Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

Theorem 3.1. Let $f(\zeta) \in \Sigma$ be a solution of the extremal problem $\text{Re}\{J[f]\} = \max \min \pmod{\min \min}$ . Then $f(\zeta)$ must satisfy a differential equation of the form $\zeta f'(\zeta)/f(\zeta) = P(\zeta)/Q(\zeta)$ for all $|\zeta| > 1$ .

Theorem 3.2. Suppose $K(\zeta)$ and $L(\zeta)$ are rational functions of degree n and are regular on $|\zeta| = 1$ . Let $Q(\zeta)$ (defined in (24)) have 2k zeros not on $|\zeta| = 1$ . Then the extremal function $f(\zeta)$ maps $|\zeta| > 1$ onto the plane cut by n - k or fewer radial slits and $f(\zeta)$ has the form (25) $$f(\zeta) = \prod_{p=1}^{m} \left(1 - \frac{\beta_p}{\zeta}\right)^{\alpha_p}, m \leqslant n - k,$$ where $$\alpha_p > 0, \sum_{p=1}^m \alpha_p = 2 \text{ and } |\beta_p| = 1.$$

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 differential equation of the form (30) $$\frac{w\phi'(w)}{\phi(w)} = \frac{l(\phi(w)) - l\left(\frac{1}{\phi(w)}\right) + n - \bar{n}}{k(\phi(w)) + k\left(\frac{1}{\phi(w)}\right) + m - \bar{m} + 2n} = \frac{q(\phi(w))}{p(\phi(w))}$$ for all w in D. Equation (30) may also be written as $$[zf'(z)/f(z)] = p(z)/q(z)$$ for $|z| > 1$ . Another property of the extremal function is obtained upon noting that if p(<t>(w)) and q((j)(w)) are finite, non-zero, and continuous at a point w0 on T then f(z), the inverse of the extremal function 4>(w), maps an arc of \z\ = 1 containing the map of WQ onto a radial line segment. In a manner analogous to Theorem 3.2 we can also prove

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 (31) $$f(z) = z \prod_{p=1}^{m} \left(1 - \frac{\beta_p}{z}\right)^{\alpha_p}, m \leqslant n - k$$ where $$\alpha_p > 0$$ , $\sum_{p=1}^{m} \alpha_p = 2$ , and $|\beta_p| = 1$ . An interesting application of Theorem 4.2 is the coefficient problem for the class $\sum^{-1}$ . In order to compute k(z) and l(z) it is necessary to rewrite the second arguments appearing in the $J_1$ functionals, namely, (32) $$\phi(w) \frac{\phi(\xi_0) + \phi(w)}{\phi(\xi_0) - \phi(w)} = -\left(\frac{2\phi(\xi_0)^2}{\phi(w) - \phi(\xi_0)} + 2\phi(\xi_0) + \phi(w)\right)$$ $$w\phi'(w) \frac{\phi(\xi_0) + \phi(w)}{\phi(\xi_0) - \phi(w)} = -\left(w\phi'(w) + 2\phi(\xi_0) \frac{w\phi'(w)}{\phi(w) - \phi(\xi_0)}\right).$$ It is well known (7) that the function (33) $$\log \frac{\phi(w) - t}{w} = -\sum_{m=1}^{\infty} \frac{1}{m} F'_m(t) w^{-m},$$ the development being valid in the neighbourhood of $w = \infty$ , generates the Faber polynomials $F_m(t)$ of degree m. Differentiating (33) with respect to t yields (34) $$\frac{1}{\phi(w) - t} = \sum_{m=1}^{\infty} \frac{1}{m} F'_m(t) w^{-m}$$ which generates the derivatives of the Faber polynomials. Also, differentiating (33) with respect to w gives (35) $$\frac{w\phi'(w)}{\phi(w) - t} = 1 + \sum_{m=1}^{\infty} F_m(t)w^{-m}.$$ Using (34), (35), and $J[\phi] = b_{\nu}$ we can now write (36) $$k(\zeta_0) = \left[ 2\phi(\xi_0)^2 \frac{1}{\nu} F'_{\nu}(\phi(\xi_0)) + b_{\nu} \right]$$ $$l(\zeta_0) = -[-\nu b_{\nu} + 2\phi(\xi_0) F_{\nu}(\phi(\xi_0))]$$ $$m = b_{\nu}$$ $$n = -\nu b_{\nu}.$$ Our problem is to find the functions of the class $\sum^{-1}$ for which $|b_r|$ is a maximum. Since $\sum^{-1}$ is a normal family we know such functions exist. We may suppose that the extremal $b_r$ is real and hence seek to maximize Re $\{b_r\}$ . Thus Theorem 4.1 applies and we have upon substituting (36) into (30) that the extremal functions f(z) must satisfy (37) $$\frac{\underline{z}f'(z)}{f(z)} = \frac{\frac{z^2}{\nu} F'_{\nu}(z) + (\nu + 1)b_{\nu} + \frac{1}{\nu z^2} \overline{F'_{\nu}\left(\frac{1}{\bar{z}}\right)}}{zF_{\nu}(z) - \frac{1}{z} F_{\nu}\left(\frac{1}{\bar{z}}\right)}, \nu \geqslant 1.$$ It is well known that $|b_0|=|a_0|\leqslant 2$ and $|b_1|=|a_1|\leqslant 1.$ For $\nu=2$ (37) becomes $$(38) \frac{zf'(z)}{f(z)} = \frac{z^3 - b_0 z^2 + 3b_2 - \overline{b_0}/z^2 + 1/z^3}{z^3 - 2b_0 z^2 + (b_0^2 - 2b_1)z - (\overline{b_0}^2 - 2\overline{b_1})/z + 2\overline{b_0}/z^2 - 1/z^3}$$ $$= \frac{p(z)}{q(z)}.$$ We know that p(z) has at least one zero on the unit circle, say at $z=z_0$ , so that $p(z_0)=0$ . If $p(z_0)=0$ , then $3b_2=b_0z_0^2+\bar{b}_0/z_0^2-z_0^3-1/z_0^3$ and hence $3|b_2|\leqslant 2|b_0|+2$ . If $b_0=0$ then $|b_2|\leqslant 2/3$ and if $b_0\neq 0$ then $|b_0|\leqslant 2$ , $|b_2|\leqslant 2$ . Upon computing the derivative of the third Faber polynomial we get $F_3'(z)=3(z^2-2b_0z+b_0^2-b_1)$ so that for $\nu=3$ , $p(z)=z^2-2b_0z+b_0^2-b_1+4b_3+\bar{b}_0^2-\bar{b}_1-2\bar{b}_0/z+1/z^2$ . As we reasoned before, p(z) has at least one root $z=z_0$ and we obtain $4|b_3|\leqslant 2|b_1|+2|b_0|^2+4|b_0|+2$ which implies, $b_0\neq 0$ , that $|b_3|\leqslant 5$ and, $b_0=0$ , that $|b_3|\leqslant 1$ . We have now the following

