Entire and Meromorphic Functions with Asymptotically Prescribed Characteristic

Albert Edrei, Wolfgang H. J. Fuchs

doi:10.4153/cjm-1965-039-7

Entire and Meromorphic Functions with Asymptotically Prescribed Characteristic

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説明

<jats:p>If <jats:italic>f</jats:italic>(<jats:italic>z</jats:italic>) is a non-constant, entire function, then Hadamard's three-circles theorem asserts that</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0008414X00039353_eqn01" /></jats:disp-formula></jats:p><jats:p>is a convex, increasing function of log r. Hence, by well-known properties of logarithmically convex functions,</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0008414X00039353_eqn02" /></jats:disp-formula></jats:p>