ON PARTIAL SOLUTIONS TO CONJECTURES FOR RADIUS PROBLEMS INVOLVING LEMNISCATE OF BERNOULLI

Given a function f analytic in open disk centred at origin of radius unity and satisfying the condition |f(z)/g(z)-1|<1 for a analytic function g with certain prescribed conditions in the unit disk, radii constants Rare determined for the values of R z f '(Rz)/f(Rz) to lie inside the domain enclosed by the curve|w(2)-1|= 1 (lemniscate of Bernoulli). This, in turn, provides a partial solution to the conjectures and problems for determination of sharp bounds R for such functions f