On the regularity of the growth of the modulus and argument of an entire function in the metric of $L^p [0, 2π]$ - Kalinets, R. Z.; Koval’chuk, Yu. A. (Ukrains’kyi Matematychnyi Zhurnal, 1998)
Under sufficiently general assumptions, we describe sets of entire functions $f$, sets of growing functions $λ$, and sets of complex-valued functions $H$ from $L^p [0, 2π]$, $p ∈ [1, + ∞]$, for which$$\left\{ {\frac{1}{{2π }}\int\limits_0^{2π } {|\log f(re^{iθ } ) - λ (r)H(θ )|^p } dθ } \right\}^{1/p} = o(λ (r)),r → ∞.$$