Asymptotic expansions for the radii of starlikeness of normalised Bessel functions

Lemma 1. For any positive integer k and real $\nu$ , $\nu > k$ , the Rayleigh function $\sigma_k(\nu)$ has the convergent Laurent expansion (2.1) $$\sigma_k(\nu) = \frac{1}{\nu^{2k-1}} \sum_{m=0}^{\infty} \frac{\sigma_m^{(k)}}{\nu^m},$$ where, for any fixed non-negative integer m, the coefficients $\sigma_m^{(k)}$ can be computed by the recurrence relation (2.2) $$\sigma_m^{(1)} = \frac{(-1)^m}{4}, \quad \sigma_m^{(k)} = \sum_{i=0}^m \sum_{j=0}^i \sum_{n=1}^{k-1} (-k)^{m-i} \sigma_j^{(n)} \sigma_{i-j}^{(k-n)}, \quad k \ge 2.$$ In particular, we have $$\sigma_m^{(2)} = \frac{(-1)^m}{16} \left( 2^{m+2} - m - 3 \right), \quad \sigma_m^{(3)} = \frac{(-1)^m}{256} \left( 3^{m+4} - 2^{m+7} + 2(m+4)(m+6) + 7 \right).$$

Lemma 2. For any positive integer k and positive real $\nu$ , the Rayleigh sum $\sigma_k(\nu)$ satisfies the inequalities (2.3) $$\sigma_k(\nu) \le \binom{2k}{k} \frac{1}{2^{2k+1}(2k-1)} \frac{1}{\nu^{2k-1}} \le \frac{1}{\nu^{2k-1}}.$$ The coefficients in our asymptotic expansions for the radii of starlikeness will be given in terms of recurrence relations involving ordinary potential polynomials of the $\sigma_m^{(k)}$ 's. For any fixed complex $\mu$ and arbitrary non-negative integer n, the ordinary potential polynomial $A_{\mu,n}$ is a polynomial in n (complex) variables and is defined by the generating function $$\left(1 + \sum_{n=1}^{\infty} x_n z^n\right)^{\mu} = \sum_{n=0}^{\infty} \mathsf{A}_{\mu,n}(x_1, x_2, \dots, x_n) z^n.$$ <span id="page-2-0"></span>Thus, in particular, $A_{\mu,0} = 1$ , $A_{\mu,1} = \mu x_1$ and $A_{\mu,2} = \mu x_2 + {\mu \choose 2} x_1^2$ . For basic properties of these polynomials, see, e.g., [Ne13, Appendix]. Let $\mathbb{D}_r = \{z \in \mathbb{C} : |z| < r\}$ , where r > 0, and let $f : \mathbb{D}_r \to \mathbb{C}$ be a normalised univalent or one-to-one function, which satisfies the conditions f(0) = 0 and f'(0) = 1, that is, f is of the form $f(z) = z + a_2 z^2 + a_3 z^3 + \ldots$ , where the coefficients $a_2, a_3, \ldots$ are real or complex numbers. The radius of univalence of the function f is the largest radius r for which f maps univalently the open disk $\mathbb{D}_r$ into some domain in the complex plane. Similarly, the radius of starlikeness of the function f is the largest radius r for which f maps $\mathbb{D}_r$ into a starlike domain with respect to the origin. Since the class of normalised starlike functions (with respect to the origin) is a subclass of univalent functions, the radius of univalence of f is clearly allways greater or equal than the radius of starlikeness of the same function f. In view of the analytic characterization of starlike functions, the radius of starlikeness is given by $$r^{\star}(f) = \sup \left\{ r > 0 : \Re\left(\frac{zf'(z)}{f(z)}\right) > 0 \text{ for all } z \in \mathbb{D}_r \right\}$$ (see, e.g., [Ne21]). Now, consider the normalised Bessel function of the first kind $\phi_{\nu}: \mathbb{D}_r \to \mathbb{C}$ , defined by (2.4) $$\phi_{\nu}(z) = 2^{\nu} \Gamma(\nu + 1) z^{1 - \frac{\nu}{2}} J_{\nu}(\sqrt{z}),$$ where $\nu > -1$ and $J_{\nu}$ is the Bessel function of the first kind of order $\nu$ [DLMF20, §10.2(ii)]. The function $\phi_{\nu}$ is also well-defined if $\nu < -1$ and $\nu$ is not a negative integer, however the condition $\nu > -1$ is important in our paper since only in this case the zeros of the Bessel function $J_{\nu}$ are all real (according to an old result of von Lommel [Wa44, Ch. XV, §15.25]) and the reality of the zeros is essential in this paper. This is because all the results in the paper [BKS14] on the radii of starlikeness, which we shall use here, hold under the condition that $\nu > -1$ . It was shown (see [BKS14, Corollary 1]) that for $\nu > -1$ the radius of starlikeness of $\phi_{\nu}$ is the smallest positive root of the equation $\sqrt{z}J'_{\nu}(\sqrt{z}) + (2-\nu)J_{\nu}(\sqrt{z}) = 0$ . We also know that the radius of starlikeness of $\phi_{\nu}$ , denoted by $r^{*}(\phi_{\nu})$ , corresponds to the radius of univalence of $\phi_{\nu}$ (see [ABY17, Theorem 2]), and that it satisfies $$\lim_{\nu \to +\infty} \frac{r^*(\phi_{\nu})}{4(\nu+1)} = 1.