Multivalent and Meromorphic Functions of Bounded Boundary Rotation

Lemma 1.1. Let $f(z) \in V_k(p)$ . Then $$\lim_{r\to 1^{-}} \int_{0}^{2\pi} \left| \operatorname{Re} \left\{ 1 + \frac{re^{i\theta}f''(re^{i\theta})}{f'(re^{i\theta})} \right\} \right| d\theta \text{ exists.}$$

Theorem 1.2. Let $f(z) \in V_k(p)$ . Then f(z) is at most max $[p, \{pk/2 - 1\}]$ valent, where $\{pk/2 - 1\}$ denotes the smallest integer greater than pk/2 - 1. Received August 12, 1973. This research is part of a Ph.D. thesis written under the direction of Professor William E. Kirwan at the University of Maryland.

Corollary 1.3. Let $f(z) \in V(p)$ with k < 2 + 2/p. Then f is at most p-valent in U. Our next goal is to obtain representation formulas for $V_k(p)$ . We will need to use the functions (1.3) $$\psi(z, z_j) = \frac{(z - z_j)(1 - \bar{z}_j z)}{z}$$ which have been employed by Hummel [6] and others.

Lemma 1.4. Let $f(z) = a_q z^q + \dots (q \ge 1)$ belong to $V_k(p)$ and have non-zero critical points $z_1, \dots z_{p-q}$ , counting multiplicities. Let $$g(z) = \int_0^z \prod_{j=1}^{p-q} \psi(z, z_j)^{-1} f'(z) dz.$$ Then g(z) has p-1 critical points all at z=0 and $g(z) \in V_k(p)$ .

Theorem 1.5. Let $f(z) = a_q z^q + \ldots \in V_k(p)$ and suppose f(z) has non-zero critical points $z_1, \ldots z_{p-q}$ , counting multiplicities. Then: (i) there is a function $\mu(t)$ of bounded variation on $[0, 2\pi]$ with $$\int_{0}^{2\pi} d\mu(t) = 2p \ and \ \int_{0}^{2\pi} |d\mu(t)| \le pk$$ and a constant A such that $$(1.4) \quad f'(z) = Az^{p-1} \prod_{i=1}^{p-q} \psi(z, z_i) \exp \left[ - \int_0^{2\pi} \log (1 - ze^{-it}) d\mu(t) \right];$$ (ii) there is a function $h(z) \in V_k$ and a constant A such that (1.5) $$f'(z) = Az^{p-1} \prod_{j=1}^{p-q} \psi(z, z_j) [h'(z)]^p;$$ (iii) there are two normalized univalent starlike functions $s_1(z)$ and $s_2(z)$ and a constant A such that $$(1.6) \quad f'(z) = Az^{p-1} \prod_{j=1}^{p-q} \psi(z, z_j) \left[ \frac{s_1(z)}{z} \right]^{\frac{1}{4}p(k+2)} \left[ \frac{s_2(z)}{z} \right]^{-\frac{1}{4}p(k-2)}.$$

Theorem 1.6. Let $f(z) = a_q z^q + \ldots \in V_k(p)$ have non-zero critical points $z_1, \ldots z_{p-q}$ , counting multiplicities. Let $R_1 = \max |z_j|$ , $R_2 = \min |z_j|$ . Then with $z = re^{i\theta}$ , $$\begin{aligned} |f'(z)| &\leq \frac{(1+r)^{\frac{1}{2}p(k-2)}}{(1-r)^{\frac{1}{2}p(k-2)}} \frac{q|a_{q}|}{\Pi|z_{j}|} r^{q-1} \prod_{j=1}^{p-q} (r+|z_{j}|) (1+r|z_{j}|) \\ |f'(z)| &\geq \frac{(1-r)^{\frac{1}{2}p(k-2)}}{(1+r)^{\frac{1}{2}p(k-2)}} \frac{q|a_{q}|}{\Pi|z_{j}|} r^{q-1} \prod_{j=1}^{p-q} (r-|z_{j}|) (1-|z_{j}|r), R_{1} < r < 1 \\ |f'(z)| &\geq \frac{(1-r)^{\frac{1}{2}p(k-2)}}{(1+r)^{\frac{1}{2}p(k-2)}} \frac{q|a_{q}|}{\Pi|z_{j}|} r^{q-1} \prod_{j=1}^{p-q} (|z_{j}|-r) (1-|z_{j}|r), 0 < r < R_{2}. \end{aligned}$$

