Bounds for radii of convexity of some $q$-Bessel functions

Theorem 1. Let $\nu > -1$ . Then the radius of convexity $r^c\left(g_{\nu}^{(2)}(z;q)\right)$ of the function $$z \mapsto g_{\nu}^{(2)}(z;q) = 2^{\nu} c_{\nu}(q) z^{1-\nu} J_{\nu}^{(2)}(z;q)$$ is the smallest positive root of the equation $$(1-\nu)^2 J_{\nu}^{(2)}(r;q) + (3-2\nu)r dJ_{\nu}^{(2)}(r;q)/dr + r^2 d^2 J_{\nu}^{(2)}(r;q)/dr^2 = 0$$ and satisfies the following inequalities $$\sqrt{\frac{4\left(1-q^{\nu+1}\right)\left(1-q\right)}{9q^{\nu+1}}} < r^{c}\left(g_{\nu}^{(2)}(z;q)\right) < \sqrt{\frac{36(q^{2}-1)\left(1-q^{\nu+1}\right)\left(1-q^{\nu+2}\right)}{q^{\nu+1}S_{\nu}(q)}},$$ $$2\sqrt[4]{\frac{(1+q)(1-q)^{2}\left(1-q^{\nu+1}\right)^{2}\left(q^{\nu+2}-1\right)}{q^{2(\nu+1)}S_{\nu}(q)}} < r^{c}\left(g_{\nu}^{(2)}(z;q)\right) < \sqrt{\frac{4\left(1-q^{\nu+1}\right)\left(1-q^{\nu+3}\right)T(q)S_{\nu}(q)}{q^{\nu+1}(1+q)\left(P_{\nu}(q)+R_{\nu}(q)\right)}},$$ where $$P_{\nu}(q) = 1458q - 729q^{\nu+2} - 1512q^{\nu+3} - 2241q^{\nu+4} - 837q^{\nu+5} - 54q^{\nu+6} + 479q^{\nu+7} + 729q^{2\nu+5},$$ $$R_{\nu}(q) = 783q^{2\nu+6} + 783q^{2\nu+7} + 152q^{2\nu+8} + 98q^6 - 675q^4 + 54q^3 + 783q^2 + 729,$$ $$S_{\nu}(q) = 31q^{\nu+3} + 81q^{\nu+2} + 50q^2 - 81q - 81$$ and <span id="page-2-0"></span> $$T(q) = (q-1)(q^3 + 2q^2 + 2q + 1).$$ Note that by multiplying by $(1-q)^{-1}$ both sides of the above inequalities and taking the limit as $q \nearrow 1$ for $\nu > -1$ we obtain the first two inequalities of [2, Theorem 6], namely: (2.1) $$\frac{2\sqrt{\nu+1}}{3} < r^c(g_\nu) < 6\sqrt{\frac{(\nu+1)(\nu+2)}{56\nu+137}}$$ and (2.2) $$2\sqrt[4]{\frac{(\nu+1)^2(\nu+2)}{56\nu+137}} < r^c(g_\nu) < \sqrt{\frac{2(\nu+1)(\nu+3)(56\nu+137)}{208\nu^2+1172\nu+1693}},$$ where $r^{c}(g_{\nu})$ stands for the radii of convexity of the normalized Bessel function <span id="page-2-1"></span> $$z \mapsto g_{\nu}(z) = 2^{\nu} \Gamma(\nu + 1) z^{1-\nu} J_{\nu}(z).$$

Theorem 2. Let $\nu > -1$ . Then the radius of convexity $r^c\left(h_{\nu}^{(2)}(z;q)\right)$ of the function $$z \mapsto h_{\nu}^{(2)}(z;q) = 2^{\nu} c_{\nu}(q) z^{1-\frac{\nu}{2}} J_{\nu}^{(2)}(\sqrt{z};q)$$ is the smallest positive root of the equation $$(2-\nu)^2 J_{\nu}^{(2)}(\sqrt{r};q) + (5-2\nu)\sqrt{r}dJ_{\nu}^{(2)}(\sqrt{r};q)/dr + rd^2 J_{\nu}^{(2)}(\sqrt{r};q)/dr^2 = 0$$ and satisfies the following inequalities $$\frac{(1-q)(1-q^{\nu+1})}{q^{\nu+1}} < r^c \left(h_{\nu}^{(2)}(z;q)\right) < \frac{8(q^{\nu+1}-1)(q^{\nu+2}-1)(1-q^2)}{q^{\nu+1}U_{\nu}(q)},$$ $$\sqrt{\frac{8(1-q)^2(1+q)(1-q^{\nu+1})^2(1-q^{\nu+2})}{q^{2\nu+2}U_{\nu}(q)}} < r^c \left(h_{\nu}^{(2)}(z;q)\right) < \frac{2(1-q^{\nu+1})(q^{\nu+3}-1)U_{\nu}(q)T(q)}{(1+q)q^{\nu+1}\left(M_{\nu}(q)+N_{\nu}(q)\right)},$$ where $$U_{\nu}(q) = \left(8q - 8q^{\nu+2} + q^{\nu+3} - 9q^2 + 8\right),$$ $$M_{\nu}(q) = 32q - 16q^{\nu+2} - 21q^{\nu+3} - 37q^{\nu+4} + 6q^{\nu+5} + 11q^{\nu+6} + 3q^{\nu+7}$$ and $$N_{\nu}(q) = 16q^{2\nu+5} + 5q^{2\nu+6} + 5q^{2\nu+7} + q^{2\nu+8} + 5q^2 - 11q^3 - 27q^4 + 12q^6 + 16.$$ Here we would like to emphasize that by multiplying by $(1-q)^{-2}$ both sides of the above inequalities and taking the limit as $q \nearrow 1$ for $\nu > -1$ we obtain the first two inequalities of [2, Theorem 7], namely: (2.3) $$\nu + 1 < r^{c}(h_{\nu}) < \frac{16(\nu + 1)(\nu + 2)}{7\nu + 23}$$ and (2.4) $$\sqrt{\frac{16(\nu+1)^2(\nu+2)}{7\nu+23}} < r^c(h_\nu) < \frac{2(\nu+1)(\nu+3)(7\nu+23)}{9\nu^2+60\nu+115},$$ where $r^{c}(h_{\nu})$ stands for the radii of convexity of the normalized Bessel function <span id="page-2-3"></span><span id="page-2-2"></span> $$z \mapsto h_{\nu}(z) = 2^{\nu} \Gamma(\nu + 1) z^{1 - \frac{\nu}{2}} J_{\nu}(\sqrt{z}).$$

Theorem 3. Let $\nu > -1$ . Then the radius of convexity $r^c\left(g_{\nu}^{(3)}(z;q)\right)$ of the function $$z \mapsto g_{\nu}^{(3)}(z;q) = c_{\nu}(q)z^{1-\nu}J_{\nu}^{(3)}(z;q)$$ is the smallest positive root of the equation $$(1-\nu)^2 J_{\nu}^{(3)}(r;q) + (3-2\nu)r dJ_{\nu}^{(3)}(r;q)/dr + r^2 d^2 J_{\nu}^{(3)}(r;q)/dr^2 = 0$$ and satisfies the following inequalities $$\sqrt{\frac{\left(1-q^{\nu+1}\right)\left(1-q\right)}{9q}} < r^{c}\left(g_{\nu}^{(3)}(z;q)\right) < \sqrt{\frac{9(q^{2}-1)\left(1-q^{\nu+1}\right)\left(1-q^{\nu+2}\right)}{qY_{\nu}(q)}},$$ $$\sqrt[4]{\frac{\left(1+q\right)\left(1-q\right)^{2}\left(1-q^{\nu+1}\right)^{2}\left(q^{\nu+2}-1\right)}{q^{2}Y_{\nu}(q)}} < r^{c}\left(g_{\nu}^{(3)}(z;q)\right) < \sqrt{\frac{\left(1-q^{\nu+1}\right)\left(1-q^{\nu+3}\right)T(q)Y_{\nu}(q)}{3q(q+1)\left(\theta_{\nu}(q)+\varphi_{\nu}(q)\right)}},$$ where $$\theta_{\nu}(q) = 261q - 18q^{\nu+2} - 504q^{\nu+3} - 620q^{\nu+4} - 504q^{\nu+5} - 18q^{\nu+6} + 67q^{2\nu+5},$$ $$\varphi_{\nu}(q) = 261q^{2\nu+6} + 261q^{2\nu+7} + 243q^{2\nu+8} + 67q^3 + 261q^2 + 243$$ and <span id="page-3-0"></span> $$Y_{\nu}(q) = 81q^{\nu+3} + 31q^{\nu+2} - 31q - 81.