Analitik fonksiyonların bazı sınır özellikleri hakkında

Abstract

This work is devoted to strenghtening of the theorems of Hardy-Littlewood type with normal majorants. Investigation of finite differences of the classes of functions defined on compact subsets of the complex plane has an important role in the modern function theory. This subject has applications in the study of smoothness of functions on the closure of their domain, of smoothness of complex homeomorphisms, of singular integrals and integrals of Cauchy type, of the Riemmann boundary value problem, in approximation theory etc. Let C be one-point compactification of the complex plane. For a set D C C let the boundary of D in C be denoted by dD and let 3D = Cfl dD. Let G C C be an open set, and let / be a function continous on G and analytic on G. Under which conditions onGcC and the majorant uj(6) (a function u: (0, +00) - » [ 0, +00) satisfying certain conditions) the following implications are true: 1) If 1/(0 - /(*)l < «(IC - *!), vc, ze dG, c î z then 1/(0 -/(*)l < Cu,{\(-z\), vc* eü,C ^z (l) where C > 1 is a constant independent of ( and z. 2) For a given point z0 e dG, if 1/(0 - /(*>)! < «(IC - *bl), vc edG,c? z0 then 1/(0 - /(«Ol < Cu(\C - *b|), VC e Ü, C * zo (2) where C > 1 is a constant independent of C- v Hardy G. H. and Littlewood J. E. [1] proved the implication. 1) for G a circle and u){8) = 8a (a = const ? (0,1]). For G a Jordan domain and u)(S) - Sa Warschawski S. E. [2] proved the implication 2), Walsh J. L. and Sewell W. E. [3] proved the implication 1) so that in both results C = 1 (Similar results are also obtained for u)(6) = S\ ln£|.) In 1942, Sewell in his monograph [4] put forward a group of open problems now called Warschawski- Walsh-Sewell problems. One of them is the generalization of the results obtained by Warshawski- Walsh-Sewell to domains more general than Jordan domains and to majorants of modulus of continuity type more general ih&nSa, S\lnS\. On this subject, certain results are obtained by Magnaradze L. G., Gagua M. B., Geronimus Y. L., Brudny Y. A., Hopenhaus I. E. and Trahimchuk Y. Y. Since 1979 the problems above are completely solved for uj{8) = Sa majorants by Schekorskii A. I. [13] " Tamrazov P. M. [11], Gehring F. W., Hayman W. K. and Hinkkanen A. [14]. In 1984, Aliyev T. H. and Tamrazov P. M. put forward the following problems: 1. Effect of nonunivalence of the function in the inequalities (1) and (2), 2. Generalization of the above results to meromorphic functions. For Sa and bilogarithmic concave majorants (that is, logw(ei) is concave) both problems are completely solved in terms of Green function by Aliyev T. H. and Tamrazov P. M. [15, 16]. Also the effect of nonunivalence in (1) and (2) is solved [17]. For normal majorants and sufficiently general set the problems 1 and 2 are solved by Aliyev T. H. [17]. Intersection of the class of normal majorants and the class of bilogarithmic majorants is not empty and they don't include each other. In this work, the problems 1 and 2 are studied for the class of normal majo rants and the inequality (2) is stregthened. A function tOf(S) continuous on positive real-axis, nondecreasing and semi- additive, which satifies the condition lim ut(8) = 0 is called modulus of conti- $-?+0 Jy ' nuity major ant. For a nondecreasing function u(S): (0, -t-oo) - > [ 0, +oo), if there exist numbers a > 1 and 7 > 0 so that w{t6) < <r*m0,V*>1 vi then w(6) is called normal majorant of class of (a, 7). For a function 0(t): [l,+oo) - » (0,+oo), if the function logV>(e*) is concave on (0, +00) then if>(t) is called bilogarithmic concave function. For a nondecreasing function <="" 4="" style="margin: 0px; padding: 0px; outline: 0px;">{t) u(6), V<5 > 0, Vt > 1 then ui(t) is called majorant having coefficient of normality ip(t). The classes of normal majorants and of nondecreasing majorants having nor mality coefficients coincide, every majorant of modulus of continuity type is a normal majorant of class (2, 1). Let Cap(if) = C{K) denote the logarithmic capacity of the set K. Let H denote the class of sets E C C having zero capacity. Let G C C be an open set. For z G G and t ? (0, +00) let C* = C*p({Ç:\(-z\{v)ij){u{zo, «)), z0 İ G,t > 0 v>t e(G) = sup e(G,z). zedG If the function / is meromorphic on G let k(f, w) denote order of value f(w) for w e G. For a point z ? C and a set K C C let us define p(z,K)=M\z-(\. vii For a domain GcC and a point Co ? G let ga(-, Co) denote generalized Green function of G. Let B C C be an open set. If two points w, C belong to same connected component Bj of B, then gsiT, C) is denoted by #&,-(«>, C)> generalized Green function of the domain jBj. If w, ( belong to different connected components of B then 5fs(w, C) = 0- IQ this case gg(w,Ç) is called generalized Green function of the open set B [22]. In the first section of the second chapter the following local result is given: Theorem [23, 16] Let G C C be a bounded open set; Q C C \ G a set containing the point z0: Q G Af; w a majorant having coefficient of normality ^;/:G->Ca mermorphic function with a finite number of poles, V denote the set of all poles of / in G. If the function / is bounded on every portion of G separated from the poles and Hm |/(C)| <«(|*-*b|), V*?ÖG\Ç then 1/(01 exp x exp L pep o fc(/> w) w:f(w)=0 V(eG\v hence if zq is an isolated point of dG also the term corresponding to the point w = z0 is also included in the sum in the inequality above. </r*m