Estimating the module function which is analytic at rectilinear strip

The complex analysis of the role and values of analytic 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 play a significant role in the applications of the theories of the functions of complex variables. It is enough to recall the Riesz – Thorin theorem or the theorem of analytical capacity. In 1966 Yu.I. Maslyakov published the results of estimations of the analytical function module inside the right half-plane, provided that there was some estimate of decreasing the function module’ on the conjugate axis (see [4]). It has been found that the estimates of this kind are fair not only for analytical continuous up to the border and limited functions, but also for functions from the classes of I.I. Privalov (see [5]), where at the border the estimate of ‘everywhere’ is replaced by “almost everywhere” and the class of functions is significantly expanded, where the result is fair (see [2; 7]). Similar estimates for the module of functions of I.I. Privalov can be obtained not only in the right half-plane, but also in a single circle (see [1]), and in a straight lane (see [6]). Classification through $N_p(\Pi)$, the analytical class in a straight lane of functions $\displaystyle{\Pi=\left\{x+iy:|y|<\frac{\pi}{2}\right\}}$ satisfies the condition:
1) $\displaystyle{\sup_{-\frac{\pi}{2}<y<+\frac{\pi}{2}}\left\{\int_{-\infty}^{+\infty}\left(ln^{+}|f(x+iy)|\right)^{p}e^{-x}dx\right\}<\infty, p>0; }$
2) $\displaystyle{\lim_{x\rightarrow\infty}\frac{e^{-x}}{\pi} \int_{-\frac{\pi}{2}}^{+\frac{\pi}{2}}ln^{+}|f(x+iy)|dy=0, }$
where $\displaystyle{\ln^{+}a=\left\{
\begin{array}{c} \ln a, \, \text{at} \, a>1; \\ 0, \, \text{at} \, 0<a\leq 1 . \\ \end{array}
\right. }$
This function class is called the class of I.I. Privalov in a straight lane.
Classification through “B” class of positive, increasing, continuous functions $\varphi(t)$ at $[0,+\infty)$ meets the conditions:
a) $\displaystyle{\varphi\left(e^t\right)}$ convex down at $t\geq0$;
b) $\displaystyle{\int_{0}^{\infty}\varphi\left(t\right)e^{-t}dt<+\infty}$;
c) $\displaystyle{\lim_{t\rightarrow+\infty}\frac{\varphi\left(e^t\right)}{t}=+\infty}$.
Theorem 1.
Let the function $f(z)\in N_{p}(\Pi),\, p>1$, and at the border of $\Pi$ aдmost everywhere satisfy the condition
$$\left|f\left(x\pm i\frac{\pi}{2}\right)\right|\leq e{-\varphi(k)},\,(-\infty<x<+\infty). $$
where $\varphi(t)\in B$.
Then everywhere in $\Pi$ the following equation is valid
$$|f(z)|\leq Ke^{-\varphi(z)}, K=e^{\varphi(1)-\varphi(0)}. $$