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$$f(t) = \sum_{j= 1}^{n} jb_jt^{j-1}\,\,,\,\;t \in \Omega \subset \mathbb{R}\;\;\; \text{an open bounded set.}$$

Is it true that there exists a constant $D$ such that $|f(t)| \le D(1 + |t|^{n-1})$?

Then, is it true that $||f||_{L^{\infty}(\Omega)} \le D(1 + ||t||_{L^{\infty}(\Omega)}^{n-1})$?

I suppose the answer is yes to both questions, can you help me to find a proof?

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Do you mean $|f(t)| \le D(1+|t|^{n-1})$? – Greg Martin Mar 27 '13 at 23:16
yes, thank you :D – user67133 Mar 27 '13 at 23:21
up vote 1 down vote accepted

Any $t^{j-1}$ in the sum satisfies $|t^{j-1}| \le 1$ if $|t|\le1$ and $|t^{j-1}| \le |t^{n-1}|$ if $|t|\ge1$. In particular, we always have $|t^{j-1}| \le 1 + |t|^{n-1}$. Therefore (by the triangle inequality) $$ |f(t)| \le \sum_{j=1}^n |jb_jt^{j-1})| \le \sum_{j=1}^n j|b_j|(1+|t|^{n-1}), $$ showing that you can take $D = \sum_{j=1}^n j|b_j|$.

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