On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

Theorem 1. Suppose $f \in S_0^*$ with $\mu(\theta) = \lim_{r \to 1} \arg f(re^{i\theta})$ , and $0 . Then <math>D_L^2 f \in H^p$ iff $S'_{\mu} \in H^p$ . If $S'_{\mu} \in N$ then $D_L^2 f \in N$ , and if $S'_{\mu} \notin N$ , then $D_L^2 f \notin H^p$ for all p > 0. PROOF. With f and $\mu$ as above, we have that $$f(z) = z \exp\left\{\frac{1}{\pi} \int_0^{2\pi} \log \frac{1}{1 - e^{-it}z} d\mu(t)\right\},$$ see [14] pp. 209–210; from above, we can form $S_{\mu}(z) = \exp\{-\int_0^{2\pi} \frac{e^{it}+z}{e^{it}-z} d\mu(t)\}$ . Now $$\begin{split} \frac{S'_{\mu}(z)}{S_{\mu}(z)} &= -2 \int_{0}^{2\pi} \frac{e^{it}}{(e^{it}-z)^{2}} \, d\mu(t) \\ &= -2 \int_{0}^{2\pi} \frac{d\mu(t)}{(e^{it}-z)} - 2z \int_{0}^{2\pi} \frac{d\mu(t)}{(e^{it}-z)^{2}} \\ &= -2 \int_{0}^{2\pi} \frac{d\mu(t)}{e^{it}-z} - 2z\pi D_{L}^{2}f. \end{split}$$ The first term of this last expression is in $H^p$ for all p, $0 , see [6] p. 39. It follows that <math>D_L^2 f \in H^p$ iff $S'_{\mu}/S_{\mu} \in H^p$ . However, by [3] Theorem 4 p. 118, $S'_{\mu}/S_{\mu} \in H^p$ iff $S'_{\mu} \in H^p$ , thus the $H^p$ part of the theorem follows. If $S'_{\mu} \in N$ , then again by [3] Corollary 4, p. 118, $S'_{\mu}/S_{\mu} \in N^+$ , hence in N. Finally, if $S'_{\mu} \notin N$ , then by the same corollary, $S'_{\mu}/S_{\mu} \notin N^+$ , hence $S'_{\mu}/S_{\mu} \notin H^p$ for all p > 0. The proof is complete. For purposes of orientation let us note the following: first, for all $f \in S$ , $\log f(z)/z \in \bigcap_{p < \infty} H^p$ . Now if $f \in S$ , $D_L^1 f = \frac{1}{\pi} \int_0^{2\pi} \frac{d\mu(t)}{(e^{\mu}-z)}$ where $\mu$ is the boundary argument function for f, where $\mu$ need not be singular [14], pp. 209–210; thus $D_L^1 f \in \bigcap_{p < 1} H^p$ . Recall that by a theorem of Hardy and Littlewood ([6] p. 88) $g' \in H^p$ , $0 implies <math>g \in H^{\frac{p}{1-p}}$ , so $C_1 = \{f : D_L^1 f \in \bigcap_{p < 1} H^p\} \subset \bigcap_{p < \infty} H^p$ , and thus any $f \in S$ is in the subclass $C_1$ of $\bigcap_{p < \infty} H^p$ . It is then natural to ask whether $S^ \subset C_2 = \{f : D_L^2 f \in \bigcap_{p < 1/2} H^p\} \subset C_1 \subset \bigcap_{p < \infty} H^p$ . Theorem 1 and the corollaries in Section 3 will show that the first inclusion does not always hold; counterexamples will be members of $S_0$ . Of course, for arbitrary starlike functions one can easily have $D_L^n f \in \bigcap_{p < \infty} H^p$ for any n; take f(z) = z, for example. See also [8] for other subclasses of $\bigcap_{p < \infty} H^p$ to which S or $S$ may belong. We close with some final remarks on the quantity $D_L^2 f$ for $f \in S^*$ . Note that $D_L^1 f(z) = f'(z)/f(z) = 1/z$ if $z \neq 0$ , and since $D_L^1 f \in \bigcap_{p < 1} H^p$ , it follows that $M_p(r,f'/f) < \infty$ as $r \to 1$ , for all p < 1. The identity $D_L^2 f(z) = f''(z)/f(z) - [f'(z)/f(z)]^2 + 1/z^2$ for $z \neq 0$ thus gives that $D_L^2 f \in H^p$ iff $M_p(r,f''/f) < \infty$ as $r \to 1$ , if $0 . So we may use <math>D_L^2 f$ and f''/f interchangeably in any statements of our theorems. Secondly, we have that $1 + zD_L^1 f = zf'(z)/f(z)$ , so $\text{Re}[1 + zD_L^1 f] = \frac{\partial}{\partial \theta} \arg f(re^{i\theta})$ where $z = re^{i\theta}$ ; this shows that $D_L^1 f$ is related to $\frac{\partial}{\partial \theta} \arg f(re^{i\theta})$ . For $D_L^2 f$ we have: PROPOSITION. Let $f \in S^*$ . If $D_L^2 f \in H^p$ for some $0 , then <math>M_p(r,g) < \infty$ as $r \to 1$ , where $g(re^{i\theta}) = \frac{\partial^2}{\partial^2 \theta} \arg f(re^{i\theta})$ . PROOF. Differentiating both sides of the identity $1 + zD_L^1 f(z) = zf'(z)/f(z)$ with respect to $\theta$ and taking real parts gives $$\operatorname{Re}[iz^{2}D_{L}^{2}f(z) + izD_{L}^{1}f(z)] = \operatorname{Re}\left[\frac{\partial}{\partial\theta} \frac{zf'(z)}{f(z)}\right] = \frac{\partial}{\partial\theta}\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right] = \frac{\partial^{2}}{\partial\theta^{2}}\operatorname{arg}f(re^{i\theta}).$$ The result follows since $D_L^1 f \in \bigcap_{p<1} H^p$ . Thus a mean growth condition of this sort on $D_L^2 f$ implies one of the same sort on $\frac{\partial^2}{\partial \theta^2} \arg f(re^{i\theta})$ . 3. Corollaries. We now use the results in [1]–[5] to relate growth classes for $S'_{\mu}$ to conditions on $\mu$ itself, and thus obtain corollaries relating $D_{t}^{2}f$ to $\mu$ directly. We first assume that $\mu$ is continuous. An example of $f \in S_0^*$ with $\mu$ continuous is given in [12], Section 5, where $\mu$ is the Lebesgue function of the standard Cantor set on $[0, 2\pi]$ normalized so that $\mu(2\pi) - \mu(0) = 2\pi$ ; similar examples will occur in Corollary 1. Note that this function has an image which contains no angular sectors with vertex at the origin and positive angular spread ([15], p. 591). Also, this function is shown to be bounded in [12]. For arbitrary $\mu$ continuous we have the standard modulus of continuity $\omega_{\mu}(t) = \sup_{|x-y| \le t} |\mu(x) - \mu(y)|$ ; since $\mu$ is continuous, $\omega_{\mu}(t) \to 0$ as $t \to 0$ , and since $\mu$ is singular, $\omega_{\mu}(t)/t \to \infty$ as $t \to 0$ ([1], p. 315). Our first corollary involves a condition on $\omega_{\mu}(t)$ :

Corollary 1. Suppose that $f \in S_0$ , $\mu$ is continuous and $\omega_{\mu}(t) = O(t^{\alpha})$ , for some $\alpha$ , $0 < \alpha < 1$ . Then $D_L^2 f \notin H^{\frac{1-\alpha}{2-\alpha}}$ . Furthermore, this is best possible in the sense that for each $\alpha$ , $0 < \alpha < 1$ , there is an $f_{\alpha} \in S_0$ with argument function $\mu_{\alpha}$ such that $\omega_{\mu_{\alpha}}(t) = O(t^{\alpha})$ and $f_{\alpha}' \in H^p$ for all $p < \frac{1-\alpha}{2-\alpha}$ . PROOF. Suppose $f \in S_0$ with $\mu$ continuous and $\omega_{\mu}(t) = O(t^{\alpha})$ , for some $\alpha$ , $0 < \alpha < 1$ . Then if $S_{\mu}$ is the associated singular inner function, we have by [1] p. 341 that $S'_{\mu} \notin H^{\frac{1-\alpha}{2-\alpha}}$ , so by our Theorem 1 the first statement follows. For the second part, let $0 < \alpha < 1$ be fixed, and define $\omega_{\alpha}(t) = t^{\alpha}/(2\pi)^{\alpha}$ . Then Ahern in [1] pp. 323-326 constructs a Cantor set "of constant ratio $2^{-\frac{1}{\alpha}}$ " whose Lebesgue function $\lambda_{\alpha}$ has a modulus of continuity $\omega_{\lambda_{\alpha}}(t)$ satisfying $\frac{1}{2}t^{\alpha}/(2\pi)^{\alpha} \le w_{\lambda\alpha}(t) \le 4t^{\alpha}/(2\pi)^{\alpha}$ , and thus the associated singular inner function $S_{\lambda_{\alpha}}$ has $S'_{\lambda_{\alpha}} \in H^p$ for all $p < \frac{1-\alpha}{2-\alpha}$ by [1], p. 346. By replacing $\lambda_{\alpha}$ by $\mu_{\alpha} = 2\pi\lambda_{\alpha}$ we obtain a new singular function $S_{\mu_{\alpha}}$ with modulus of continuity $O(t^{\alpha})$ where $S'_{\mu_{\alpha}} \in H^p$ for all $p < \frac{1-\alpha}{2-\alpha}$ ([3], Theorem 4 can be used to see this last statement). Now define $f_{\alpha}(z) = z \exp\left\{\frac{1}{\pi}\int_{0}^{2\pi}\log\frac{1}{1-e^{-it}z}d\mu_{\alpha}(t)\right\}$ ; then $f \in S_0$ with $f'_{\alpha} \in H^p$ for all $p < \frac{1-\alpha}{2-\alpha}$ by Theorem 1; it is easy to see that $\mu_{\alpha}$ is the boundary argument function for $f_{\alpha}$ . We are done. COMMENTS. 1) The corollary says that in some sense the more smoothly the set of arguments of f is distributed, the worse the behavior of $D_L^2 f$ . - 2) This shows the existence of $f \in S^*$ for which $f \notin C_2 = \{f : f'' \in \bigcap_{p < 1/2} H^p\} \in \bigcap_{p < \infty} H^p$ . - 3) Given any subclass of S, a problem of considerable interest has been to find extremal functions for integral means for functions f in the class, as well as for $f^{(n)}$ and $D_L^n f$ . Baernstein's theorem ([7] p. 215) says that the Koebe function K(z) is extremal for $M_p(r,f)$ , $f \in S$ , $0 , but for <math>f^{(n)}$ , $n \ge 1$ the Koebe function does not necessarily play this role. If p > 2/5, then $M_p(r,f^{(n)}) = O(M_p(r,K^{(n)}))$ , $r \to 1$ for all n [10]. It is also known that $M_p(r,f^{(n)}) \le M_p(r,K^{(n)})$ , 0 , <math>0 < r < 1, $n \ge 1$ for all f in the close to convex class [13]. For starlike functions the bound $M_2^2(r,D_L^1f) = O(M_2^2(r,D_L^1K))$ was obtained in [8] and in [9] it was shown that $M_p(r,zf'(z)/f(z)) \le M_p(r,zK'(z)/K(z))$ for all/ starlike, 0 < p < oo. Now D\K G H^p for all p < 1 /2, but clearly £ cannot be extremal forMp(r,Djf),f G 5\ for any/? < 1/2 by this corollary. It is interesting that this happens even though log/(z)/z is subordinate to log K(z)/z for all/ G 5\ ([7], p. 213). We next remove the requirement that /i be continuous and focus on restrictions on the set[Q,2n]-E=\Jk(ak,bk).

