A value-distribution criterion for the class $L~{\rm log} L$ and some related questions

Counting functions and majorization for Jensen measures

“…We next establish an identity for integral means of / in terms of this counting function. This generalizes Theorems 2 and 9 of [6]. Classical identities of Cartan and of Hardy and Stein occur as special cases.…”

“…// α, z, f e B" then The Green function G Ω is harmonic on Ω \ {/(0)}, while G Ω is harmonic on Ω p \{/(α)} and 0 on Ω\Ω p . It follows from Lemma 2 and the majorization principle (6) that N(w; /, v a ) < h(w) on Ω \ Ω p . Since Λ has a logarithmic pole at /(α) it now follows from the maximum principle that N(w; f, v a ) < h(w) on Ω p , and hence N(w; /, v a ) < h(w) on all of Ω.…”

“…Let Ω be a region containing spec/ such that areaΩ < area(spec/) + ε. By the Hardy-Stein identity (10), Lehto's principle of majorization (6), and the proposition above we have…”

Compact composition operators on 𝐿¹

“…A crucial role was played in [ 11 ] by a special case of C. S. Stanton's remarkable formula for integral means [3], [13], [14]. The ideas of [11], along with the general version of Stanton's formula, provide the key to the main result of this paper.…”

“…We can now state: Stanton's Formula [3], [13], [14]. If g is subharmonic onD, then for any holomorphic self-map b of D and any 0 < r < 1 :…”