Phragman-Lindelöf theorems on the growth of regular functions

Phragman - Lindelöf theorems on the growth of regular functions - statements that a function of a complex variable $F(z)$ ${\ displaystyle F (z)}$ $F (z)$ regular in some infinite region $D$ ${\ displaystyle D}$ $D$ and continuous in ${\overline {D}}$ ${\ displaystyle {\ overline {D}}}$ $\ overline {D}$ as well as limited at the border $\partial D$ ${\ displaystyle \ partial D}$ $\ partial D$ areas of $D$ ${\ displaystyle D}$ $D$ , or limited everywhere in ${\overline {D}}$ ${\ displaystyle {\ overline {D}}}$ $\ overline {D}$ or inside $D$ ${\ displaystyle D}$ $D$ grows fast enough - the "faster", the smaller the area $D$ ${\ displaystyle D}$ $D$ .

Content

Phragman - Lindelöf theorem on the upper half-plane

Let the function $F(z)$ ${\ displaystyle F (z)}$ regular in the half-plane $Rez>0$ ${\ displaystyle Rez> 0}$ and continuous in the half-plane $Rez\geqslant 0$ ${\ displaystyle Rez \ geqslant 0}$ , and $\mid F(iy)\mid <C_{0}$ ${\ displaystyle \ mid F (iy) \ mid <C_ {0}}$ , $-\infty <y<\infty$ ${\ displaystyle - \ infty <y <\ infty}$ . Then or $\mid F(z)\mid <C_{0}$ ${\ displaystyle \ mid F (z) \ mid <C_ {0}}$ for all $z$ ${\ displaystyle z}$ , $Rez\geqslant 0$ ${\ displaystyle Rez \ geqslant 0}$ or function $F(z)$ ${\ displaystyle F (z)}$ has in the half plane $Rez\geqslant 0$ ${\ displaystyle Rez \ geqslant 0}$ order $\rho$ ${\ displaystyle \ rho}$ not less than one.

Explanation

Number $\rho$ ${\ displaystyle \ rho}$ called the order of the whole function $F(z)$ ${\ displaystyle F (z)}$ , if a $\rho =\varlimsup _{t\to \infty }{\frac {\ln \ln M_{F}(r)}{\ln r}}$ ${\ displaystyle \ rho = \ varlimsup _ {t \ to \ infty} {\ frac {\ ln \ ln M_ {F} (r)} {\ ln r}}}$ . In other words, the whole function is of order $\rho$ ${\ displaystyle \ rho}$ if for any $\epsilon >0$ ${\ displaystyle \ epsilon> 0}$ there is a constant $C_{\epsilon }$ ${\ displaystyle C _ {\ epsilon}}$ and the sequence of increasing to $\infty$ ${\ displaystyle \ infty}$ positive numbers $r_{k}$ ${\ displaystyle r_ {k}}$ such that

\max _{0\leqslant \varphi \leqslant 2\pi }\mid F(re^{i\varphi })\mid \leqslant C_{\epsilon }exp(r^{\rho +\epsilon })

{\ displaystyle \ max _ {0 \ leqslant \ varphi \ leqslant 2 \ pi} \ mid F (re ^ {i \ varphi}) \ mid \ leqslant C _ {\ epsilon} exp (r ^ {\ rho + \ epsilon} )}

,

$r>0$ ${\ displaystyle r> 0}$ ,

\max _{0\leqslant \varphi \leqslant 2\pi }\mid F(r_{k}e^{i\varphi })\mid \geqslant C_{\epsilon }exp(r_{k}^{\rho +\epsilon })

{\ displaystyle \ max _ {0 \ leqslant \ varphi \ leqslant 2 \ pi} \ mid F (r_ {k} e ^ {i \ varphi}) \ mid \ geqslant C _ {\ epsilon} exp (r_ {k} ^ {\ rho + \ epsilon})}

,

$k=1,2,$ ${\ displaystyle k = 1,2,}$ .

Proof

The proof is in the book ^[1] .

Notes

↑ Methods of interpolation of functions and some of their applications, 1971 , p. 37.

Literature

Ibragimov I. I. Methods of interpolation of functions and some of their applications. - M .: Nauka, 1971. - 518 p.

[_6de3554ae3b26a7b-1] Methods of interpolation of functions and some of their applications, 1971 , p. 37.