LETTER TO THE EDITOR

In my note (Sh. Yarmukhamedov, “On the growth of functions harmonic in a cylinder and growing on its boundary together with the normal derivative”), published in DAN, vol. 152, no. 3, 1963, inaccuracies were allowed in the formulation of the theorems.

Theorem 1 is true under the following conditions:

1. Inside the half-cylinder

\[ |U|<M\exp\left[e^{(\rho_1-\varepsilon)x}\right]. \]

1. On the boundary

\[ |U|+\left|\frac{\partial}{\partial n}U\right| <M_1\exp\left(ae^{\rho x}\right),\qquad \rho<\rho_1<2\rho,\qquad \alpha<a\cos\frac{\pi\rho}{2\rho_1}. \]

In the description of the contour, \(\rho_1\) should be replaced by \(\rho\).

In Theorem 2 the same condition inside must also be satisfied by the gradient of the function. Apparently, the condition on the gradient is superfluous; however, we have no proof of this.

In Theorem 3, \(o(e^{\rho_1 x})\) should be replaced by \(e^{(\rho_1-\varepsilon)x}\) \((\rho<\rho_1-\varepsilon)\), and \(a_1\) by \(a_1+\varepsilon_0\) \((\varepsilon_0>0)\); in the boundary condition, \(\dfrac{\partial U}{\partial n}\) should be replaced by \(\operatorname{grad} U\).

Sh. Yarmukhamedov