Theorem 1.7. Let $f(z) = a_q z^q + \ldots \in V_k(p)$ have p-q non-zero critical points $z_1, \ldots z_{p-q}$ , counting multiplicities. The f(z) is q-valently convex for $|z| < r_q$ , where rq is the least positive root of $$\frac{p}{2} \left[ \left( 1 + \frac{k}{2} \right) \left( \frac{1-r}{1+r} \right) + \left( 1 - \frac{k}{2} \right) \left( \frac{1+r}{1-r} \right) \right] - \sum_{j=1}^{p-q} \frac{|z_j| (1-r^2)}{(|z_j|-r)(1-|z_j|r)} = 0.$$

Corollary 1.8. Let $f(z) = a_p z^p + \ldots \in V_k(p)$ . Then f(z) is p-valently convex for $|z| < \frac{1}{2}(k - (k^2 - 4)^{1/2})$ and this result is sharp. 2. Coefficient Bounds for $V_{\kappa}(p)$ . Goodman [4] has conjectured that if $f(z) = \sum_{n=1}^{\infty} a_n z^n$ is at most p-valent in U, then for $n \ge p+1$ , $$(2.1) |a_n| \leq \sum_{j=1}^{p} \frac{2j(n+p)!}{(n^2-j^2)(p+j)!(p-j)!(n-p-1)!} |a_j|.$$ The conjecture (2.1) has been verified for certain subclasses of p-valent functions. If f(z) belongs to the class K(p) of p-valent close-to-convex functions defined by Livingston [9], then (2.1) is known for n = p + 1 with no restriction on $a_1, \ldots a_p$ [9] and for $n \ge p + 1$ , provided $a_1 = \ldots = a_{p-2} = 0$ [11]. We recall that if $g(z) = z + \sum_{n=2}^{\infty} b_n z^n \in V_k$ , then $|b_2| \le k/2$ , $|b_3| \le (k^2 + 2)/6$ [8] and $|b_4| \le (k^3 + 8k)/4$ ! [15], with equality if $$g(z) = \frac{1}{k} \left[ \left( \frac{1+z}{1-z} \right)^{k/2} - 1 \right].$$ We will first consider functions $f(z) \in V_k(p)$ with all critical points at the origin.

Theorem 2.1. Let $f(z) = a_p z^p + \ldots \in V_k(p)$ . Then $$(p+1)|a_{p+1}| \le p^2 k|a_p|$$ $$(p+2)|a_{p+2}| \le \left(\frac{p^2k^2}{2} + p\right)p|a_p|$$ $$(p+3)|a_{p+3}| \le \frac{p^2k}{6} (p^2k^2 + 6p + 2)|a_p|.$$ All of these results are sharp, with equality for $F'(z) = p \ a_p[g'(z)]^p$ , where $$g(z) = \frac{1}{k} \left[ \left( \frac{1+z}{1-z} \right)^{k/2} - 1 \right].$$

Lemma 2.2. Let $g(z) = z + b_2 z^2 + \ldots \in V_k(p)$ . Then for any integer $p \ge 1$ , $|3pb_3 - 2p(p-1)b_2| \le p^2k^2/2 - p$ , with equality for $$g(z) = \frac{1}{k} \left[ \left( \frac{1 + iz}{1 - iz} \right)^{k/2} - 1 \right].$$

Theorem 2.3. Let $f(z) = a_{p-1}z^{p-1} + ... \in V_k(p)$ . Then $$(p+1)|a_{p+1}| \leq p^2 k |a_p| + (p-1)|a_{p-1}| (p^2 k^2/2 - p + 1).$$

Lemma 2.4. Let $g(z) = z + b_2 z^2 + \ldots \in V_k$ have real coefficients. Then if $p \ge 2$ , $|1 + 3pb_3 - 2p(p+1)b_2| \le p^2k^2/2 - p - 1$ , and this result is sharp.

