Extreme control for a functional on classes of analytical functions

Let S be the class of holomorphic univalent functions fz) normalized by conditions f(0) = 0, f(0) = 1 in a unit circle E = {z: |z| < 1} functions fz), rated conditions f(0)=0, f(0)=1. Let S (p = 1, 2, ...) is a subclass of the class S of functions possessingp-multiple symmetry of rotation with respect to zero, that is, such that j . 2nk \ 2nk \e J= f(z), k = 1,2,...,p -1. The subclass S is distinguished as an independent class of functions, and S =S. We consider Loewner's equation =-ф,т)^!^!^), ,0) = z т Ц (т)-? (z,т) |z| < 1, 0 <т<да , in which control function ^(x), |ц(т)|=1, is continuous or piecewise-continuous on [0,<»). Functions f (z) = lim e ?(z, т) which we call limiting for solutions of the Loewner equation form a т^да dense subclass of the class S . In this article the problem of finding control functions leading to boundary functions of the f (z) functional I = ln in Loewner's equation on classes S and S is solved by the parametrical z method. The set of values of this functional does not depend on arg z therefore, from now on we suppose z = r, 0 < r < 1. Executing some transformations over Loewner's equation, introducing the designations I?( , t)| = p( , t), ?( , т)Ц(т) = p( , ) y ( , ) and substituting p = I ---1 and y = | I , we have 1 1 + s i -1 1 f (r ln [g(s,t)ds - -ln(1 - r " ) p J p V ) t 11 1 - р where g (s, t) = ----, ст=-. t +1 s' 1 + r" The condition g'(s,t) = 0 yields t(s) = 0 and t(s) = да. The solution t(s) = 0 leads to extreme control functions ^=1 , providing a minimum to the studied functional. Function f (z) =- e S , as applied to the functional I, is a boundary function at which the func- ( + z") " tional reaches the minimum value. As t(s) = да, we find extreme control functions ^ = (-1) , leading to a maximum of the functional I. The boundary function f (z) =- e S pro- ( z") " vides a maximum to the functional I. Setting everywhere p = 1, we find extreme control functions for the functional I on the class S. Keywords: Boundary function, class of univalent holomorphic functions, the maximum value of the functional, minimal value of the functional, Loewner's equation, extreme control function.

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