$$ Note that by multiplying by $(1-q)^{-1}$ both sides of the above inequalities and taking the limit as $q \nearrow 1$ for $\nu > -1$ we obtain the following inequalities (2.5) $$\frac{\sqrt{\nu+1}}{3} < r^c(g_\nu) < 3\sqrt{\frac{(\nu+1)(\nu+2)}{56\nu+137}}$$ and (2.6) $$\sqrt[4]{\frac{(\nu+1)^2(\nu+2)}{56\nu+137}} < r^c(g_\nu) < \sqrt{\frac{(\nu+1)(\nu+3)(56\nu+137)}{2(208\nu^2+1172\nu+1693)}}.$$

Theorem 4. Let $\nu > -1$ . Then the radius of convexity $r^c\left(h_{\nu}^{(3)}(z;q)\right)$ of the function <span id="page-3-1"></span> $$z \mapsto h_{\nu}^{(3)}(z;q) = c_{\nu}(q)z^{1-\frac{\nu}{2}}J_{\nu}^{(3)}(\sqrt{z};q)$$ is the smallest positive root of the equation $$(2-\nu)^2 J_{\nu}^{(3)}(\sqrt{r};q) + (5-2\nu)\sqrt{r}dJ_{\nu}^{(3)}(\sqrt{r};q)/dr + rd^2 J_{\nu}^{(3)}(\sqrt{r};q)/dr^2 = 0$$ and satisfies the following inequalities $$\frac{(1-q)(1-q^{\nu+1})}{4q} < r^c \left( h_{\nu}^{(3)}(z;q) \right) < \frac{2(q^{\nu+1}-1)(q^{\nu+2}-1)(q^2-1)}{qE_{\nu}(q)}.$$ $$\sqrt{\frac{(1-q)^2(1+q)(1-q^{\nu+1})^2(q^{\nu+2}-1)}{2q^2E_{\nu}(q)}} < r^c \left(h_{\nu}^{(3)}(z;q)\right) < \frac{(1-q^{\nu+1})(q^{\nu+3}-1)E_{\nu}(q)T(q)}{2q(1+q)\left(K_{\nu}(q)+L_{\nu}(q)\right)},$$ where $$E_{\nu}(q) = (8q^{\nu+3} - q^{\nu+2} + q - 8)$$ $$K_{\nu}(q) = 5q + 11q^{\nu+2} - 21q^{\nu+3} - 34q^{\nu+4} - 21q^{\nu+5} + 11q^{\nu+6}$$ and <span id="page-3-2"></span> $$L_{\nu}(q) = q^{2\nu+5} + 5q^{2\nu+6} + 5q^{2\nu+7} + 16q^{2\nu+8} + 5q^2 + q^3 + 16.$$ Here we would like to emphasize that by multiplying by $(1-q)^{-2}$ both sides of the above inequalities and taking the limit as $q \nearrow 1$ for $\nu > -1$ , we obtain the next two inequalities (2.7) $$\frac{\nu+1}{4} < r^c(h_\nu) < \frac{4(\nu+1)(\nu+2)}{7\nu+23}$$ and <span id="page-3-3"></span>(2.8) $$\sqrt{\frac{(\nu+1)^2(\nu+2)}{7\nu+23}} < r^c(h_\nu) < \frac{(\nu+1)(\nu+3)(7\nu+23)}{2(9\nu^2+60\nu+115)}.$$ It is important to mention that by making a comparison among of above obtained inequalities we have that the left-hand sides of (2.5) and (2.6) are weaker than the left hand sides of (2.1) and (2.2), respectively. However, the right-hand sides of (2.5) and (2.6) improve the right-hand sides of (2.1) and (2.2), respectively. On the other hand, the left-hand sides of (2.7) and (2.8) are weaker than the left-hand sides of (2.3) and (2.4), while the right-hand sides of (2.7) and (2.8) improve the right-hand sides of (2.3) and (2.4).