Theorem 2.5. Let $f(z) = a_{p-1}z^{p-1} + \ldots \in V_k(p) (p > 2)$ have real coefficients. Then $$(p+1)|a_{p+1}| \le p^2 k|a_p| + (p-1)|a_{p-1}|(p^2 k^2/2 - p - 1)$$ and there is a function in $V_k(p)$ for which equality holds.

Theorem 3.1. Let $f(z) \in V_k(p)$ . Then $$\alpha = \lim_{r \to 1} (1 - r)^{\frac{1}{2}p(k+2)} M(r, f')$$ exists. If $\alpha > 0$ , there is a unique $\theta_0$ so that $$\alpha = \lim_{r \to 1} (1 - r)^{\frac{1}{2}p(k+2)} |f'(re^{i\theta_0})|.$$

Theorem 3.2. Letf(z) = TA anz n € F\(/>). Then* $$\lim_{n\to\infty}\frac{|a_n|}{(n)^{\frac{1}{2p(k+2)-2}}}=\frac{\alpha}{\Gamma(\frac{1}{2}p(k+2))},$$ where a is the constant of Theorem 3.1.

Theorem 4.1. Let $f(z) \in V_k^*(p)$ . Then: $$p + 1 \le N(\infty, f') \le (pk + 2p + 4)/4$$ and $$0 \le N(0, f') \le p(k-2)/4.$$

Lemma 4.2. Let f(z) be meromorphic for |z| < R, $f'(z) \neq 0$ on |z| = R. If $$\int_0^{2\pi} \left| \operatorname{Re} \left\{ 1 + \frac{zf''(z)}{f'(z)} \right\} \right| d\theta < 2\pi [M - N(\infty, f) + 1],$$ where [ ] denotes the greatest integer function, then f is at most M valent and at least max $[2N(\infty, f) - M, 1]$ valent for $|z| \le R$ .

Corollary 4.3. Let $f(z) \in V^*(p)$ have q poles in U. Then f(z) is at least $\max [q+1-pk/2, 1]$ valent and at most pk/2+q-1 valent in U. We note that if k < 2 + 2/p, then for r sufficiently near 1, $$\frac{1}{2\pi} \int_{0}^{2\pi} \left| \operatorname{Re} \left\{ 1 + \frac{re^{i\theta} f''(re^{i\theta})}{f'(re^{i\theta})} \right\} \right| d\theta$$ and hence f(z) belongs to the class $K^*(p)$ of meromorphic close-to-convex functions of order p defined by Livingston [10]. The following result is similar to Theorem 1.5 and its proof will be omitted.

Theorem 4.4. Let $f(z) \in V_k^*(p)$ and suppose f'(z) has zeros at $\alpha_1 \ldots \alpha_n$ and poles at $\beta_1, \ldots, \beta_{n+p+1}$ , counting multiplicities. Then there are two univalent starlike functions $s_1(z)$ and $s_2(z)$ such that $$\frac{f'(z) = \frac{1}{z^{p+1}} \left[ \prod_{j=1}^{n+p+1} \psi(z, B_j) \right]^{-1} \prod_{j=1}^{n} \psi(z, \alpha_j) \left[ \frac{s_1(z)}{z} \right]^{\frac{1}{4}p(k-2)}}{\left[ \frac{s_2(z)}{z} \right]^{\frac{1}{4}p(k+2)}}.$$ We note that Theorem 4.4 gives distortion theorems analogous to Theorems 1.6 and 1.7, but we do not state them here.

Theorem 4.5. Let $f(z) \in V_k^*(p)$ . Then $\alpha = \lim_{r \to 1} (1-r)^{\frac{1}{2}p(k-2)} M(r,f')$ exists. For k > 2, if $\alpha > 0$ , there is a unique $\theta_0$ such that $$\alpha = \lim_{r \to 1} (1 - r)^{\frac{1}{2}p(k-2)} |f'(re^{i\theta_0})|.$$

Theorem 4.6. Let f{z) = Y,n=-Q anz , with k > 2 + 2/p. Then if a denotes the constant of Theorem 4.5, $$\lim_{n\to\infty} \frac{|a_n|}{n^{\frac{1}{2}p(k-2)-2}} = \frac{\alpha}{\Gamma[\frac{1}{2}p(k-2)]}.$$