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Let$f$and$g$be nonlinear entire functions. The relations between the dynamics of$f⊗g$and$g⊗f$are discussed. Denote byℐ (·) and$F$(·) the Julia and Fatou sets. It is proved that if$z$∈$C$, then$z$∈ℐ8464 ($f⊗g$) if and only if$g(z)$∈ℐ8464 ($g⊗f$); if$U$is a component of$F$($f$○$g$) and$V$is the component of$F$($g$○$g$) that contains$g(U)$, then$U$is wandering if and only if$V$is wandering; if$U$is periodic, then so is$V$and moreover,$V$is of the same type according to the classification of periodic components as$U$. These results are used to show that certain new classes of entire functions do not have wandering domains.

Let ($A$_{$0, A$}_{$1$}) be a compatible pair of quasi-Banach spaces and 1et$A$be a corresponding space of real interpolation type such that$A$_{$0$}∩$A$_{$1$}is not dense in$A$. Upper and lower estimates are obtained for the distance of any element$f$of$A$from$A$_{$0$}∩$A$_{$1$}. These lead to formulae for the distance in a large number of concrete situations, such as when$A$_{$0$}∩$A$_{$1$}=$L$^{$∞$}and$A$is either weak-$L$^{q}, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.

If$n$is a non-negative integer, then denote by ∂^{-$n$}$H$^{∞}the space of all complex-valued functions$f$defined on $\mathbb{D}$ such that$f$,$f$^{(1)},$f$^{(2)},...,$f$^{($n$)}belong to$H$^{∞}, with the norm $$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$$ We prove bounds on the solution in the corona problem for ∂^{-$n$}$H$^{∞}. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space$H$^{∞}.