Computational Methods and Function Theory 7 (2007), No. 1, 151--165
Copyright Heldermann Verlag 2007
Gershon Kresin
kresin@netvision.net.il, kresin@yosh.ac.il , College of Judea and Samaria, Department of Computer Sciences and Mathematics, 44837 Ariel, Israel.
Vladimir Maz'ya
vlmaz@math.ohio-state.edu , Ohio State University, Department of Mathematics, 231 W 18th Avenue, Columbus, OH 43210, U.S.A.
We consider analytic functions $f$ in the unit disk $\mathbb{D}$ with Taylor coefficients $c_0, c_1, \dots $ and derive estimates with sharp constants for the $l_q$-norm (quasi-norm for 0 < q < 1) of the remainder of their Taylor series, where $q \in (0, \infty ]$. As the main result, we show that given a function $f$ with $\Re f $ in the Hardy space $h_1(\mathbb{D})$ of harmonic functions on $\mathbb{D}$, the inequality $$ \left(\sum_{n=m}^\infty |c_n z^n|^q \right)^{1/q} \leq \frac{2r^m}{(1-r^q)^{1/q}} \,\|\Re f \|_{h_1} $$ holds with the sharp constant, where $r =|z|<1$, $m\geq 1$. This estimate implies sharp inequalities for $l_q$-norms of the Taylor series remainder for bounded analytic functions, analytic functions with bounded $\Re f$, analytic functions with $\Re f$ bounded from above, as well as for analytic functions with $\Re f >0$. In particular, we prove that $$ \left( \sum_{n=m}^\infty |c_n z^n|^q \right)^{1/q} \leq \frac{2r^m}{(1-r^q)^{1/q}}\sup_{|\zeta|<1} \Re(f(\zeta)-f(0)). $$ As corollary of the above estimate with $\|\Re f\|_{h_1}$ in the right-hand side, we obtain some sharp Bohr type modulus and real part inequalities.
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