Estimating the module function which is analytic at rectilinear strip

The compiex anaiysis of the roie and vaiues of anaiytic function’s estimates is carried out within the range if a respective estimate is known at the boundary or part of the boundary of this area. Such estimations piay a significant roie in the appiications of the theories of the functions of compiex variabies. It is enough to recaii the Riesz - Thorin theorem or the theorem of anaiyticai capacity. In l966 Yu.I. Masiyakov pubiished the resuits of estimations of the anaiyticai function moduie inside the right haif-piane, provided that there was some estimate of decreasing the function moduie’ on the conjugate axis (see [4]). It has been found that the estimates of this kind are fair not oniy for anaiyticai continuous up to the border and iimited functions, but aiso for functions from the ciasses of I.I. Privaiov (see [5]), where at the border the estimate of ‘everywhere’ is repiaced by “aimost everywhere” and the ciass of functions is significantiy expanded, where the resuit is fair (see [2; 7]). Simiiar estimates for the moduie of functions of I.I. Privaiov can be obtained not oniy in the right haif-piane, but aiso in a singie circie (see [l]), and in a straight iane (see [6]). Ciassification through N p (П ), the anaiyticai ciass in a straight iane of functions í y П =ìx +iy :tü2 ýsatisfies the condition: î þ +¥ + p - x l) sup-ty +t2 2-xí1(ln f (x +iy ))ïî_ ¥+t2e dxý¥,ïþ p >O ;2) iim et1ln+ t f (x +iy )dy =O , x ®¥ - 2 ìin a, at a >l;where in+a =í îO, at O a l. This function ciass is caiied the ciass of I.I. Privaiov in a straight iane. Ciassification through “B” ciass of positive, increasing, continuous functions j(t) at[0, +¥) meets the conditions: a) p(et )convex down at t ³0 ; ¥ b) 1p(t )e-t dt +¥;0c) iim t ®+¥ p ( et ) t = +¥. Theorem 1. Let the function f (z)ÎNp (П ), р > 1, and at the border of П aimost everywhere satisfy the condition f (x ±i =töe-φ(ExE)ç 2 ÷, (-¥x +¥). è ø where φ (t )B. Then everywhere in П the foiiowing equation is vaiid f ( z ) K e-φ(EzE), K= eφ(1)-φ( 0).