By Norair Arakelian (auth.), N. Arakelian, P. M. Gauthier, G. Sabidussi (eds.)

Hermann Weyl thought of price distribution concept to be the best mathematical success of the 1st half the 20 th century. the current lectures exhibit that this pretty idea continues to be becoming. a massive instrument is complicated approximation and a few of the lectures are dedicated to this subject. Harmonic approximation began to flourish astonishingly quickly in the direction of the top of the twentieth century, and the newest improvement, together with approximation manifolds, are offered here.

Since de Branges proven the Bieberbach conjecture, the first challenge in geometric functionality idea is to discover the ideal worth of the Bloch consistent. After greater than part a century with no growth, a step forward was once lately completed and is gifted. different issues also are offered, together with Jensen measures.

A priceless creation to at the moment lively parts of complicated research and capability idea. might be learn with revenue through either scholars of study and examine mathematicians.

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20, 21]) shows that the Kelvin transform preserves harmonicity: if u E ti{O), then u* E ti(O). 10) m=O and the series is Weierstrass convergent on JRn \ B(xo, R) for each R > r. 37 Uniform and tangential harmonic approximation We indicate a proof in the case where B(xo, r) = B(0,1). The function u* is harmonic on B(0,1) \ {O} and u*(x) = o(U(x,O» as x -t O. 7]. 8, there exist polynomials 1m in tim such that u* = L 1m, so = IlxI1 2- nu*(x*) = L 00 u(x) =L 00 IlxIJ2- n I m (x*) IlxI12-n-2mIm(x) (Jlxll > 1).

Math. Soc. Transl. Ser. 2 122 (1984), 85-97. [A5] N. U. Arakelian, Entire and analytic functions of limited growth with an infinite number of deficient values, Izv. Acad. Nauk Armjan. SSR Ser. Mat. 5 (6) (1970), 486-506. [A6] N. U. Arakelian, Approximation complexe et proprietes des fonctions analytiques, Actes, Congres International des MatMmaticiens (Nice 1970), t. 2, Gauthier-Villars, Paris, 1971, 595-600. [A 7] N. U. Arakelian, On uniform and asymptotic approximation on the real axis by entire functions of restricted growth, Mat.

2) (with A = l/p) by the formula log Bp(z} = -z dt, 10roo t (n(t}) t- z z E C \ [0, oo}. ~/2 the estimate 1I0g Bp(z)l roo t(tn(t} + Izl} dt = log M(lzl, Bp}. ~/2. ,,/2. ,,/(28) it satisfies 1 -2"l z IP 1 :s log Ilmp (JJz) I < 2"lzIP. 26) 22 N. U. 4 and a E (0,(3), where (3::; min{(3p,rr/(2s)}. ,,((28). 23). 25). 22), as IF(z) where = [F(z) F(z) - G(z)]lBp(pz) = + J(z), Jp(pz) - J(z). {3). 4. 16). ",. 5) one can replace the function J by IF. 2) then completes the proof. 3 Indices of functions of order zero In this subsection we present an analytic construction of entire functions of order zero, having a given finite set of complex numbers as its index values.