On the integral means of univalent, meromorphic functions

Theorem 1. Let $\Phi$ be a convex nondecreasing function on $(-\infty, \infty)$ . Then for all $f \in \Sigma(p)$ and 0 < r < 1, $$\int_{-\pi}^{\pi} \varPhi(\pm \, \log \, |f(re^{i\theta})|) d\theta \leqq \int_{-\pi}^{\pi} \varPhi(\pm \, \log \, |F_{p}(re^{i\theta})|) d\theta$$ where $${F}_{\scriptscriptstyle p}(z) = \frac{(1\,+\,z)^{\scriptscriptstyle 2}}{\left(1\,-\,\frac{z}{p}\right)\!(1\,-\,pz)} \;.$$

Theorem 2. Let $f \in \Sigma(p)$ , then for all $\lambda$ , $-\infty < \lambda < \infty$ , and 0 < r < 1, $$(2.14) \qquad \int_{-\pi}^{\pi} |f(re^{i\theta})|^{\lambda} d\theta \leq \int_{-\pi}^{\pi} |F_{p}(re^{i\theta})|^{\lambda} d\theta.$$ 3. Applications of Theorem 2.

Theorem 3. Let $f \in \Sigma(p)$ and 0 < r < 1, then for |z| = r. (3.1) $$F_{p}(-r) \leq |f(z)| \leq |F_{p}(r)|.$$ REMARK. Inequality (3.1) was obtained earlier by Libera and the author [4] for the class $\Lambda^*(p) \subset \Sigma(p)$ .

Theorem 4. Let $f \in \varSigma(p)$ and $f(z) = 1 + \sum_{n=1}^\infty a_n z^n$ for |z| < p, then $$\frac{(1-p)^2}{p} \le |a_1| \le \frac{(1+p)^2}{p}.$$ The inequalities are sharp.

Theorem 5. Let $f \in \Sigma(p)$ and $w \in \Delta$ , $w \neq p$ , then $$(3.3) \qquad \frac{1}{(1-|w|^2)} \frac{(1-|a|)^2}{|a|} \leq \left| \frac{f'(w)}{f(w)} \right| \leq \frac{1}{(1-|w|^2)} \frac{(1+|a|)^2}{|a|}$$ where $a = (p - w)/(1 - p\bar{w})$ . Moreover, given $w \in \Delta$ , $w \neq p$ , there exists a function $f \in \Sigma(p)$ for which equality is obtained on the right side of (3.3) and similarly for the left side of (3.3).