Theorem 4.3. If $\phi(w) = w + b_0 + b_1/w + \dots$ belongs to the class $\sum^{-1}$ then $$|b_{\nu}| \leqslant \frac{1}{\nu} \binom{2\nu}{\nu+1}$$ , $\nu = 1, 2, 3$ , for $b_0 \neq 0$ and $$|b_1| \leqslant 1$$ , $|b_2| \leqslant 2/3$ , $|b_3| \leqslant 1$ , for $b_0 = 0$ . All these inequalities are sharp. Unfortunately the above trend does not continue for $\nu > 3$ , that is, the sharp bounds are not obtained by solving $p(z_0) = 0$ for $b_{\nu}$ and substituting in the maximum values for the other coefficients. Springer (8) has shown for the more general class of univalent functions with $b_0 = 0$ that $|b_3| \le 1$ and has conjectured that $$|b_{\nu}| \leqslant \frac{1}{\nu} \left( 2 \frac{\nu}{n} \right),$$ $n \ge 2$ , $nk = \nu + 1$ , k an integer, where n is the least prime divisor of $\nu + 1$ . We can show, as did Springer for his class, the following: Let $D_n$ be the domain consisting of the whole w plane except for $n(n \ge 2)$ symmetric radial slits, each of length $(2)^{2/n}$ , originating at the origin. Let $\phi_n(w) = w + c_1^{(n)}/w + c_2^{(n)}/w + \ldots$ be the function mapping $D_n$ conformally on |z| > 1. The inverse of $\phi_n(w)$ is essentially $w = f_n(z) = z(1 + z^{-n})^{2/n}$ . If n is a divisor of $\nu + 1$ then $f_n(z)$ satisfies (37). As Springer shows, the $c_{\nu}^{(n)}$ are the largest when n is the least prime divisor of $\nu + 1$ . If $b_0 \neq 0$ then $w = f_1(z) = z(1+z^{-1})^2$ satisfies (37). As we have found, this function is extremal for $|b_2|$ and $|b_3|$ . Thus one might suspect in the general case with $b_0 \neq 0$ that $$|b_{\nu}| \leqslant \frac{1}{\nu} \binom{2\nu}{\nu+1}$$ which is also the bound given by Loewner for the coefficients of the inverse schlicht functions in the unit circle.