$$ Moreover, by using some Euler–Rayleigh inequalities, it was proved (see also [ABY17, Theorem 2]) that the asymptotic formula (2.5) $$r^{\star}(\phi_{\nu}) = 4(\nu+1)\left(1 - \frac{1}{\nu} + \mathcal{O}\left(\frac{1}{\nu^2}\right)\right)$$ holds as $\nu \to +\infty$ . The following theorem gives a complete asymptotic expansion for the radii of starlikeness of the function $\phi_{\nu}$ .

Theorem 1. The radius of starlikeness $r^*(\phi_{\nu})$ has the asymptotic expansion (2.6) $$r^{\star}(\phi_{\nu}) \sim 4\nu \left( 1 + \sum_{k=1}^{\infty} \frac{\varepsilon_k}{\nu^k} \right) = 4\nu \left( 1 + \frac{1}{\nu^2} - \frac{4}{\nu^4} + \frac{2}{\nu^5} + \frac{44}{\nu^6} - \cdots \right),$$ as $\nu \to +\infty$ , where the coefficients $\varepsilon_k$ can be determined recursively by (2.7) $$\varepsilon_k = (-1)^{k+1} - \sum_{n=1}^{k-1} (-1)^{k-n} \varepsilon_n - \sum_{n=2}^{k+1} 4^n \sum_{m=0}^{k-n+1} \sigma_{k-n-m+1}^{(n)} \mathsf{A}_{n,m}(\varepsilon_1, \varepsilon_2, \dots, \varepsilon_m).$$ Remark 1. It is readily seen from (2.2), by using an induction argument in k, that $4^k \sigma_m^{(k)}$ is an integer for any $k \ge 1$ and $m \ge 0$ . Consequently, the coefficients $\varepsilon_k$ are all integer valued. <span id="page-3-0"></span>Remark 2. By employing a simple algebraic manipulation, the asymptotic expansion (2.6) may be re-arranged to the alternative form $$r^*(\phi_{\nu}) \sim 4(\nu+1) \left( 1 + \sum_{k=1}^{\infty} \frac{\delta_k}{\nu^k} \right) = 4(\nu+1) \left( 1 - \frac{1}{\nu} + \frac{2}{\nu^2} - \frac{2}{\nu^3} - \frac{2}{\nu^4} + \cdots \right),$$ which shares a closer resemblance to (2.5). The coefficients $\delta_k$ can be computed via the simple recursion $\delta_1 = -1$ and $\delta_k = \varepsilon_k - \delta_{k-1}$ for $k \ge 2$ . Next, consider the normalised Bessel function of the first kind $\varphi_{\nu}: \mathbb{D}_r \to \mathbb{C}$ , defined by (2.8) $$\varphi_{\nu}(z) = 2^{\nu} \Gamma(\nu + 1) z^{1-\nu} J_{\nu}(z),$$ where $\nu > -1$ . This function is also well-defined if $\nu < -1$ and $\nu$ is not a negative integer, however the condition $\nu > -1$ is again essential because of the reality of the zeros of $J_{\nu}$ . It is known (cf. [Br60, Theorem 3] or [BKS14, Corollary 1]) that for $\nu > -1$ the radius of starlikeness of the function $\varphi_{\nu}$ is the smallest positive root of the equation $zJ'_{\nu}(z) + (1-\nu)J_{\nu}(z) = 0$ . We also know that the radius of starlikeness of $\varphi_{\nu}$ , denoted by $r^*(\varphi_{\nu})$ , corresponds to the radius of univalence of $\varphi_{\nu}$ (see [Wi62, Theorem 2]), and, as $\nu \to +\infty$ , it satisfies [Wi62, Theorem 2] (2.9) $$r^{\star}(\varphi_{\nu}) = \sqrt{2\nu} \left( 1 + \frac{1}{4\nu} + \mathcal{O}\left(\frac{1}{\nu^2}\right) \right).$$ While studying the asymptotic behaviour of the quantity $r^(\varphi_{\nu})$ , we observed that the recurrence relation for the asymptotic expansion coefficients is much simpler and presentable if one considers instead the square of $r^(\varphi_{\nu})$ . Therefore, in the following theorem, we provide a complete asymptotic expansion for $(r^*(\varphi_{\nu}))^2$ .

