“…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].…”
supporting
confidence: 59%
“…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].…”
mentioning
confidence: 91%
“…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.…”
“…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].…”
supporting
confidence: 59%
“…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].…”
mentioning
confidence: 91%
“…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.…”
“…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.…”
Let Li denote the lemniscate | n?-i(z-f»)| =L Assume the poles f, are inscribed in the disc \z\ Sa. Let Zo = w~'/ .?_if" Conditions for the convexity of ¿i are established in terms of a and Zo. Sharp bounds are derived for real f».
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