sb2$(r,f) is the L$\\sb2$ norm of $\\rm log\\vert f(re\\sp{i\\theta})\\vert,$ and that $\\varphi$(x) is a positive increasing function satisfying $\\rm \\int\\sp\\infty{dx\\over \\varphi(x)} < \\infty.$ Then we prove that there exists a set F with finite logarithmic measure such that$$\\rm{\\lim\\limits\\sb{r\\to\\infty\\atop r\\notin F}}\\ {m\\sb2 (r,f) \\over T(r,f) \\lbrack\\varphi (log\\ T(r,f))\\rbrack\\sp{1\\over 2}} = 0.\\leqno(**)$$The relation $(**)$ is shown to be sharp. We also prove several other theorems of $\\rm {m\\sb2(r,f)\\over T(r,f)},$ and study the upper bounds for $\\rm log\\ M(r,f)\\over T(r,f)$ for analytic functions on the unit disc.","Made available in DSpace on 2011-05-07T13:48:49Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210882.pdf: 1555747 bytes, checksum: aba4d56126f329024b515f8e99739c98 (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:59:27Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9210882","(UMI)AAI9210882","http://hdl.handle.net/2142/22708"],"dc:language":["eng"],"dc:rights":["Copyright 1991 Kwon, Ki-Ho"],"dc:subject":["Mathematics"],"dc:title":["Growth comparisons for certain Nevanlinna theory functionals"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}