Hausdorff’s moment sequences and exponential convolutions

Lemma 1. Let $h_1, h_2 \in A_0^+$ and let $\omega_1 \in G(h_1), \omega_2 \in G(h_2)$ . Then $\omega_1 \omega_2 \in G(h_1 h_2)$ .

Lemma 2. Let $\Phi(z, w) = \sum_{n,k=0}^{\infty} \alpha_{n,k} z^n w^k$ be a formal power series in two variables with $\alpha_{n,k} \geq 0$ (n, k = 0, 1, ...), and let $g, h \in A_0^+$ . Then for any $\omega \in G(h)$ and N = 1, 2, ..., $$|\{f\}_N| \le \{F\}_N \tag{1}$$ where $$f(z) = \Phi(z, (g * \omega)(z)) \tag{2}$$ and $$F(z) = \Phi(z, (g * h)(z)). \tag{3}$$ The equality in (1) holds for $\{F\}_N \neq 0$ iff $\arg\{(g \omega)^k\}_{N-n} = \text{constant}$ and $|\{(g \omega)^k\}_{N-n}| = \{(g h)^k\}_{N-n}$ provided that $\alpha_{n,k}\{(g h)^k\}_{N-n} \neq 0$ , $0 < k + n \le N$ . Received July 11, 1995, revised September 4, 1995. ¹⁹⁹¹ Mathematics Subject Classification. Primary 30C99, 30C45, 30E05, 30C50. Key words and phrases: Hausdorff's moment problem, analytic functions, convolutions, exponential compositions, coefficient bounds, univalent functions. 32 GRINSHPAN

Lemma 3. Let $g, h \in A_0^+$ and $\omega \in G(h)$ . Let $\Phi(z, w) = ze^w$ , and let f and F be defined by (2) and (3), respectively. Then M(r, f)/F(r) and $|f(re^{i\theta})|/F(r)$ for any $\theta \in [0, 2\pi)$ are non-increasing functions of r in (0, 1). If $|f(re^{i\theta})| = F(r)$ for some $r \in (0, 1)$ and $\theta \in [0, 2\pi)$ , then $f(z) = e^{i\theta}F(e^{-i\theta}z)$ for each $z \in E$ .

Lemma 4. (see e.g., [15, Ch. 1]) If $\eta(x)$ and $\varphi(x)$ are continuous and $\nu(x)$ is of bounded variation in [a,b], and if $$\mu(x) = -\int_{x}^{b} \varphi(t)d\nu(t), \qquad x \in [a, b],$$ then $$\int_a^b \eta(x)d\mu(x) = \int_a^b \eta(x)\varphi(x)d\nu(x).$$

Lemma 5. If $\nu(x)$ is of bounded variation in [0,1], then $$\lim_{x \to 0+} x \int_{x}^{1} \frac{\nu(t)}{t^{2}} dt = \nu(0^{+}). \tag{6}$$

Theorem 1. A necessary and sufficient condition that the convolution of a fixed function $g \in A_0$ and an arbitrary function $f \in A_0$ have the expression $$(f * g)(z) = \int_0^1 f(zx)d\mu(x), \qquad z \in E, \tag{9}$$ where $$\mu(x) = -\int_{x}^{1} \frac{d\nu(t)}{t}, \qquad x \in (0,1], \quad [x\mu(x)]_{x=0} = \nu(0) - \nu(0+), \tag{10}$$ and $\nu(t)$ is of bounded variation in [0,1], is that the sequence $$\{\beta_n\}_0^{\infty}, \quad \beta_n = \sum_{k=0}^n \binom{n}{k} \left| \sum_{l=0}^{n-k} (-1)^l \binom{n-k}{l} \{g\}_{n+1-l} \right|,$$ (11) be bounded. Each function $\mu$ defined by (10) generates such a function $g \in A_0$ by (9) with $f(z) = z^n$ , n = 1, 2, ...

Lemma 6. Let $\mu(x)$ , $|\mu| < \infty$ , be nondecreasing in (0,1], and let $\int_{0+}^{1} x d\mu(x) < \infty$ . Then the limit value $[x\mu(x)]_{x=0+}$ exists and equals 0.

