Radii of starlikeness and convexity of some $q$-Bessel functions

Theorem 1. Let $\nu > -1$ and $s \in \{2,3\}$ . The following statements hold: a) If $\alpha \in [0,1)$ and $\nu > 0$ , then $r_{\alpha}^* \left( f_{\nu}^{(s)} \right) = x_{\nu,\alpha,1}$ , where $x_{\nu,\alpha,1}$ is the smallest positive root of the equation $$r \cdot dJ_{\nu}^{(s)}(r;q)/dr - \alpha \nu J_{\nu}^{(s)}(r;q) = 0.$$ Moreover, if $\alpha \in [0,1)$ and $\nu \in (-1,0)$ , then $r_{\alpha}^*\left(f_{\nu}^{(s)}\right) = x_{\nu,\alpha}$ , where $x_{\nu,\alpha}$ is the unique positive root of the equation $$ir \cdot dJ_{\nu}^{(s)}(ir;q)/dr - \alpha \nu J_{\nu}^{(s)}(ir;q) = 0.$$ b) If $\alpha \in [0,1)$ , then $r_{\alpha}^*\left(g_{\nu}^{(s)}\right) = y_{\nu,\alpha,1}$ , where $y_{\nu,\alpha,1}$ is the smallest positive root of the equation $$r \cdot dJ_{\nu}^{(s)}(r;q)/dr - (\alpha + \nu - 1)J_{\nu}^{(s)}(r;q) = 0.$$ c) If $\alpha \in [0,1)$ , then $r_{\alpha}^*\left(h_{\nu}^{(s)}\right) = z_{\nu,\alpha,1}$ , where $z_{\nu,\alpha,1}$ is the smallest positive root of the equation $$r \cdot dJ_{\nu}^{(s)}(r;q)/dr - (2\alpha + \nu - 2)J_{\nu}^{(s)}(r;q) = 0.$$ Our second result concerns the radii of convexity.

Theorem 2. Let $\nu > -1$ and $s \in \{2,3\}$ . The following statements hold: a) If $\nu > 0$ and $\alpha \in [0,1)$ , then the radius of convexity of order $\alpha$ of the function $f_{\nu}^{(s)}(\cdot;q)$ is the smallest positive root of the equation $$1 + \frac{r \cdot d^2 J_{\nu}^{(s)}(r;q)/dr^2}{dJ_{\nu}^{(s)}(r;q)/dr} + \left(\frac{1}{\nu} - 1\right) \frac{r \cdot dJ_{\nu}^{(s)}(r;q)/dr}{J_{\nu}(r;q)} = \alpha.$$ Moreover, we have $r_{\alpha}^{c}(f_{\nu}^{(2)}) < j_{\nu,1}'(q) < j_{\nu,1}(q)$ and $r_{\alpha}^{c}(f_{\nu}^{(3)}) < l_{\nu,1}'(q) < l_{\nu,1}(q)$ , where $j_{\nu,1}(q)$ , $l_{\nu,1}(q)$ , $j_{\nu,1}'(q)$ and $l_{\nu,1}'(q)$ are the first positive zeros of the functions $J_{\nu}^{(2)}(\cdot;q)$ , $J_{\nu}^{(3)}(\cdot;q)$ , $z \mapsto dJ_{\nu}^{(2)}(z;q)/dz$ and $z \mapsto dJ_{\nu}^{(3)}(z;q)/dz$ . b) If $\nu > -1$ and $\alpha \in [0,1)$ , then the radius of convexity of order $\alpha$ of the function $g_{\nu}^{(s)}(\cdot;q)$ is the smallest positive root of the equation $$1 - \nu + r \frac{(2 - \nu) \cdot dJ_{\nu}^{(s)}(r;q)/dr + r \cdot d^2 J_{\nu}^{(s)}(r;q)/dr^2}{(1 - \nu)J_{\nu}^{(s)}(r;q) + r \cdot dJ_{\nu}^{(s)}(r;q)/dr} = \alpha.$$ Moreover, we have $r_{\alpha}^{c}(g_{\nu}^{(2)}) < \alpha_{\nu,1}(q) < j_{\nu,1}(q)$ and $r_{\alpha}^{c}(g_{\nu}^{(3)}) < \gamma_{\nu,1}(q) < l_{\nu,1}(q)$ , where $\alpha_{\nu,1}(q)$ and $\gamma_{\nu,1}(q)$ are the first positive zeros of the functions $z \mapsto z \cdot dJ_{\nu}^{(2)}(z;q)/dz + (1-\nu)J_{\nu}^{(2)}(z;q)$ and $z \mapsto z \cdot dJ_{\nu}^{(3)}(z;q)/dz + (1-\nu)J_{\nu}^{(3)}(z;q)$ . c) If $\nu > -1$ and $\alpha \in [0,1)$ , then the radius of convexity of order $\alpha$ of the function $h_{\nu}^{(s)}(\cdot;q)$ is the smallest positive root of the equation $$1 - \frac{\nu}{2} + \frac{\sqrt{r}}{2} \frac{(3-\nu) \cdot dJ_{\nu}^{(s)}(\sqrt{r};q)/dr + \sqrt{r} \cdot d^2 J_{\nu}^{(s)}(\sqrt{r};q)/dr^2}{(2-\nu)J_{\nu}^{(s)}(\sqrt{r};q) + \sqrt{r} \cdot dJ_{\nu}^{(s)}(\sqrt{r};q)/dr} = \alpha.