Theorem 6. Let $\Phi$ be a convex nondecreasing function on $(-\infty, \infty)$ . Then for all $f \in \Sigma(p, q)$ and 0 < r < 1, $$\int_{-\pi}^\pi \varPhi(\pm \log|f(re^{i\theta})|)d\theta \leqq \int_{-\pi}^\pi \varPhi(\pm \log|G_{(p,q)}(re^{i\theta})|)d\theta \text{ .}$$ Proof. We first consider the inequality $$(4.1) \qquad \qquad \int_{-\pi}^{\pi} \varPhi(\log|f(re^{i\theta})|) d\theta \leqq \int_{-\pi}^{\pi} \varPhi(\log|G_{(p,q)}(re^{i\theta})|) d\theta \; .$$ Let $$u(w) = \begin{bmatrix} -\log \mid f^{-1}(w) \mid , & w \in f(\Delta) \ 0 , & \text{otherwise} \end{cases}$$ and $$v(w) = \begin{bmatrix} -\log |G_{(p,q)}^{-1}(w)| \;, & w \in G_{(p,q)}(\varDelta) \ 0 \;, & \text{otherwise} \;. \end{cases}$$ Arguing as in Theorem 1, inequality (4.1) will be proven if we can prove that $$(4.2) (u^ - v^)(s) \le 0 , s \in R .$$ For this purpose we let $$d_f = \inf_{w \notin f(J)} |w| \qquad \text{and} \qquad D_f = \sup_{w \notin f(J)} |w|$$ and $$R = (-\infty, -D_f) \cup [-D_f, -d_f] \cup (-d_f, 0) \cup [0, +\infty)$$ . Case (i). $s \in [0, +\infty)$ . This case is exactly as in Theorem 1. Case (ii). $s \in (-\infty, -D_f)$ . The argument is the same as the corresponding case in Theorem 1. Case (iii). $s \in (-d_f, 0)$ . Since $\{w \colon |w| < d_f\} \subset f(\Delta)$ , we have that $u(w) + \log |w-1|$ is harmonic in $|w| < d_f$ and $v(w) + \log |w-1|$ is subharmonic in $|w| < d_f$ . (The term $\log |w-1|$ is only necessary when 1 < d.) It follows that (u-v) is superharmonic for $|w| < d_f$ and therefore $$(u^-v^)(s)=\int_{-\pi}^{\pi}(u-v)(|s|e^{i\theta})d\theta \le 2\pi(u-v)(0)=0$$ . Case (iv). $s[-D_f - d_f]$ . The argument in this case is the same as the argument given in case (iii) of the proof of Theorem 1. This then proves (4.2) and hence (4.1). The inequality $$\int_{-\pi}^{\pi} \varPhi(-\log|f(re^{i\theta})|)d\theta \leqq \int_{-\pi}^{\pi} \varPhi(-\log|G_{\scriptscriptstyle(p,q)}(re^{i\theta})|)d\theta$$ is obtained as in Theorem 1 except that (52) of [1] is now replaced by $$\begin{aligned} \int_{-\pi}^{\pi} \log^+\left(\rho \left|f(re^{i\theta})\right|\right) d\theta &= 2\pi\biggl[\log\rho + \log^+\frac{r}{q} - \log^+\frac{r}{p}\biggr] \ &+ \int_{-\pi}^{\pi} \log^+\frac{1}{\rho \left|f(re^{i\theta})\right|} d\theta \;. \end{aligned}$$ 177 We have the following as an immediate consequence of Theorem 6.

Theorem 7. Let $$f \in \Sigma(p,q), \ 0 < r < 1, \ -\infty < \lambda < \infty, \ then$$ $$\int_{-\pi}^{\pi} |f(re^{i\theta})|^{\lambda} d\theta \leqq \int_{-\pi}^{\pi} |G_{(p,q)}(re^{i\theta})|^{\lambda} d\theta \ .$$ 5. Applications of Theorem 7. Arguing as in Theorem 3 we obtain the following.

Theorem 8. Let $$f\in \Sigma(p,q),$$ then for $|z|=r.$ $$|G_{(p,q)}(-r)|\leqq |f(z)|\leqq |G_{(p,q)}(r)| \ .$$

Theorem 9. Let $f \in \Sigma(p,q)$ and $f(z) = 1 + \sum_{n=1}^{\infty} a_n z^n$ , |z| < p, then (5.1) $$\frac{|p-q|(1-pq)}{pq} \le |a_1| \le \frac{(p+q)(1+pq)}{pq} .$$ Both inequalities are sharp.

Theorem 10. Let $f \in \Sigma(p, q)$ with $f(z_0) = 0$ , $|z_0| = q$ , then for $w \in \Delta$ , $w \neq z_0$ , $w \neq p$ , (5.2) $$\frac{1}{1-|w|^2} \left[ \frac{||a|-|b||(1-|a||b|)}{|a||b|} \right] \leq \left| \frac{f'(w)}{f(w)} \right|$$ $$\leq \frac{1}{1-|w|^2} \left[ \frac{(|a|+|b|)(1+|a||b|)}{|a||b|} \right]$$ where $$|a|=\left|\frac{p-w}{1-p\bar{w}}\right| \quad and \quad |b|=\left|\frac{z_{\scriptscriptstyle 0}-w}{1-\bar{w}z_{\scriptscriptstyle 0}}\right|$$ . The left hand side of (5.2) is sharp and the right side is sharp at least for |w| < q.