Distortion theorems for lemniscates and level loci of Green's functions

Extension and some applications of the coincidence theorems

“…However, Corollary 3(c) combined with Theorem 8 contains three out of the four estimates of Theorem 4 in [7]. In addition our Corollary 4 is identical with Corollary 1 in [7], which in turn implies Corollaries 2-4 of [7, p. 66].…”

“…Inequality (8) In the particular case when c>0 and ¿" are lemniscates Corollary 3 follows also from two theorems due to Shaffer [7,Theorems 3 and 4], where inequality (8) is deduced for a region containing {w| | vvj S 3} but where the number 3 cannot be replaced by a smaller number. However, Corollary 3(c) combined with Theorem 8 contains three out of the four estimates of Theorem 4 in [7].…”

“…However, Corollary 3(c) combined with Theorem 8 contains three out of the four estimates of Theorem 4 in [7]. In addition our Corollary 4 is identical with Corollary 1 in [7], which in turn implies Corollaries 2-4 of [7, p. 66]. We thus conclude that the analytic method which is applicable to a wide class of functions even though it does not yield the strongest possible results in the case of lemniscates is sufficient to yield exact results concerning the sufficient conditions for the convexity of lemniscates.…”

On the convexity of lemniscates

“…It was proved by Erdös, Herzog and Piranian [l ] that Li is convex if all the f,-are inscribed in a disc of radius a^sin 7r/8/(l+sin ir/8). This estimate was improved by the author [3] to a:S21/2 -1 =.414. It is the object of this note to improve these bounds; a sharp result is obtained for the case of a real polynomial.…”