$$ Moreover, we have $r_{\alpha}^{c}(h_{\nu}^{(2)}) < \beta_{\nu,1}(q) < j_{\nu,1}(q)$ and $r_{\alpha}^{c}(h_{\nu}^{(3)}) < \delta_{\nu,1}(q) < l_{\nu,1}(q)$ , where $\beta_{\nu,1}(q)$ and $\delta_{\nu,1}(q)$ are the first positive zeros of the functions $z \mapsto z \cdot dJ_{\nu}^{(2)}(z;q)/dz + (2-\nu)J_{\nu}^{(2)}(z;q)$ , and $z \mapsto z \cdot dJ_{\nu}^{(3)}(z;q)/dz + (2-\nu)J_{\nu}^{(3)}(z;q)$ . We note that these theorems are natural q-extension to Jackson and Hahn-Exton q-Bessel functions of the results obtained in [4] and [6]. While the ideas of the proofs are similar, here we need some specific q-extensions of results about the Bessel functions which are of independent interest, such as Lemmas 1, 6 and 9. Finally, we state a result, which is the q-extension of the first part of [7, Theorem 1] for the Jackson q-Bessel function.

Theorem 3. If $\nu > -1$ , then the function $h_{\nu}(\cdot;q) = h_{\nu}^{(2)}(\cdot;q)$ is starlike and all of its derivatives are close-to-convex in $\mathbb{D}$ if and only if $\nu \geq \max\{\nu_0(q), \nu^(q)\}$ , where $\nu_0(q)$ is the unique root of the equation $dh_{\nu}^{(2)}(z;q)/dz\Big|_{z=1} = h_{\nu}'(1;q) = 0$ , and $\nu^(q)$ is the unique root of the equation $j_{\nu,1}(q) = 1$ . The paper is organized as follows. Section 2 contains the preliminary results together with their proofs, while in Section 3 we present the proofs of the main results. In Section 4 we present some consequence of the Hadamard factorizations to Rayleigh sums of the q-Bessel functions under discussion and formulate some open problems.

Lemma 1. If $\nu > -1$ , then $z \mapsto \mathcal{J}_{\nu}^{(2)}(z;q) = 2^{\nu}c_{\nu}(q)z^{-\nu}J_{\nu}^{(2)}(z;q)$ and $z \mapsto \mathcal{J}_{\nu}^{(3)}(z;q) = c_{\nu}(q)z^{-\nu}J_{\nu}^{(3)}(z;q)$ are entire functions of order $\rho = 0$ . Consequently, their Hadamard factorization for $z \in \mathbb{C}$ are of the form (2.1) $$\mathcal{J}_{\nu}^{(2)}(z;q) = \prod_{n>1} \left( 1 - \frac{z^2}{j_{\nu,n}^2(q)} \right), \quad \mathcal{J}_{\nu}^{(3)}(z;q) = \prod_{n>1} \left( 1 - \frac{z^2}{l_{\nu,n}^2(q)} \right),$$ where $j_{\nu,n}(q)$ and $l_{\nu,n}(q)$ are the nth positive zeros of the functions $J_{\nu}^{(2)}(\cdot;q)$ and $J_{\nu}^{(3)}(\cdot;q)$ . Proof. Since <span id="page-3-1"></span> $$\mathcal{J}_{\nu}^{(2)}(z;q) = 2^{\nu} c_{\nu}(q) z^{-\nu} J_{\nu}^{(2)}(z;q) = \sum_{n \geq 0} \frac{(-1)^n z^{2n} q^{n(n+\nu)}}{2^{2n} (q;q)_n (q^{\nu+1};q)_n},$$ $$\mathcal{J}_{\nu}^{(3)}(z;q) = c_{\nu}(q) z^{-\nu} J_{\nu}^{(3)}(z;q) = \sum_{n \geq 0} \frac{(-1)^n z^{2n} q^{\frac{1}{2}n(n+1)}}{(q;q)_n (q^{\nu+1};q)_n},$$ it follows that the growth orders of the even entire functions $z \mapsto \mathcal{J}_{\nu}^{(2)}(z;q)$ and $z \mapsto \mathcal{J}_{\nu}^{(3)}(z;q)$ are zero. Namely, we have that $$\begin{split} \lim_{n \to \infty} \frac{n \log n}{\log(q;q)_n + \log\left(q^{\nu+1};q\right)_n + 2n \log 2 - n(n+\nu) \log q} &= 0, \\ \lim_{n \to \infty} \frac{n \log n}{\log(q;q)_n + \log\left(q^{\nu+1};q\right)_n - \frac{1}{2}n(n+1) \log q} &= 0, \end{split}$$ since as $n \to \infty$ we have $(q;q)_n \to (q;q)_\infty < \infty$ and $(q^{\nu+1};q)_n \to (q^{\nu+1};q)_\infty < \infty$ . On the other hand, we know that the zeros $j_{\nu,n}(q)$ , $n \in \mathbb{N}$ , and $l_{\nu,n}(q)$ , $n \in \mathbb{N}$ , are real and simple, according to [14, Theorem 4.2] and [18, Theorem 3.4], and with this the rest of the proof of (2.1) follows by applying Hadamard's Theorem [20, p. 26]. 2.2. Quotients of power series. We will also need the following result, see [8, 22]:

