By A.M. Jerbashian

It is a exact ebook on the topic of the idea of services of a-bounded variety within the half-plane of the complicated aircraft, that's built by way of program of the Liouville integro-differential operator. moreover, the publication comprises advancements of a number of effects corresponding to the Phragmen-Lindelof precept and Nevanlinna Factorization within the Half-Plane, and provides a brand new, similar definition of the classical Hardy areas within the half-plane. The final bankruptcy of the booklet offers an software of the developed idea in addition to M.M.Djrbashian’s thought of Nevanlinna kind periods within the disc within the spectral thought of linear operators. this can be a resolution of an issue time and again acknowledged through M.G.Krein and being of designated curiosity for a very long time.

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Additional info for Functions of a-Bounded Type in the Half-Plane (Advances in Complex Analysis and Its Applications)

Example text

Let a G (—1,1) and ( = ^-{-ir] e G~ be fixed. Then the function Re (pa{wX) ^5 harmonic in G~ and continuous in G~. Proof. 9) + a)^^{w + tC) at the origin. 6) the similar formula for ^o{wX) (= 0) multiplied by \rj\^. This gives ^aiw,C) = \[aJ\rj\ \v\-hiw]-' ~ [a + \rj\+iw]-H{a^ ^ + / - \rjnda ^ {W- \v\ - H ~ ' -W- \v\ + H " ' } K - IvHdcT. But \a^ — \rj\^\ < CaW — \rj\\ (0 < a < +oo), where Ca depends only on a and \r]\. Therefore, estimating the real parts of integrands and using the Lebesgue theorem on dominated convergence, we conclude that Re cp'^{w,Q is continuous in G"~ and /'+00 lim R e ^ ; K 0 - 2 M / a _ I la ^ r ^ d a .

16) for any a e (—1, +oo) and p > 0. Now assume that ^ < h < \r]\ and 0 < p < max{/i, |ry| - h}. 2) 1 /"^^ = WT^^—T /'' (l^|-| p. Hence ^ rW-^\ogK{i-ih^pe'\0\dd 1 (['-' • /'^'^ r(i + a) l^y_|,, 7;,+,; {\v\-\t\r (i^. 17) ^ ^ for a G (—l,+oo), 0 < /i < |7y| and 0 < p < max{/i, |r/| — /i}. One can be convinced that also 27r T ^ - log |6,(C + pe^^ C)M^ - W 27ryo log |6a(C, 01 1 p^ r(l+a) a > 0.

3*. 20) Zk ^ = 0. 21) R^-\-oo 4. 1. Below we shall prove a representation which will be used later, in Ch. 5, for investigation of boundary properties of our Blaschke type products. 1. Let a G (—1,1) be any number, and let {wk} C G~ be a sequence of complex numbers such that simultaneously ^^\lm Wk\^^'^ <-^00 k and ^ |Im tt;^! < + 0 0 . 3) for any e > 0. Besides, whatever be a G (—1,1) there exists a sequence dn iO by which fi{t) = lim / W-"" log Bcc{u-i6n,{wk}) BQ{u-i8n,{wk}) du, -cxD

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