We propose a new approach for studying asymptotic behaviour of pth means of the logarithmic potential and classes of analytic and subharmonic functions in the unit disc. In particular, we generalize a criterion due to G. MacLane and L. Rubel of boundedness of the L2-norm of log |B|, where B is a Blaschke product, in several directions. We describe growth and decrease of pth means, p ∈ (1,∞), for nonpositive subharmonic functions in the unit disc. As a consequence, we obtain a complete description of the asymptotic behaviour of pth logarithmic means of bounded analytic functions in the unit disc in terms of its zeros and the boundary measure. We also prove sharp upper estimates of pth means of analytic and subharmonic functions of finite order in the unit disc.

中文翻译：

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单位圆盘中解析函数和次谐波函数的 pth 均值的渐近行为和零点角分布

我们提出了一种新的方法来研究单位圆盘中对数势的 pth 均值和解析函数和次谐波函数的类的渐近行为。特别是，我们在几个方向上概括了由于 G. MacLane 和 L. Rubel 的 log |B| 的 L2 范数的有界性的标准，其中 B 是 Blaschke 产品。我们描述了单位圆盘中非正次谐波函数的 pth 均值 p ∈ (1,∞) 的增长和下降。因此，我们获得了单位圆盘中的有界解析函数的 pth 对数均值在其零点和边界测度方面的渐近行为的完整描述。我们还证明了单位圆盘中有限阶解析函数和次谐波函数的 pth 均值的尖锐上估计。