Corollary 2. Suppose/ G S\$ tfftd d/i is supported on E, where [0,27r] — E — Uk(^a k,bk), andpiO) = limr\_^i arg/(r^). Lef 0 < 7 < 1. 77ien - «) #"E\ \h - a\|⁷ < oo, then Djf G 7/¹ ?. - fcJ //E\ |&\ flikl log | ^ < oo, rten D£/ G M PROOF. Both of these results follow immediately from [5] Theorem 1, p. 284 and our Theorem 1. • The corollary may be viewed as saying that the faster the lengths of intervals of constant boundary argument for/ go to zero, the better the behavior of Djf. Finally, we turn to boundary argument functions fi which are essentially step functions: assume that 0 < a\ < b\ = ai < bi = «3 < h = • • • < 2TT and that \i has a jump of 7rAfc at ak, where A^ > 0. Thus the measure dji is purely atomic with weights 7rAjt at dk\ let us call such an argument function purely atomic also. Recall that if/ G SQ has purely atomic boundary argument function, then the image of/ contains a maximal sector of angle 7r\k for each k [actually, the existence of such a sector of argument nX^k is an equivalent condition for having a jump nXk in //] (see [15], p. 591). We also must have £\ Xk — 2.*$

Corollary 3. Suppose/ G SQ and p,(0) = limr\_+i arg/(rel9) is purely atomic with jumps nXk at ak. Let 0 < 7 < 1/2. Then i/T,k^l < °°> we nave ^if ^ H^P> for a^ p<l/2. PROOF. This follows from [4], see also [1] p. 346. • This corollary may be viewed as saying that if the wedge arguments in the image of / go to zero "faster than 1/fc² ", then the behavior of Djf is the best possible over the class SQ. It is perhaps interesting in this regard that ([14] p. 211 ) if a is the largest wedge argument, then Moo(r,/) > C(l — r)~^a for 0 < r < 1, so the smaller the wedge the tamer the maximum modulus is allowed to be. We conclude this section by noting that in [1] are other results stating conditions on E such that 5^ G H^p for some/? < 1/2, these relate to the "type" of E defined by p(e) = Lebesgue measure of {6 : \0—E\ < e} and to functions related to uj^t), see pp. 344-345. These conditions can then be related to / G SQ as we have done in the corollaries. 4. The rate of change of the argument function. In this final section we move from considerations of the quantity Djf to ^ arg/(re^l °). For any starlike function/ with boundary argument function $\mu$ the relation $$\begin{split} \frac{\partial}{\partial \theta} \arg f(re^{i\theta}) &= \mathrm{Re} \bigg[ \int_0^{2\pi} \frac{e^{it} + re^{i\theta}}{e^{it} - re^{i\theta}} \, d\mu(t) \bigg] \\ &= \int_0^{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - t) + r^2} \, d\mu(t) = P_{\mu}(r, \theta) \end{split}$$ holds, where the next to last expression is the familiar Poisson integral of $\mu$ . If $f \in S_0$ , then $\mu$ is singular, so, for example, we have $\lim_{r\to 1} P_\mu(r,\theta) = 0$ a.e., with respect to Lebesgue measure and $\lim_{r\to 1} P_\mu(r,\theta) = \infty$ a.e. $[d\mu]$ , see [2] or [11], p. 77. For any $\beta>0,\ 0< r<1$ , we may define $E(r,\beta)=\{\theta:P_\mu(r,\theta)>\beta\}$ ; if $\mu$ has compact support of measure zero (as it will for $f\in S_0$ ) then $|E(r,\beta)|\to 0$ as $r\to 1$ , where $|E(r,\beta)|$ denotes the Lebesgue measure of $E(r,\beta)$ , see [2], p. 1. In [2] are found bounds from above and below on the rate at which $|E(r,\beta)|\to 0$ as $r\to 1$ ; thus we have bounds on the rate at which $|\{\theta:\frac{\partial}{\partial \theta}\arg f(re^{i\theta})>\beta\}|\to 0$ as $r\to 1$ . Note that the fact that $|E(r,\beta)|\to 0$ for any $\beta$ as $r\to 1$ says that $f\in S_0^*$ is never starlike of order $\beta$ for any $\beta>0$ . In Theorem 2 below we state some bounds from below on the rate of decay to zero of $|\{\theta: \partial/\partial\theta \arg f(re^{i\theta}) > \beta\}|$ . These bounds are not stated in the most general form, greater generality for bounds from above or below may be obtained by referring to Theorems 2 and 4 in [2].

Theorem 2. Suppose $f \in S_0^*$ . Then for any $\beta > 0$ there is a constant $C = C(\beta, f)$ such that $\left|\left\{\theta: \frac{\partial}{\partial \theta} \arg f(re^{i\theta}) > \beta\right\}\right| \geq C\sqrt{1-r}$ as $r \to 1$ . If in addition f has continuous boundary argument $\mu$ with modulus of continuity $\omega_{\mu}(t) = O(t^{\alpha})$ for some $\alpha$ , $0 < \alpha < 1$ , then for any $\beta$ there is a constant $C(f, \beta)$ such that $$\left|\left\{\theta: \frac{\partial}{\partial \theta} \arg f(re^{i\theta}) > \beta\right\}\right| \geq C(1-r)^{\frac{1-\alpha}{2-\alpha}}, \ as \ r \to 1.$$ PROOF. This is [2] Theorems 4 and 5; see also [1] for further details on the calculation of the quantity $\delta(r)$ .