Results & Lemmas (5)
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THEOREM 3.1.
THEOREM 3.1. Let fix) Ç YL be a solution of the extremal problem Re /[/] = maximum minimum). Then /(f) must satisfy a differential equation…
THEOREM 3.1. Let fix) Ç YL be a solution of the extremal problem Re {/[/]} = maximum {minimum). Then /(f) must satisfy a differential equation of the form r/'G-y/o-) = POO/GOO M M in > i.
THEOREM 3.2.
THEOREM 3.2. Suppose K(Ç) and L(f) are rational functions of degree n and are regular on |f| = 1. Let Q(f) defined in (24)) have 2k zeros…
THEOREM 3.2. Suppose K(Ç) and L(f) are rational functions of degree n and are regular on |f| = 1. Let Q(f) {defined in (24)) have 2k zeros not on |f| = 1. Then the extremal function /(f) maps |f | > 1 onto the plane cut by n — k or fewer radial slits and /(f) has the form (25) /(f) = n ^ l - ~ J , m < n - k , where m av > 0> 23 av = 2 and \fiv\ = 1. V=l
THEOREM 4.1.
THEOREM 4.1. Let <t>(w) 6 ^ - 1 be a solution of the extremal problem Re /[</>] = maximum for w Ç D, then <j>(w) must satisfy a…
THEOREM 4.1. Let <t>(w) 6 ^ - 1 be a solution of the extremal problem Re {/[</>]} = maximum for w Ç D, then <j>(w) must satisfy a differential equation of the form l{<t>(w)) - l\=^==) +n - n (on) w<t>'(w) = \<j)(w)' q(<t>(w)) k((j>(w)) + k[ ) + m — m + 2n \(j>(w)>/ for all w in D. Equation (30) may also be written as [#'(*)//(*)] = p(z)/q(z) for
THEOREM 4.2.
THEOREM 4.2. Let k(z) and l(z), z = 4>(w), be rational functions in z of degree n and be regular on = 1. Let q(z) (defined in (30)) have 2k…
THEOREM 4.2. Let k(z) and l(z), z = 4>(w), be rational functions in z of degree n and be regular on \z\ = 1. Let q(z) (defined in (30)) have 2k zeros not on \z\ = 1. Thenf(z), the inverse of the extremal function <t>{w), maps \z\ > 1 onto the w plane cut by n — k or fewer radial slits and f(z) has the form https://doi.org/10.4153/CJM-1962-045-8 Published online by Cambridge University Press
THEOREM 4.3. · coeff
THEOREM 4.3. If <j>(w) = w + 60 + 61/w +... belongs to the class Yl~l then IM <~ ^ J,^ = 1,2, 3, for^o^O V + 1/ M < 1, M < 2/3, |ô3| < 1,…
THEOREM 4.3. If <j>(w) = w + 60 + 61/w + . . . belongs to the class Yl~l then IM <~\ ^ J ,^ = 1,2, 3, for^o^O V \V + 1/ M < 1, M < 2/3, |ô3| < 1, for bo = 0. ^4 ZZ JÂese inequalities are sharp. Unfortunately the above trend does not continue for v > 3, that is, the sharp bounds are not obtained by solving p(zo) = 0 for bv and substituting in the maximum values for the other coefficients. Springer (8) has shown for the more general class of univalent functions with bo = 0 that \bz\ < 1 and has co