Theorem 2. A necessary and sufficient condition that the convolution of a fixed function $g \in A_0$ and an arbitrary function $f \in A_0$ has the expression $$(f * g)(z) = \int_0^1 f(zx)d\mu(x), \qquad z \in E, \tag{13}$$ where $\mu \in \mathcal{M}$ , is that the sequence $\{g\}_n$ , $n \geq 1$ is completely monotonic. Each function $\mu \in \mathcal{M}$ generates such a function $g \in A_0$ by (13) with $f(z) = z^n$ , $n = 1, 2, \ldots$

Corollary 1. Under the conditions of Theorem 2, $\mu \in \mathcal{M}_0$ iff the sequence $\{g\}_n$ , $n \geq 1$ , is minimal completely monotonic. Proof. Let the sequence $\{g\}_n$ , $n \geq 1$ , be minimal completely monotonic and $\mu \notin \mathcal{M}_0$ . This means that $\mu \in \mathcal{M}$ with $[x\mu(x)]_{x=0} < 0$ . Let $\tilde{\mu}(x) = \mu(x)$ , $x \in (0,1]$ , and $[x\tilde{\mu}(x)]_{x=0} = 0$ ; then $\tilde{\mu} \in \mathcal{M}$ . The function $\tilde{\mu}$ generates a function $\tilde{g} \in A_0$ by (13) with $f(z) = z^n$ , $n = 1, 2, \ldots$ . We have $\{\tilde{g}\}_1 = \{g\}_1 + [x\mu(x)]_{x=0}, \{\tilde{g}\}_n = \{g\}_n, n \geq 2$ . Since the sequence $\{\tilde{g}\}_n$ , $n \geq 1$ , is completely monotonic and $[x\mu(x)]_{x=0} < 0$ , we have reached a contradiction. Now let us assume that $\mu \in \mathcal{M}_0$ and the sequence $\{g\}_n$ , $n \geq 1$ , is not minimal completely monotonic. Without loss of generality, we can assume that $\mu$ is normalized as in Remark 2. Let $g(z) = \tilde{g}(z) + \varepsilon z$ , $\varepsilon > 0$ , $z \in E$ , where both sequences $\{g\}_n$ and $\{\tilde{g}\}_n$ , $n \geq 1$ , are completely monotonic. According to Theorem 2, $\{\tilde{g}\}_n = \int_0^1 x^n d\tilde{\mu}(x)$ , $n \geq 1$ , where $\tilde{\mu} \in \mathcal{M}$ with $[x\tilde{\mu}(x)]_{x=0} \leq 0$ . Therefore, $\{g\}_n = \int_0^1 x^n d\tilde{\mu}(x)$ , $n \geq 1$ , where $\tilde{\mu}(x) = \mu(x)$ , $x \in (0,1]$ , and $[x\tilde{\mu}]_{x=0} = [x\tilde{\mu}(x)]_{x=0} - \varepsilon \leq 0$ . Since $\tilde{\mu} \in \mathcal{M}$ and $\mu$ is unique, $\tilde{\mu}(x) = \mu(x)$ , $x \in (0,1]$ , and $[x\tilde{\mu}(x)]_{x=0} = \varepsilon = 0$ . We have reached a contradiction again. Every admissible function g, either from Theorem 1 or Theorem 2, generates some linear integral operator $I(f) \colon A_0 \to A_0$ . For example, if $g(z) = \log(1/(1-z))$ and $\mu(x) = \log x \in \mathcal{M}_0$ in Theorem 2, we get the classical operator of Alexander [1] $$I(f)(z) = \int_0^z \frac{f(\zeta)}{\zeta} d\zeta. \tag{19}$$ See Miller and Mocanu [13], for example, for material concerning various generalizations of Alexander's operator. We shall consider exponential convolutions of the form $$f(z) = z \exp\{(g * \omega)(z)\}$$ (20) where the function $g \in A_0^+$ and has a completely monotonic sequence of Taylor coefficients. Let $\mathcal{E}C_h(g)$ be the class of all functions f defined by (20) where $\omega \in G(h)$ for some fixed function $h \in A_0^+$ . According to Theorem 2, for every such pair of functions g and h, we have the corresponding exponential integral operator $$J(\omega)(z) = z \exp\left\{\int_0^1 \omega(zx) d\mu(x)\right\}: G(h) \to \mathcal{E}C_h(g)$$ where $\mu \in \mathcal{M}$ . 38 GRINSHPAN If a function $\mu \in \mathcal{M}$ with $[x\mu(x)]_{x=0} = -a \leq 0$ related to some admissible function g by (13) is given, then we use the notation $EC_h(\mu|a)$ for the class $\mathcal{E}C_h(g)$ . Let $EC_h(\mu) = \bigcup_{a\geq 0} EC_h(\mu|a)$ . Clearly, each function $f_a \in EC_h(\mu|a)$ can be represented in the form $$f_a(z) = f(z) \exp\{a\{\omega\}_1 z\}, \qquad z \in E,$$ where $f \in EC_h(\mu|0)$ and $\omega \in G(h)$ . Thus, the case when $x\mu(x)$ is right-continuous at x = 0 is the most important one. The function $$F(z) = z \exp\left\{ \int_0^1 h(zx)d\mu(x) \right\}$$ (21) is the envelope function for the class $EC_h(\mu)$ . Lemmas 2 and 3 give some extremal properties of this function on $EC_h(\mu)$ . Let $\mu \in \mathcal{M}$ and $$L(\mu) = \begin{cases} 0 & \text{if } \int_0^1 x d\mu(x) = 0, \\ \infty & \text{if } \int_0^1 x^2 d\mu(x) \neq 0, \\ 1 & \text{otherwise.} \end{cases}$$ (22) Note that if $L(\mu) < \infty$ , then the corresponding function g(z) from (13) is a linear function az with $a \ge 0$ .

