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2sinz assumption canonical product chapter compact set complex constant conclude cz+d deduce defined desired conclusion DOCTOR OF PHILOSOPHY establish the following exponential type f 0(lnr f 2sin(Az+B f_(z_ finite logarithmic length finite order fonctions entieres form an A-set function f(z function with real Hadamard product Hellerstein Hellerstein-Williamson Theorem Hence holomorphic implies Imz^O integral ir/A is_ an entire Laguerre-Polya class Let f(z LI-CHIEN SHEN logarithmic derivative lower half plane Math mod TT n n n n Nevanlinna Nevanlinna theory non-real constant non-real entire non-real meromorphic function non-real zeros numbers o_f_ observe obtain Polya-Saxer theorem polynomial positive integer proof is complete proof of Theorem real axis real constant real entire function real poles real zeros Rolle's theorem satisfies the hypothesis sinAc*0 sinz strictly non-real meromorphic substitute Theorem 1.2 thesis trigonometric polynomial TT(z unit circle upper half plane zeros of f zeros of H(z zeros of_ zsinaz