Theorems concerning mean values of analytic functions

1.1. The theorems proved in this paper concern mean values of the type (1.11)ϕ = ϕ(x, y) = ϕp (x, y)= 1/2y ∫y -y |f(x + iw) |p dw, wheref(z) =f (x +iy ) is an analytic function regular in a stripα <x <β of finite breadth. We call this strip, S, its frontier, formed by the linesx =α andx =β , F(S), and the closed stripα ≦x ≦β , or S+ F(S), S. We denote generally by S' a closed strip interior to S, i.e. a stripα +δ ≦x ≦β . —δ , where 0 <δ < ½ (β —α ). The numberp is, in general, any positive constant; but in 7 it is restricted to even integral values. We suppose throughout the paper thatf(z) is regular in S and If(z) I continuous in any finite part of S, and sometimes (as in Theorem 1) thatf(z) is regular in S. Throughout 1-5 we suppose, in addition, thatf(z) satisfies what we call condition E, viz. thatf(z) =O (eekIyI ).