Lemma 2. Consider the power series $$f(x) = \sum_{n\geq 0} a_n x^n$$ and $g(x) = \sum_{n\geq 0} b_n x^n$ , where $a_n \in \mathbb{R}$ and $b_n > 0$ for all $n \ge 0$ . Suppose that both series converge on (-r,r), for some r > 0. If the sequence $\{a_n/b_n\}_{n\ge 0}$ is increasing (decreasing), then the function $x \mapsto f(x)/g(x)$ is increasing (decreasing) too on (0,r). The result remains true for the power series $$f(x) = \sum_{n \ge 0} a_n x^{2n}$$ and $g(x) = \sum_{n \ge 0} b_n x^{2n}$ . 2.3. Zeros of polynomials and entire functions, and the Laguerre-Pólya class. In this subsection we provide the necessary information about polynomials and entire functions with real zeros. An algebraic polynomial is called hyperbolic if all its zeros are real. We note that the simple statement that two real polynomials p and q posses real and interlacing zeros if and only if any linear combinations of p and q is a hyperbolic polynomial is sometimes called Obrechkoff's theorem. We formulate the following specific statement that we shall need, see [1].

Lemma 3. Let $p(x) = 1 - a_1x + a_2x^2 - a_3x^3 + \cdots + (-1)^n a_nx^n = (1 - x/x_1) \cdots (1 - x/x_n)$ be a hyperbolic polynomial with positive zeros $0 < x_1 \le x_2 \le \cdots \le x_n$ , and normalized by p(0) = 1. Then, for any constant C, the polynomial q(x) = Cp(x) - xp'(x) is hyperbolic. Moreover, the smallest zero $\eta_1$ belongs to the interval $(0, x_1)$ if and only if C < 0. The proof of this result is straightforward; it is enough to apply Rolle's theorem and then count the sign changes of the linear combination at the zeros of p. We refer to [9, 10] for further results on monotonicity and asymptotics of zeros of linear combinations of hyperbolic polynomials. A real entire function $\psi$ belongs to the Laguerre-Pólya class $\mathcal{LP}$ if it can be represented in the form $$\psi(x) = cx^m e^{-ax^2 + \beta x} \prod_{k \ge 1} \left( 1 + \frac{x}{x_k} \right) e^{-\frac{x}{x_k}},$$ with $c, \beta, x_k \in \mathbb{R}$ , $a \ge 0$ , $m \in \mathbb{N} \cup \{0\}$ , $\sum x_k^{-2} < \infty$ . Similarly, $\phi$ is said to be of type I in the Laguerre-Pólya class, written $\varphi \in \mathcal{LP}I$ , if $\phi(x)$ or $\phi(-x)$ can be represented as $$\phi(x) = cx^m e^{\sigma x} \prod_{k>1} \left( 1 + \frac{x}{x_k} \right),$$ with $c \in \mathbb{R}$ , $\sigma \geq 0$ , $m \in \mathbb{N} \cup \{0\}$ , $x_k > 0$ , $\sum 1/x_k < \infty$ . The class $\mathcal{LP}$ is the complement of the space of hyperbolic polynomials in the topology induced by the uniform convergence on the compact sets of the complex plane while $\mathcal{LP}I$ is the complement of the hyperbolic polynomials whose zeros posses a preassigned constant sign. Given an entire function $\varphi$ with the Maclaurin expansion $$\varphi(x) = \sum_{k>0} \gamma_k \frac{x^k}{k!},$$ its Jensen polynomials are defined by $$g_n(\varphi; x) = g_n(x) = \sum_{k=0}^n \binom{n}{k} \gamma_k x^k.$$ Jensen proved the following relation in [17]:

Lemma 4. The function $\varphi$ belongs to $\mathcal{LP}$ ( $\mathcal{LP}I$ , respectively) if and only if all the polynomials $g_n(\varphi; x)$ , $n \in \mathbb{N}$ , are hyperbolic (hyperbolic with zeros of equal sign). Moreover, the sequence $g_n(\varphi; z/n)$ converges locally uniformly to $\varphi(z)$ . Further information about the Laguerre-Pólya class can be found in [21, 23] while [11] contains references and additional facts about the Jensen polynomials in general and also about those related to the Bessel function. The following result is a key tool in the proof of Theorems 1 and 2.

