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Let $\left(z_{\nu }\right)$ be a sequence in the upper half plane. If $1\<p\le \infty$ and if $$y_{\nu }^{1/p}f(z_{\nu })=a_{\nu },\nu =1,2,...(*)$$ has solution $f\left(z\right)$ in the class of Poisson integrals of $L^p$ functions for any sequence $(a_{\nu })\in {\ell }^p$, then we show that $\left(z_{\nu }\right)$ is an interpolating sequence for $H^{\infty }$. If $f(z_{\nu })=a_{\nu }$, $\nu =1,2,...$ has solution in the class of Poisson integrals of BMO functions whenever $(a_{\nu })\in {\ell }^{\infty }$, then $\left(z_{\nu }\right)$ is again an interpolating sequence for $H^{\infty }$. A somewhat more general theorem is also proved and a counterexample for the case $p\le 1$ is described.