Theorem 3. Let $h \in A_0^+$ , $\mu \in \mathcal{M}$ , and let F be defined by (21). Then, for any $f(z) = z \exp\{\int_0^1 \omega(xz) d\mu(x)\} \in EC_h(\mu)$ with $\omega \in G(h)$ , - (a) $|\{f(z)\}_N| \le \{F\}_N, N \ge 2$ , - (b) $|f(z)| \le F(|z|), z \in E$ . For some natural $N \geq 2$ , let $\{h\}_n > 0$ (n = 1, ..., k) where $k = \min\{N - 1, L(\mu)\}$ and $L(\mu) \neq 0$ is defined by (22). Then the equality in (a) holds iff $\{\omega\}_n = \{h(ze^{i\theta})\}_n$ for every n = 1, ..., k and for some $\theta \in [0, 2\pi)$ . If for some $z = re^{i\theta} \in E - \{0\}$ |f(z)| = F(r), then for each $z \in E$ , $f(z) = e^{i\theta}F(e^{-i\theta}z)$ .

Theorem 4. $EC(\log x^2) = S^*$ .

Theorem 5. For every $q \in (0,1)$ , $PS_q = EC(\mu_q)$ where $\mu_q \in \mathcal{M}_0$ and is defined by $$\mu_q(x) = \log q^{2k}, \qquad x \in [q^k, q^{k-1})$$ (30) for k = 0, 1, ....

Theorem 6. Let $\mu \in \mathcal{M}_0$ , $f \in EC(\mu)$ , and let $F_{\mu}$ be defined by (27). Then - (a) $|\{f\}_n| \le \{F_\mu\}_n, \ n \ge 2$ , - (b) $M(r,f)/F_{\mu}(r)$ and $|f(re^{i\theta})|/F_{\mu}(r)$ for any $\theta \in [0,2\pi)$ are nonincreasing functions of r in (0,1). If |z| = r < 1, - (c) $-F_{\mu}(-r) \le |f(z)| \le F_{\mu}(r)$ , - $(\mathrm{d}) \left| \arg \frac{f(z)}{z} \right| \leq \max_{|\zeta| = r} \left| \arg \frac{F_{\mu}(\zeta)}{\zeta} \right| = \max_{\theta \in [0, 2\pi)} \int_0^1 \frac{xr \sin \theta}{1 2xr \cos \theta + x^2 r^2} d\mu(x).$ The equality sign in (a) occurs iff $$f(z) = e^{-i\theta} F_{\mu}(e^{i\theta}z) \tag{32}$$ for some $\theta \in [0, 2\pi)$ . The equality $|f(z)| = F_{\mu}(|z|)$ for some $z \in E - \{0\}$ holds iff f is defined by (32) with $\theta = -\arg z$ . The equality $|f(z)| = -F_{\mu}(-|z|)$ for some $z \in E - \{0\}$ holds iff f is defined by (32) with $\theta = \pi - \arg z$ . Proof. The statements (a) and (b), and the right-hand inequality in (c) follow from Theorem 3 and Lemma 3. If $\int_0^1 x d\mu(x) = 0$ , then $f(z) = F_{\mu}(z) = z$ . Otherwise, according to Theorem 3, the equality in (a) for some $n \geq 2$ implies that $|\{\omega\}_1| = 1$ , where $\omega \in W$ is defined by (23). It follows that in this case $\omega(z) = \omega^*(e^{i\theta}z)$ for some $\theta \in [0, 2\pi)$ . The representation (25) implies that for each $\omega \in W$ $$-\frac{|z|}{1+|z|} \le \operatorname{Re}\{\omega(z)\}, \qquad z \in E,$$ with equality iff $w(z) = w^*(e^{i\theta}z)$ where $\theta = \pi - \arg z$ (see e.g., [4, Ch. 7]). Using this, (23), and (27), we obtain the left-hand inequality in (c). Of course, the right-hand inequality in (c) also can be proved in this way. 42 GRINSHPAN For every $\omega \in W$ , we have from (25) $$\left|\operatorname{Im}\{\omega(z)\}\right| \leq \max_{\alpha \in [0,2\pi)} \left|\operatorname{Im}\left\{\omega^*\left(e^{i\alpha}z\right)\right\}\right|.$$ Using this inequality, (23), and (27), we get (d).