Lemma 5. Let $\nu > -1$ and a < 0. Then the functions $z \mapsto (2a + \nu)J_{\nu}^{(2)}(z;q) - z \cdot dJ_{\nu}^{(2)}(z;q)/dz$ and $z \mapsto (2a + \nu)J_{\nu}^{(3)}(z;q) - z \cdot dJ_{\nu}^{(3)}(z;q)/dz$ can be represented in the form $$c_{\nu}(q)\left((2a+\nu)J_{\nu}^{(2)}(z;q) - z \cdot dJ_{\nu}^{(2)}(z;q)/dz\right) = 2\left(\frac{z}{2}\right)^{\nu}\phi_{\nu}(z;q),$$ $$c_{\nu}(q)\left((2a+\nu)J_{\nu}^{(3)}(z;q) - z \cdot dJ_{\nu}^{(3)}(z;q)/dz\right) = 2z^{\nu}\psi_{\nu}(z;q),$$ where $\phi_{\nu}(\cdot;q)$ and $\psi_{\nu}(\cdot;q)$ are entire functions which belongs to the Laguerre-Pólya class $\mathcal{LP}$ . Moreover, the smallest positive zero of $\phi_{\nu}(\cdot;q)$ does not exceed the first positive zero $j_{\nu,1}(q)$ , while the smallest positive zero of $\psi_{\nu}(\cdot;q)$ is less than $l_{\nu,1}(q)$ . Proof. It is clear from the infinite product representation of $z \mapsto \mathcal{J}_{\nu}^{(2)}(z;q) = 2^{\nu}c_{\nu}(q)z^{-\nu}J_{\nu}^{(2)}(z;q)$ that this function belongs to the Laguerre-Pólya class of entire functions (since the exponential factors in the infinite product are canceled because of the symmetry of the zeros $\pm j_{\nu,n}(q)$ , $n \in \mathbb{N}$ , with respect to the origin). This implies that the function $z \mapsto \mathcal{J}_{\nu}^{(2)}(2\sqrt{z};q) = \tilde{\mathcal{J}}_{\nu}(z;q)$ belongs to $\mathcal{LP}I$ . Then it follows form Lemma 4 that its Jensen polynomials $$g_n(\tilde{\mathcal{J}}_{\nu}(\cdot;q);\zeta) = \sum_{k=0}^n \binom{n}{k} \frac{k!}{(q;q)_k (q^{\nu+1};q)_k} q^{k(\nu+k)} (-\zeta)^k$$ are all hyperbolic. However, observe that the Jensen polynomials of $\tilde{\phi}_{\nu}(z;q) = \phi_{\nu}(2\sqrt{z};q)$ are simply $$g_n(\tilde{\phi}_{\nu};\zeta) = ag_n(\tilde{\mathcal{J}}_{\nu}(\cdot;q);\zeta) - \zeta g'_n(\tilde{\mathcal{J}}_{\nu}(\cdot;q);\zeta).$$ Lemma 3 implies that all zeros of $g_n(\tilde{\phi}_{\nu};\zeta)$ are real and positive and that the smallest one precedes the first zero of $g_n(\tilde{\mathcal{J}}_{\nu}(\cdot;q);\zeta)$ . In view of Lemma 4, the latter conclusion immediately yields that $\tilde{\phi}_{\nu} \in \mathcal{LP}I$ and that its first zero precedes $j_{\nu,1}(q)$ . Finally, the first part of the statement of the lemma follows after we go back from $\tilde{\phi}_{\nu}(\cdot;q)$ to $\phi_{\nu}(\cdot;q)$ by setting $\zeta = \frac{z^2}{4}$ . Similarly, because of Lemma 1 the function $z \mapsto \mathcal{J}_{\nu}^{(3)}(z;q) = c_{\nu}(q)z^{-\nu}J_{\nu}^{(3)}(z;q)$ belongs to the Laguerre-Pólya class of entire functions, which implies that the function $z \mapsto \mathcal{J}_{\nu}^{(3)}(\sqrt{z};q) = \overline{\mathcal{J}}_{\nu}(z;q)$ belongs to $\mathcal{LP}I$ . Then it follows from Lemma 4 that its Jensen polynomials $$g_n(\overline{\mathcal{J}}_{\nu}(\cdot;q);\zeta) = \sum_{k=0}^n \binom{n}{k} \frac{k!}{(q;q)_k (q^{\nu+1};q)_k} q^{\frac{1}{2}k(k+1)} (-\zeta)^k$$ are all hyperbolic. However, observe that the Jensen polynomials of $\tilde{\psi}_{\nu}(z;q) = \psi_{\nu}(\sqrt{z};q)$ are simply $$g_n(\tilde{\psi}_{\nu};\zeta) = ag_n(\overline{\mathcal{J}}_{\nu}(\cdot;q);\zeta) - \zeta g'_n(\overline{\mathcal{J}}_{\nu}(\cdot;q);\zeta).$$ Lemma 3 implies that all zeros of $g_n(\tilde{\psi}_{\nu};\zeta)$ are real and positive and that the smallest one precedes the first zero of $g_n(\overline{\mathcal{J}}_{\nu}(\cdot;q);\zeta)$ . In view of Lemma 4, the latter conclusion immediately yields that $\tilde{\psi}_{\nu} \in \mathcal{LP}I$ and that its first zero precedes $l_{\nu,1}(q)$ . Thus, the second part of the statement of this lemma follows after we go back from $\tilde{\psi}_{\nu}(\cdot;q)$ to $\psi_{\nu}(\cdot;q)$ by setting $\zeta=z^2$ . The following result is an immediate consequence of Lemma 5 and is the q-extension to Jackson and Hahn-Exton q-Bessel functions of the well known result that if $\nu > -1$ and c is a constant such that $c + \nu > 0$ , then the Dini function $z \mapsto zJ'_{\nu}(z) + cJ_{\nu}(z)$ has only real zeros and its first positive zero does not exceed the first positive zero of $J_{\nu}$ , see [26, p. 597] and [16, p. 11].

Lemma 6. If $\nu > -1$ and c is a constant such that $c + \nu > 0$ , then the Jackson q-Dini function $z \mapsto z \cdot dJ_{\nu}^{(2)}(z;q)/dz + cJ_{\nu}^{(2)}(z;q)$ has only real zeros and its first positive zero does not exceed $j_{\nu,1}(q)$ . Similarly, under the same assumptions the Hahn-Exton q-Dini function $z \mapsto z \cdot dJ_{\nu}^{(3)}(z;q)/dz + cJ_{\nu}^{(3)}(z;q)$ has only real zeros and its first positive zero does not exceed $l_{\nu,1}(q)$ . 2.4. The Hadamard factorization of the derivatives of q-Bessel functions. The following infinite product representations are natural q-extensions of the well-known Hadamard factorization for the derivative of Bessel functions of the first kind.