Theorem 2. The square of the radius of starlikeness $r^*(\varphi_{\nu})$ has the asymptotic expansion $$(2.10) (r^*(\varphi_{\nu}))^2 \sim 2\nu \left(1 + \sum_{k=1}^{\infty} \frac{\rho_k}{\nu^k}\right) = 2\nu \left(1 + \frac{1}{2\nu} + \frac{1}{2\nu^2} - \frac{1}{2\nu^3} - \frac{1}{2\nu^4} + \cdots\right),$$ as $\nu \to +\infty$ , where the coefficients $\rho_k$ can be determined recursively by $$\rho_k = (-1)^{k+1} - \sum_{n=1}^{k-1} (-1)^{k-n} \rho_n - \sum_{n=2}^{k+1} 2^{n+1} \sum_{m=0}^{k-n+1} \sigma_{k-n-m+1}^{(n)} \mathsf{A}_{n,m}(\rho_1, \rho_2, \dots, \rho_m).$$ A simple corollary of Theorem 2 and Exercise 8.4 of Olver [Ol97, p. 22] is the following generalisation of (2.9). Corollary 1. The radius of starlikeness $r^*(\varphi_{\nu})$ has the asymptotic expansion $$r^{\star}(\varphi_{\nu}) \sim \sqrt{2\nu} \left( 1 + \sum_{k=1}^{\infty} \frac{\pi_k}{\nu^k} \right) = \sqrt{2\nu} \left( 1 + \frac{1}{4\nu} + \frac{7}{32\nu^2} - \frac{39}{128\nu^3} - \frac{405}{2048\nu^4} + \cdots \right),$$ as $\nu \to +\infty$ , where the coefficients $\pi_k$ can be computed via the recurrence relation $$\pi_k = \frac{1}{2}\rho_k - \frac{1}{2}\sum_{n=1}^{k-1} \pi_n \pi_{k-n}.$$ Remark 3. A third type of normalised Bessel function of the first kind that is studied frequently in the literature is $$\theta_{\nu}(z) = (2^{\nu}\Gamma(\nu+1)J_{\nu}(z))^{\frac{1}{\nu}},$$ <span id="page-4-0"></span>with $\nu > 0$ . It is known (see [Br60, Theorem 2] or [BKS14, Corollary 1]) that the radius of starlikeness of $\theta_{\nu}$ , denoted by $r^{\star}(\theta_{\nu})$ , is the smallest positive zero of $J'_{\nu}$ . Accordingly, by a result of Olver [Ol54], $$r^{\star}(\theta_{\nu}) \sim \nu - \frac{\alpha}{2^{\frac{1}{3}}} \nu^{\frac{1}{3}} + \frac{2^{\frac{1}{3}}}{10} \left( \frac{3}{2} \alpha^2 + \frac{1}{\alpha} \right) \frac{1}{\nu^{\frac{1}{3}}} + \frac{1}{100} \left( \frac{\alpha^3}{7} - 4 + \frac{1}{\alpha^3} \right) \frac{1}{\nu} + \cdots$$ as $\nu \to +\infty$ , where $\alpha$ denotes the first negative zero of the derivative of the Airy function Ai. Olver derived his expansions for the zeros of $J'_{\nu}$ through an asymptotic inversion of $J'_{\nu}$ in its transition region. The asymptotic nature of the expansions was not demonstrated rigorously, however, it is possible to fill this gap by the methods of the present work. The details are not considered here.