Estimates of the conformal modulus of a set of domains and covering theorems for univalent functions

Abstract

where the integrals are taken, for example, along linear intervals. Let O,(r darZ~. Q(,~-~ a~,az) dwz-(3) Let ])(a~,az) denote a doubly connected domain of the ~-plane which is symmetric with respect to the circle Iwl=~ and whose inner boundary continuum for a=o is the interval o<ur<9,~--maxI~,,~zl, for O<a<~-(consists of three analytic arcs),which join the point ~(a~,a~) respectively, with the points o~ a~az and are the closures of the critical trajectories of the differential Q(w~o,,~z)d~ z , and for a=~ is the interval ~r-~:,-~z<~<~o Let the function M(~,az) be defined by the conditions: KIg)/K(9) for ~=o, 2(I~' ~ l~'dllr f~ 0(~'{' K'(~)/K~x),, = 9'*.AZ2.L for CX--~ ~+?~?z ~ ,--Z " THEOREM 1. The maximum of the con.formal modulus in the set ~{o,a~az}~t ,-t c~ 9 g; ), where a~--~ ~z= ~z~-~ o<~<i (~=~z),o4~ ~ equals ~(a~,oz) and is realized only by the domain ]) (~ a,) . Proof. By Theorem 5 of [1] there exists only one extremal domain of the considered problem and this domain is symmetric with respect to the circle I~l=i. Let the indicated domain be denoted by ]) and its coniormal modulus by ~ . First we consider the case 0(~<~ .

Problem of the maximum of the n-th diameter in the hyperbolic plane

“…In [4] one gives the conditions which determine the zero ~=6 of differential (I) in circle and the hyperbolic capacity C(k~('~,~) These conditions analytically express the fact that the set ~\~ for differential (i) is an annular domain for this differential and that each trajectory of differential (i) Subsequently, one has given other proofs and one has obtained refinements of this result. Thus, it has been proved [6,9] *In the proof of Proposition 1 of [6], at the computation of one of the elliptic integrals, a mistake has been made and, therefore, the expression given in [6] for ~<~)(~, ~/2-~) is not correct.…”

“…Namely, we prove the following theorem (using the notations from the Introduction).Equality in(6) holds only in the case E=ie ~L3s ~, being a real number.For the proof of Theorem 2.1 one makes use of the geometric characteristicsof the extremal configuration of the given problem, obtained by the method of boundary variations of[13] and of the geometric properties of the continua of least hyperbolic capacity, established in[4].…”

Problem of the maximum of the product of the conformal radii of nonoverlapping domains

“…* Regarding the above-mentioned problem on the capacity, in the case n = 3 this problem is completely solved: for the zeros of the associated quadratic differential and for the capacity of the continuum E(5~,6z~6 ~) one has obtained explicit expressions in terms of the Jacobi elliptic functions (see [15] and also [2, Theorems 1.5 and 1.6]). 9 Subsequently, the solutions of problems 1-3 in the above indicated cases have been obtained also by other methods and one has obtained various generalizations of these results.…”

“…Therefore, for solving problem 1 in the case of (15) these ~(~), corresponding to the extremal configuraNow it is easy to see that polynomial tion I)~)K=I,...)4, has the form P (~) = (z -~o)(Z-6~ C~o) ) (~ T 6z(~0)) (g-~a (~o)), ( 16 ) where ~o is some of the zeros of P(~) . Indeed, let ~0 be one of the zeros of P(~) .…”

“…Since C~p E~I~I,~), as a function of ~, is expressed explicitly in terms of the Jacobi elliptic functions (see [15] and also [ …”