Lemma 7. If $\nu > 0$ , then $z \mapsto (2^{\nu}/\nu)c_{\nu}(q)z^{1-\nu} \cdot dJ_{\nu}^{(2)}(z;q)/dz$ and $z \mapsto (1/\nu)c_{\nu}(q)z^{1-\nu} \cdot dJ_{\nu}^{(3)}(z;q)/dz$ are entire functions of order $\rho = 0$ . Consequently, their Hadamard factorization for $z \in \mathbb{C}$ are of the form <span id="page-5-1"></span> $$(2.2) dJ_{\nu}^{(2)}(z;q)/dz = \frac{\nu \left(\frac{1}{2}z\right)^{\nu-1}}{2c_{\nu}(q)} \prod_{n>1} \left(1 - \frac{z^2}{j_{\nu,n}^{\prime 2}(q)}\right), dJ_{\nu}^{(3)}(z;q)/dz = \frac{\nu z^{\nu-1}}{c_{\nu}(q)} \prod_{n>1} \left(1 - \frac{z^2}{l_{\nu,n}^{\prime 2}(q)}\right),$$ where $j'_{\nu,n}(q)$ and $l'_{\nu,n}(q)$ are the nth positive zeros of $z \mapsto dJ^{(2)}_{\nu}(z;q)/dz$ and $z \mapsto dJ^{(3)}_{\nu}(z;q)/dz$ . Proof. We have that $$\frac{1}{\nu} 2^{\nu} c_{\nu}(q) z^{1-\nu} \cdot dJ_{\nu}^{(2)}(z;q) / dz = \frac{1}{\nu} \sum_{n > 0} \frac{(-1)^{n} (2n+\nu) z^{2n} q^{n(n+\nu)}}{2^{2n} (q;q)_{n} (q^{\nu+1};q)_{n}},$$ $$\frac{1}{\nu}c_{\nu}(q)z^{1-\nu}\cdot dJ_{\nu}^{(3)}(z;q)/dz = \frac{1}{\nu}\sum_{n\geq 0}\frac{(-1)^{n}(2n+\nu)z^{2n}q^{\frac{1}{2}n(n+1)}}{(q;q)_{n}\left(q^{\nu+1};q\right)_{n}},$$ and $$\begin{split} \lim_{n \to \infty} \frac{n \log n}{\log(q;q)_n + \log\left(q^{\nu+1};q\right)_n + 2n \log 2 - n(n+\nu) \log q - \log(2n+\nu)} &= 0, \\ \lim_{n \to \infty} \frac{n \log n}{\log(q;q)_n + \log\left(q^{\nu+1};q\right)_n - \frac{1}{2}n(n+1) \log q - \log(2n+\nu)} &= 0, \end{split}$$ since as $n \to \infty$ we have $(q;q)_n \to (q;q)_\infty < \infty$ and $(q^{\nu+1};q)_n \to (q^{\nu+1};q)_\infty < \infty$ . Moreover, we know that the zeros $j'_{\nu,n}(q)$ , $n \in \mathbb{N}$ , and $l'_{\nu,n}(q)$ , $n \in \mathbb{N}$ , are real for $\nu > 0$ , according to Lemma 6, and with this the rest of the proof of (2.2) follows by applying Hadamard's Theorem [20, p. 26]. 2.5. The Hadamard factorization of q-Dini functions. The following infinite product representations are q-extensions of the known Hadamard factorization for Dini functions $z \mapsto zJ'_{\nu}(z) + (1-\nu)J_{\nu}(z)$ and $z \mapsto zJ'_{\nu}(z) + (2-\nu)J_{\nu}(z)$ , see [5, Theorem 1].

Lemma 8. If $\nu > -1$ , then $z \mapsto dg_{\nu}^{(2)}(z;q)/dz$ , $z \mapsto dh_{\nu}^{(2)}(z;q)/dz$ , $z \mapsto dg_{\nu}^{(3)}(z;q)/dz$ and $z \mapsto dh_{\nu}^{(3)}(z;q)/dz$ are entire functions of order $\rho = 0$ . Consequently, their Hadamard factorization for $z \in \mathbb{C}$ are of the form $$dg_{\nu}^{(2)}(z;q)/dz = \prod_{n\geq 1} \left(1 - \frac{z^2}{\alpha_{\nu,n}^2(q)}\right), \quad dh_{\nu}^{(2)}(z;q)/dz = \prod_{n\geq 1} \left(1 - \frac{z}{\beta_{\nu,n}^2(q)}\right),$$ $$dg_{\nu}^{(3)}(z;q)/dz = \prod_{n \geq 1} \left(1 - \frac{z^2}{\gamma_{\nu,n}^2(q)}\right), \quad dh_{\nu}^{(3)}(z;q)/dz = \prod_{n \geq 1} \left(1 - \frac{z}{\delta_{\nu,n}^2(q)}\right),$$ where $\alpha_{\nu,n}(q)$ and $\beta_{\nu,n}(q)$ are the nth positive zeros of $z \mapsto z \cdot dJ_{\nu}^{(2)}(z;q)/dz + (1-\nu)J_{\nu}^{(2)}(z;q)$ and $z \mapsto z \cdot dJ_{\nu}^{(2)}(z;q)/dz + (2-\nu)J_{\nu}^{(2)}(z;q)$ , while $\gamma_{\nu,n}(q)$ and $\delta_{\nu,n}(q)$ are the nth positive zeros of $z \mapsto z \cdot dJ_{\nu}^{(3)}(z;q)/dz + (1-\nu)J_{\nu}^{(3)}(z;q)$ and $z \mapsto z \cdot dJ_{\nu}^{(3)}(z;q)/dz + (2-\nu)J_{\nu}^{(3)}(z;q)$ . Proof. We have that $$\begin{split} \frac{dg_{\nu}^{(2)}(z;q)}{dz} &= 2^{\nu}c_{\nu}(q)z^{-\nu}\left(z\cdot\frac{dJ_{\nu}^{(2)}(z;q)}{dz} + (1-\nu)J_{\nu}^{(2)}(z;q)\right) = \sum_{n\geq 0}\frac{(-1)^{n}(2n+1)z^{2n}q^{n(n+\nu)}}{2^{2n}(q;q)_{n}\left(q^{\nu+1};q\right)_{n}},\\ \frac{dh_{\nu}^{(2)}(z;q)}{dz} &= 2^{\nu-1}c_{\nu}(q)z^{-\nu/2}\left(\sqrt{z}\cdot\frac{dJ_{\nu}^{(2)}(\sqrt{z};q)}{dz} + (2-\nu)J_{\nu}^{(2)}(\sqrt{z};q)\right) = \sum_{n\geq 0}\frac{(-1)^{n}(n+1)z^{n}q^{n(n+\nu)}}{2^{2n}(q;q)_{n}\left(q^{\nu+1};q\right)_{n}},\\ \frac{dg_{\nu}^{(3)}(z;q)}{dz} &= c_{\nu}(q)z^{-\nu}\left(z\cdot\frac{dJ_{\nu}^{(3)}(z;q)}{dz} + (1-\nu)J_{\nu}^{(3)}(z;q)\right) = \sum_{n\geq 0}\frac{(-1)^{n}(2n+1)z^{2n}q^{\frac{1}{2}n(n+1)}}{(q;q)_{n}\left(q^{\nu+1};q\right)_{n}},\\ \frac{dh_{\nu}^{(3)}(z;q)}{dz} &= \frac{1}{2}c_{\nu}(q)z^{-\nu/2}\left(\sqrt{z}\cdot\frac{dJ_{\nu}^{(3)}(\sqrt{z};q)}{dz} + (2-\nu)J_{\nu}^{(3)}(\sqrt{z};q)\right) = \sum_{n\geq 0}\frac{(-1)^{n}(n+1)z^{n}q^{\frac{1}{2}n(n+1)}}{(q;q)_{n}\left(q^{\nu+1};q\right)_{n}},\\ \text{and}\\ \lim_{n\to\infty}\frac{n\log n}{\log(q;q)_{n} + \log\left(q^{\nu+1};q\right)_{n} + 2n\log 2 - n(n+\nu)\log q - \log(2n+1)} = 0,\\ \lim_{n\to\infty}\frac{n\log n}{\log(q;q)_{n} + \log\left(q^{\nu+1};q\right)_{n} + 2n\log 2 - n(n+\nu)\log q - \log(2n+1)} = 0,\\ \lim_{n\to\infty}\frac{n\log n}{\log(q;q)_{n} + \log\left(q^{\nu+1};q\right)_{n} - \frac{1}{2}n(n+1)\log q - \log(2n+1)} = 0, \end{split}$$ $\lim_{n\to\infty}\frac{n\log n}{\log(q;q)_n+\log\left(q^{\nu+1};q\right)_n-\frac{1}{2}n(n+1)\log q-\log(n+1)}=0,$ since as $n\to\infty$ we have $(q;q)_n\to(q;q)_\infty<\infty$ and $\left(q^{\nu+1};q\right)_n\to\left(q^{\nu+1};q\right)_\infty<\infty$ . Moreover, we know that the zeros $\alpha_{\nu,n}(q),\ \beta_{\nu,n}(q),\ \gamma_{\nu,n}(q),\ \delta_{\nu,n}(q),\ n\in\mathbb{N},$ are real for $\nu>-1$ , according to Lemma 6, and with this the rest of the proof follows by applying Hadamard's Theorem [20, p. 26]. 2.6. Interlacing of zeros of q-Bessel functions and their derivatives. The next result complements the other interlacing properties of the zeros of Jackson and Hahn-Exton q-Bessel functions, see [14, Theorem 4.3] and [18, Theorem 3.7]. This preliminary result is necessary in the proof of the first part of Theorem 2.