Corollary 2. The Koebe domain for the class $EC(\mu)$ is the open disk centered at zero with the radius $$r(\mu) = \exp\left\{-\int_0^1 \frac{x}{1+x} d\mu(x)\right\}.$$ In particular, we have the well-known value $r(\log x^2) = 1/4$ for the class $S^*$ and a new result $r(\mu_q) = \exp\left\{2(\log q)\sum_{k=0}^{\infty}\frac{q^k}{1+q^k}\right\}$ for the class $PS_q$ where $\mu_q$ is defined by (30). Remark 3. Note that for all $f \in EC(\mu)$ the radius of starlikeness of f is greater than or equal to that of $F_{\mu}$ . Equation (26) gives that the starlike radius $r_{\mu}$ of $EC(\mu)$ either is defined by the transcendental equation $\min_{\theta} \operatorname{Re}\{\int_{0}^{1} K(xr_{\mu}e^{i\theta})d\mu(x)\} = -1$ where K is the Koebe function, or $r_{\mu} = 1$ . Earlier, the starlike radius of $PS_{q}$ was found in [10]. In fact, the classes $EC(\mu)$ , $\mu \in \mathcal{M}_0$ , give some natural extension of the class $PS_q$ , which in turn is a q-extension of the class $S^*$ . Some properties of this natural extension are easier to prove than those in the q-case. It allows us to establish some new q-properties and to prove known ones in a simple way. An approach based on a natural extension that includes a q-extension of some mathematical object may be effective in other cases also. Let GS be the class of bounded functions $f \in A_0$ satisfying the conditions: f'(0) = 1, $f(z)/z \neq 0$ ( $z \in E$ ), and $\inf\{|w| : w \in \partial f(E)\} > 0$ . This class was introduced by the author and Saff in [8] without the normalization f'(0) = 1. Some estimates of the functional $\sup_{z \in E} |\arg \frac{f(z)}{z}|$ on the class GS are given in [8]. These results are related to the problem of estimating the argument of approximate conformal mappings of simply connected domains onto the unit disk. The statements (c) and (d) of Theorem 6 and Theorem 7 provide a new stimulus for this problem.

Theorem 7. Let $\mu \in \mathcal{M}_0$ , then $EC(\mu) \subset GS$ iff $$\int_0^1 \frac{x d\mu(x)}{1-x} < \infty. \tag{33}$$

Theorem 8. Let $\mu \in \mathcal{M}_0$ and for some real numbers $x_k$ , $k = 1, \ldots, n$ $$\min_{\theta \in [0,\pi]} \sum_{k=1}^{n} x_k \left( k \int_0^1 x^k d\mu(x) \right)^2 \sin k\theta = 0.$$ (39) For each $f \in EC(\mu)$ , let $$\mathbf{M}(f) = \sum_{k=1}^{n} x_k k |\beta_k|^2 \tag{40}$$ where the logarithmic coefficients $\beta_k$ are defined by (38). Finally, let $F_{\mu}$ be defined by (27). Then $F_{\mu}$ maximizes the functional $\mathbf{M}(f)$ on the class $EC(\mu)$ . If the sum in (39) is not identically zero, then only the function $F_{\mu}$ and its rotations maximize this functional on $EC(\mu)$ .