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Let $a_n$,$n=0,1,2...$ are real numbers.$f(x)=a_0+a_1x+a_2x^2...$ is a real power series with radius of convergence $R>0$. Suppose there exist $M>0$ such that for all real $x$ with $|x|<R$ we have $|f(x)|<M$.Is it true that $|f(z)|$ is bounded in the complex disc of radius $R$?At first,I think log is a counterexample but it has complex coefficient.

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I don't understand what you mean about $\log$. Whether or not the coefficients are complex depends on where you do the expansion. Since the principal branch of $\log$ is real on the positive real line, any power series on the positive real line will have real coefficients. –  Jonas Meyer Dec 13 '12 at 15:51

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Hint: What is the power series for $\dfrac{1}{1+x^2}$?

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