Lemma 9. Between any two consecutive roots of the function $z \mapsto J_{\nu}^{(s)}(z;q)$ the function $z \mapsto dJ_{\nu}^{(s)}(z;q)/dz$ has precisely one zero when $\nu > -1$ and $s \in \{2,3\}$ . Proof. The proofs for Jackson q-Bessel and Hahn-Exton q-Bessel are very similar, and thus we will give the proof only for s=2. For the simplicity in this proof we will use the notation $\mathcal{J}_{\nu}(z;q)=2^{\nu}c_{\nu}(q)z^{-\nu}J_{\nu}(z;q)$ instead of $\mathcal{J}_{\nu}^{(2)}(z;q)=2^{\nu}c_{\nu}(q)z^{-\nu}J_{\nu}^{(2)}(z;q)$ . Moreover, we will use simply $J_{\nu}'(z;q)$ instead of $dJ_{\nu}^{(2)}(z;q)/dz$ . Since $\mathcal{J}_{\nu}(\cdot;q)$ belongs to the Laguerre-Pólya class of entire functions, it follows that it satisfies the Laguerre inequality (see [25]) (2.3) $$\left( \mathcal{J}_{\nu}(z;q)^{(n)} \right)^{2} - \left( \mathcal{J}_{\nu}(z;q) \right)^{(n-1)} \left( \mathcal{J}_{\nu}(z;q) \right)^{(n+1)} > 0,$$ where $\nu > -1$ and $z \in \mathbb{R}$ . On the other hand, we have that <span id="page-6-1"></span> $$\begin{split} \mathcal{J}_{\nu}'(z;q) &= 2^{\nu} c_{\nu}(q) z^{-\nu-1} \left( z J_{\nu}'(z;q) - \nu J_{\nu}(z;q) \right), \\ \mathcal{J}_{\nu}''(z;q) &= 2^{\nu} c_{\nu}(q) z^{-\nu-2} \left( z^2 J_{\nu}''(z;q) - 2\nu z J_{\nu}'(z;q) + \nu(\nu+1) J_{\nu}(z;q) \right), \end{split}$$ and thus the Laguerre inequality (2.3) for n=1 is equivalent to $$2^{2\nu}c_{\nu}(q)z^{-2\nu-2}\left(z^{2}\left(J_{\nu}'(z;q)\right)^{2}-z^{2}J_{\nu}(z;q)J_{\nu}''(z;q)-\nu J_{\nu}^{2}(z;q)\right)>0.$$ This implies that $$(J'_{\nu}(z;q))^2 - J_{\nu}(z;q)J''_{\nu}(z;q) > \nu J^2_{\nu}(z;q)/z^2 > 0$$ for $\nu > 0$ and $z \in \mathbb{R}$ , that is, the function $z \mapsto J'_{\nu}(z)/J_{\nu}(z)$ is decreasing on $(0, \infty) \setminus \{j_{\nu,n}(q) | n \in \mathbb{N}\}$ . Recall that the zeros $j_{\nu,n}(q)$ , $n \in \mathbb{N}$ , of the Jackson q-Bessel function are real and simple, according to [14, Theorem 4.2], and thus $J'_{\nu}(z;q)$ does not vanish in $j_{\nu,n}(q)$ , $n \in \mathbb{N}$ . Thus, for a fixed $k \in \mathbb{N}$ the function $z \mapsto J'_{\nu}(z)/J_{\nu}(z)$ takes the limit $\infty$ when $z \setminus j_{\nu,k-1}(q)$ , and the limit $-\infty$ when $z \nearrow$ $j_{\nu,k}(q)$ . Moreover, since $z \mapsto J'_{\nu}(z)/J_{\nu}(z)$ is decreasing on $(0,\infty) \setminus \{j_{\nu,n}(q) | n \in \mathbb{N}\}$ it results that in each interval $(j_{\nu,k-1}(q),j_{\nu,k}(q))$ its restriction intersects the horizontal line only once, and the abscissa of this intersection point is exactly $j'_{\nu,k}(q)$ . Here we used the convention that $j_{\nu,0}(q) = 0$ . 2.7. Starlikeness of entire functions in the open unit disk. The next result (see [24, Theorem 2]) is the key tool in the proof of Theorem 3.

Lemma 10. Let $f: \mathbb{D} \to \mathbb{C}$ be a transcendental entire function of the form <span id="page-7-0"></span> $$f(z) = z \prod_{n>1} \left(1 - \frac{z}{z_n}\right),\,$$ where all $z_n$ have the same argument and satisfy $|z_n| > 1$ . If f is univalent in $\mathbb{D}$ , then (2.4) $$\sum_{n>1} \frac{1}{|z_n| - 1} \le 1.$$ In fact (2.4) holds if and only if f is starlike in $\mathbb{D}$ and all of its derivatives are close-to-convex there.